Helical magnetic fields and semi-classical asymptotics of the lowest eigenvalue

B. Helffer, A. Kachmar

Abstract

We study the 3D Neuman magnetic Laplacian in the presence of a semi-classical parameter and a non-uniform magnetic field with constant intensity. We determine a sharp two term asymptotics for the lowest eigenvalue, where the second term involves a quantity related to the magnetic field and the geometry of the domain. In the special case of the unit ball and a helical magnetic field, the concentration takes place on two symmetric points of the unit sphere.

1 Main results

Let $\Omega\subset\mathbb{R}^{3}$ be an open and bounded set with a smooth boundary $\partial\Omega$ . Let us consider a smooth magnetic field $\mathbf{B}:\overline{\Omega}\to\mathbb{R}^{3}$ (so $\mathbf{B}$ should be closed) which will always be assumed to satisfy

\forall\,x\in\Omega,\quad|\mathbf{B}(x)|=b

(1.1)

where $b>0$ is a constant. Without loss of generality, we assume from now on that $b=1$ . Let ${\bf A}(x)$ be a magnetic potential such that

{\rm curl}\,{\bf A}=\mathbf{B}\;.

(1.2)

We are interested in the analysis of the lowest eigenvalue $\lambda_{1}({\bf A},h)$ of the Neumann realization of the Schrödinger operator in $\Omega$ with magnetic field

P_{\bf A}^{h}:=\Delta_{h,{\bf A}}=\sum_{j=1}^{3}(hD_{x_{j}}+A_{j}(x))^{2}\;.

(1.3)

We introduce the following assumptions.

Assumption 1.1 (C1).

The set of boundary points where $\mathbf{B}$ is tangent to $\partial\Omega$ , i.e.

\displaystyle\Gamma:=\{x\in\partial\Omega\,\big{|}\,\mathbf{B}\cdot\mathbf{N}(x)=0\},

(1.4)

is a regular submanifold of $\partial\Omega$ :

\kappa_{n,\mathbf{B}}(x):=|d^{T}(\mathbf{B}\cdot\mathbf{N})(x)|\neq 0\;,\;\forall x\in\Gamma\;.

(1.5)

Here $d^{T}$ is the differential defined on functions on $\partial\Omega$ and $\mathbf{N}(x)$ is the unit inward normal of $\Omega$ .

Assumption 1.2 (C2).

The set of points where $\mathbf{B}$ is tangent to $\Gamma$ is finite.

These assumptions are rather generic and for instance satisfied for ellipsoids, when $\mathbf{B}$ is constant. When $|\mathbf{B}|$ is constant, the above assumptions hold for the sphere with a helical magnetic field (see Sec. 3).

Let us introduce the constant $\widehat{\gamma}_{0,\mathbf{B}}$ involving the “magnetic curvature” in (1.5), which is defined by

\widehat{\gamma}_{0,\mathbf{B}}:=\inf_{x\in\Gamma}\widetilde{\gamma}_{0,\mathbf{B}}(x),

(1.6)

where

\widetilde{\gamma}_{0,\mathbf{B}}(x):=2^{-2/3}\widehat{\nu}_{0}\delta_{0}^{1/3}|\kappa_{n,\mathbf{B}}(x)|^{2/3}\Big{(}1-(1-\delta_{0})|\mathbf{T}(x)\cdot\mathbf{B}(x)|^{2}\Big{)}^{1/3}\;.

(1.7)

Here $\mathbf{T}(x)$ is the oriented, unit tangent vector to $\Gamma$ at the point $x$ , $\delta_{0}\in]0,1[$ and $\widehat{\nu}_{0}>0$ are spectral quantities relative to the De Gennes and Montgomery operators which will be introduced in (4.2) and (4.5).

When $\mathbf{B}$ is constant, the following two-term asymptotics of $\lambda_{1}(B)$ has been established by Helffer-Morame [HelMo4] and Pan [Pan3].

Theorem 1.3.

Let us assume that $\mathbf{B}$ is constant. Then, if $\Omega$ and $\mathbf{B}$ satisfy (C1)-(C2), there exists $\eta>0$ such that the lowest eigenvalue $\lambda_{1}^{N}({\bf A},h)$ satisfies as $h\rightarrow 0$

\lambda_{1}^{N}({\bf A},h)=\Theta_{0}h+\widehat{\gamma}_{0,\mathbf{B}}h^{\frac{4}{3}}+{\mathcal{O}}(h^{\frac{4}{3}+\eta})\,.

(1.8)

The aim of this paper is to prove that Theorem 1.3 also holds under the weaker assumption that $|\mathbf{B}|$ is constant.

Theorem 1.4.

Under the assumptions (C1)-(C2), if $|\mathbf{B}|$ is constant, then the asymptotics in (1.8) holds for the lowest eigenvalue $\lambda_{1}^{N}({\bf A},h)$ .

An interesting example of a non-constant magnetic field but with a constant intensity is the helical magnetic field occurring in the theory of liquid crystals. Up to the action of an orthogonal matrix, it can be expressed as follows [Pan6]

\mathbf{B}={\rm curl}\,\mathbf{n}_{\tau}=-\tau\mathbf{n}_{\tau},\quad\mathbf{n}_{\tau}=\Big{(}\frac{1}{\tau}\cos(\tau x_{3}),\frac{1}{\tau}\sin(\tau x_{3}),0\Big{)}.

(1.9)

Here $\tau>0$ is a given constant. In this situation ( $\mathbf{B}=-\tau\mathbf{n}_{\tau}$ ), [Pan6] derived an upper bound on the eigenvalue $\lambda_{1}^{N}({\bf A},h)$ , which is consistent with Theorem 1.4. Our contribution is valid for a more general class of magnetic fields with constant intensity and also determines the asymptotically matching lower bound of the lowest eigenvalue.

Discussion and applications

The inspection of the eigenvalue $\lambda_{1}^{N}({\bf A},h)$ is vital in understanding the transition between superconducting and normal states in the Ginzburg-Landau model [FoHe2]. In this context, the magnetic field is typically constant. Accurate estimates of the lowest eigenvalue $\lambda_{1}^{N}({\bf A},h)$ under constant magnetic fields [HelMo3, HelMo4] led to a precise understanding of the transition between superconducting and normal states [FoHe1, FS].

Non-homogeneous magnetic fields with constant intensity are encountered in the Landau–de Gennes theory of liquid crystals, which is the analog of the Ginzburg-Landau theory of superconductivity. Here a transition between smectic and nematic phases occurs. Our main result, Theorem 1.4, yields an accurate estimate of the lowest eigenvalue $\lambda_{1}^{N}({\bf A},h)$ for magnetic fields with constant intensity, and by analogy with [FoHe1], we expect it to yield a precise description of the transition between surface smectic and nematic states (see [Pan2]).

At the threshold of the phase transition, both superconductive and smectic states nucleate on the surface of the domain (near the curve $\Gamma$ introduced in (3.7)). The paper [Pan5] contains a nice discussion of this interesting analogy. The analysis of 3D surface superconductivity is the subject of the papers [Pan3, FKP, FMP], while surface smectics are rigorously studied in [HePa2, FKPa]. It would be interesting to complete this analysis by providing more accurate estimates at the threshold, where the linear analysis (such as the one in this paper) becomes handy.

The analysis in this paper concerns the lowest eigenvalue. In the presence of a constant magnetic field, and a “single well” assumption (i.e. the minimum in (1.6) is non-degenerate and attained at a unique point), accurate estimates of the low-lying eigenvalues were obtained recently in [HR]. In our setting of a non-homogeneous magnetic field, the example of the ball under the helical magnetic field suggests the presence of multiple wells (see Remark 3.5).

The interaction between magnetic fields and 3D domains is interesting in other situations. In particular, for the Robin problem, we observe pure magnetic wells on the surface of the domain [HKR], and in the case of a constant magnetic field, strong diamgnetism does not hold for the ball [Mi].

Organization and outline of the proof

The proof of Theorem 1.4 is split into two parts. In the first part, we establish a lower bound of the lowest eigenvalue, by comparing the quadratic form via a simpler form related to a new model operator. Comparing with the constant magnetic field in [HelMo4], we prove that the model operator in our setting is a perturbation of the one considered in [HelMo4].

The second part of the proof is devoted to an upper bound of the lowest eigenvalue, already studied for $\mathbf{B}$ in (1.9) [Pan6], but we revisit it since our formulation is not the same as [Pan6]. The upper bound follows after computing the quadratic form of a suitable trial state, having the same structure as the constant magnetic field case in [HelMo4, Pan3]. However, there are additional terms in the computations due to the varying magnetic field, which require a careful handling.

The model operator takes into consideration two phenomena. First, after decomposing our domain into small cells and working in a small cell near the domain’s boundary, we have to express the integrals in a flat geometry, which requires a careful expansion of the Riemannian metric in particular. This part is essentially the same as for the constant magnetic field case in [HelMo4].

Then, we have to express the magnetic potential in adapted coordinates, in each small cell, and apply a Taylor expansion and a gauge transformation to obtain a “normal” form, i.e. a simpler effective magnetic potential. In this part, we deviate from the constant magnetic field situation and find additional terms in the effective magnetic potential. Interestingly, we can still show that the analysis with this magnetic potential is somehow independent of those additional terms and treat the new model as a perturbation of the model with a constant magnetic field.

The paper is organized as follows. In Section 2 we introduce the adapted coordinates in a small “boundary” cell. In Section 3, we analyze the case of the unit ball with the “helical” magnetic field occurring in liquid crystals and verify that Assumptions 1.1 and 1.2 hold. Interestingly, after computing the energy in (1.6), we notice that this example shows a phenomenon of multiple “wells” induced by the “magnetic” geometry.

In Section 4, we review two standard 1D operators that we need in defining the quantities appearing in (1.6) and the statement in Theorem 1.4. Then, in Section 5, we introduce a new model, specific to our case of a varying magnetic field with a constant intensity, and analyze it through a perturbation argument.

With the model in Section 5, we can adjust the proof in [HelMo4] and prove Theorem 1.4. The first step is to localize the ground states near the boundary, which is the content of Section 6. Then, the approximation of the quadratic form and the magnetic potential are the subject of Section 7, which allows us, in the subsequent Section 8, to obtain a lower bound on the lowest eigenvalue.

Finally, Section 9 is devoted to the computation of the energy of a trial state, which yields an upper bound of the lowest eigenvalue, and thereby completes the proof of Theorem 1.4.

2 Adapted coordinates

We recall a rather standard choice of coordinates in the neighborhood of $\Gamma$ .

2.1 Description of the coordinates

Let $g_{0}$ be the Riemannian metric on $\mathbb{R}^{3}$ , which induces a Riemanian metric $G$ on $\partial\Omega$ . Given two vector fields $\mathbf{X},\mathbf{Y}$ of $\mathbb{R}^{3}$ , we denote by

\mathbf{X}\cdot\mathbf{Y}=\langle\mathbf{X},\mathbf{Y}\rangle:=g_{0}(\mathbf{X},\mathbf{Y})\,.

(2.1)

Consider a direct frame $(\mathbf{V}(x),\mathbf{T}(x),\mathbf{N}(x))_{x\in\Gamma}$ along $\Gamma$ such that

•

$\mathbf{T}(x)$ is an oriented unit tangent vector of $\Gamma$ ;
•

$\mathbf{V}(x):=\mathbf{T}(x)\times\mathbf{N}(x)$ , hence determining an oriented normal to the curve $\Gamma$ in the tangent space to $\partial\Omega$ .

For $m\in\Gamma$ , let $\Lambda_{m}$ be the geodesic that passes through $m$ and is normal to $\Gamma$ . Let $x_{0}\in\Gamma$ . In some neighborhood $\mathcal{N}_{x_{0}}\subset\overline{\Omega}$ of $x_{0}$ , we can introduce new coordinates $(r,s,t)$ as follows:

•

For $x\in\mathcal{N}_{x_{0}}$ , $p(x)\in\partial\Omega$ is defined by ${\rm dist}(x,p(x))=t(x):={\rm dist}(x,\partial\Omega)$ ;
•

For $x\in\mathcal{N}_{x_{0}}$ , $\gamma(x)\in\Gamma$ is defined by ${\rm dist}_{\partial\Omega}(p(x),\gamma(x))={\rm\;dist\;}_{\partial\Omega}(p(x),\Gamma)$ , where ${\rm dist}_{\partial\Omega}$ denotes the (geodesic) distance in $\partial\Omega$ ;
•

$\Gamma$ is parameterized by arc-length $s$ so that $s=s_{0}$ defines $x_{0}$ , and for $x\in\mathcal{N}_{x_{0}}$ , $s=s(x)$ defines $\gamma(x)$ ;
•

For $x\in\mathcal{N}_{x_{0}}$ , the geodesic $\Lambda_{p(x)}$ passing through $p(x)$ is parameterized by arclength $r$ , so that $r=0$ defines $\gamma(x)$ and $r=r(x)$ defines $p(x)$ .

In this way, we observe that $\Phi_{x_{0}}$

\mathcal{N}_{x_{0}}\ni x\mapsto\Phi_{x_{0}}(x):=(r(x),s(x),t(x))\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}_{+}

(2.2)

is a local diffeomorphism. Thus, we can pick a sufficiently small $\epsilon_{0}>0$ such that

(r,s,t)\in(-\epsilon_{0},\epsilon_{0})\times(-\epsilon_{0}+s_{0},s_{0}+\epsilon_{0})\times(0,\epsilon_{0})\to x=\Phi_{x_{0}}^{-1}(r,s,t)

(2.3)

is a diffeomorphism, whose image is a neighborhood of $x_{0}\in\Gamma$ parameterized by $(r,s,t)$ . Within these coordinates, $t=0$ means that we are on $\partial\Omega$ , and $r=t=0$ means we are on the curve $\Gamma$ . We can then compute

|d^{T}(\mathbf{B}\cdot\mathbf{N})(x)|=|\partial_{r}(\mathbf{B}\cdot\mathbf{N})|_{r=0}|\quad(x\in\Gamma)\,.

(2.4)

It is convenient to express the magnetic field along $\Gamma$ as follows

\mathbf{B}(x)=\sin\theta\,\mathbf{T}(x)+\cos\theta\,\mathbf{V}(x)\quad\big{(}x=\Phi_{x_{0}}^{-1}(0,s,0)\in\Gamma\big{)},

(2.5)

where $\theta=\theta(s)\in[-\frac{\pi}{2},\frac{\pi}{2}]$ is the angle defined by

\theta=\arcsin\big{(}\mathbf{B}(x)\cdot\mathbf{T}(x)\big{)}.

(2.6)

2.2 The metric in the new coordinates

Let us consider an arbitrary point $x_{0}\in\Gamma$ and a neighborhood $\mathcal{N}_{x_{0}}\subset\overline{\Omega}$ of $x_{0}$ such that the adapted coordinates introduced in (2.2) and (2.3) are valid. Modulo a translation, we can center the coordinates at $x_{0}$ so that $(r=0,s=0,t=0)$ are the coordinates of $x_{0}$ in the new frame. In the sequel, we follow closely the presentation of [HelMo4, Sec. 8] mainly following the first chapter of [DHKW] (see also the volume two of Spivak’s book [Sp]).

We label the new coordinates as follows

(y_{1},y_{2},y_{3})=(r,s,t)\,,

(2.7)

and the Riemanian metric $g_{0}$ becomes [HelMo4, Eq. (8.26)]

g_{0}=dy_{3}\otimes dy_{3}+\sum_{1\leq i,j\leq 2}\big{[}G_{ij}-2y_{3}K_{ij}+y_{3}^{2}L_{ij}\big{]}dy_{i}\otimes dy_{j}

(2.8)

where:

•

$G:=\sum\limits_{1\leq i,j\leq 2}G_{ij}dy_{i}\otimes dy_{j}$ is the first fundamental form on $\partial\Omega$ ;
•

$K:=\sum\limits_{1\leq i,j\leq 2}K_{ij}dy_{i}\otimes dy_{j}$ is the second fundamental form on $\partial\Omega$ ;
•

$L:=\sum\limits_{1\leq i,j\leq 2}L_{ij}dy_{i}\otimes dy_{j}$ is the third fundamental form on $\partial\Omega$ .

The matrix $g$ of the metric $g_{0}$ takes the form

g:=(g_{ij})_{1\leq i,j\leq 3}=\left(\begin{array}[]{lll}g_{11}&g_{12}&0\\ g_{21}&g_{22}&0\\ 0&0&1\end{array}\right)

(2.9)

whose inverse is

g^{-1}=(g^{ij})_{1\leq i,j\leq 3}=\left(\begin{array}[]{lll}g^{11}&g^{12}&0\\ g^{21}&g^{22}&0\\ 0&0&1\end{array}\right)\,.

(2.10)

We will express these matrices in a more pleasant form involving, in particular, the curvatures on the boundary. To that end, let $s\mapsto\gamma(s)$ be an arc-length parameterization of $\Gamma$ near $x_{0}$ , so that $|\dot{\gamma}(s)|=1$ , $\gamma(0)=x_{0}$ and $\mathbf{T}(\gamma(s))=\dot{\gamma}(s)$ . We can introduce the geodesic and normal curvatures at $\gamma(s)$ , $\kappa_{g}(\gamma(s))$ and $\kappa_{n}(\gamma(s))$ , as follows

\ddot{\gamma}(s)=-\kappa_{g}(\gamma(s))\mathbf{V}(\gamma(s))+\kappa_{n}(\gamma(s))\mathbf{N}(\gamma(s))\,.

(2.11)

The choice of our coordinates $(r,s)$ ensures that the metric $G$ is diagonal on $\partial\Omega$ [HelMo4, Lem. 8.2]

G=dr\otimes dr+\alpha(r,s)ds\otimes ds,

(2.12)

with

\alpha(r,s)=1-2\kappa_{g}\big{(}\gamma(s)\big{)}r+{\cal O}(r^{2})\;,\quad\alpha(0,s)=1\,,

(2.13)

and

\frac{\partial\alpha}{\partial s}(0,s)=0\;.

(2.14)

Then, with (2.7), we have for the determinant of the matrix of $g$ (see [HelMo4, Eq. (8.29) & (8.30)]),

|g|=\alpha(r,s)-2t\big{[}\alpha(r,s)K_{11}(r,s)+K_{22}(r,s)\big{]}+t^{2}\varepsilon_{3}(r,s,t)\,,

(2.15)

and

(g^{ij})_{1\leq i,j\leq 2}=\left(\begin{array}[]{ll}1&0\\ 0&\alpha^{-1}(r,s)\end{array}\right)+2t\left(\begin{array}[]{ll}K_{11}(r,s)&\alpha^{-1}K_{12}(r,s)\\ \alpha^{-1}K_{21}(r,s)&\alpha^{-2}K_{22}(r,s)\end{array}\right)+t^{2}R\,,

(2.16)

where $\varepsilon_{3}$ and $R$ are smooth functions.

2.3 The operator and quadratic form

We continue to work in the setting of Subsection 2.2 . We introduce the following neighborhood of $x_{0}$

V_{x_{0}}=\Phi_{x_{0}}^{-1}(\tilde{V}_{x_{0}})\,,

(2.17)

where (recall (2.7))

\tilde{V}_{x_{0}}=\{(y_{1},y_{2},y_{3})\in(-\epsilon_{0},\epsilon_{0})\times(-\epsilon_{0},\epsilon_{0})\times(0,\epsilon_{0})\}\,.

(2.18)

Given a function $u:V_{x_{0}}\to\mathbb{C}$ , we assign to it the function $\tilde{u}:V_{x_{0}}\to\mathbb{C}$ defined by

\tilde{u}(y_{1},y_{2},y_{3})=u(\Phi_{x_{0}}^{-1}(y_{1},y_{2},y_{3}))\,.

(2.19)

By the considerations in Subsection 2.2 on the Riemanian metric, if $u\in L^{2}(V_{x_{0}},dx)$ , then $\tilde{u}\in L^{2}(\tilde{V}_{x_{0}},|g|^{1/2}dy)$ and

\int_{V_{x_{0}}}|u(x)|^{2}dx=\int_{\tilde{V}_{x_{0}}}|\tilde{u}(y)|^{2}|g|^{1/2}dy\,.

(2.20)

Moreover, assuming $u$ supported in $V_{x_{0}}$ , we have the quadratic form formula [HelMo4, Eq. (8.27)]

	$\displaystyle q_{{\bf A}}^{h}(u)$	$\displaystyle:=\int_{V_{x_{0}}}\|(h\nabla-i{\bf A})u\|^{2}dx$		(2.21)
		$\displaystyle=\int_{\tilde{V}_{x_{0}}}\Big{[}\|(hD_{y_{3}}-\tilde{A}_{3})\tilde{u}\|^{2}+\sum_{1\leq i,j\leq 2}g^{ij}(hD_{y_{i}}-\tilde{A}_{i})\tilde{u}\cdot\overline{(hD_{y_{j}}-\tilde{A}_{j})\tilde{u}}\Big{]}\|\,g\|^{1/2}dy$		(2.21)

where the new magnetic potential $\tilde{\bf A}=(\tilde{A}_{1},\tilde{A}_{2},\tilde{A}_{3})$ is assigned to ${\bf A}=(A_{1},A_{2},A_{3})$ by the relation

A_{1}dx_{1}+A_{2}dx_{2}+A_{3}dx_{3}=\tilde{A}_{1}dy_{1}+\tilde{A}_{2}dy_{2}+\tilde{A}_{3}dy_{3}\,,

(2.22)

and after performing a (local) gauge transformation, we may assume that

\tilde{A}_{3}=0\,.

(2.23)

The operator $P_{{\bf A}}^{h}$ in (1.3) can be expressed in the new coordinates as follows [HelMo4, Eq. (8.28)]

P_{{\bf A}}^{h}=(hD_{y_{3}}-\tilde{A}_{3})^{2}+\frac{h}{2i}|g|^{-1}\frac{\partial}{\partial y_{3}}|g|(hD_{y_{3}}-\tilde{A}_{3})\\ +|g|^{-1/2}\sum_{1\leq i,j\leq 2}(hD_{y_{j}}-\tilde{A}_{j})|g|^{1/2}g^{ij}(hD_{y_{i}}-\tilde{A}_{i})\,.

(2.24)

3 Helical magnetic fields

3.1 Preliminaries

Let $\tau>0$ and consider the magnetic potential

{\bf A}(x)=\mathbf{n}_{\tau}(x):=\Big{(}\frac{1}{\tau}\cos(\tau x_{3}),\frac{1}{\tau}\sin(\tau x_{3}),0\Big{)}\,,

(3.1)

which generates the magnetic field

\mathbf{B}(x)={\rm curl}\,{\bf A}(x)=-\tau{\bf A}(x)

(3.2)

with constant intensity

|\mathbf{B}(x)|=1\,.

(3.3)

We will verify that Assumptions C1-C2 hold for this particular magnetic field in the case where $\Omega$ is the unit ball. In particular, with in mind that $\widehat{\gamma}_{0,\mathbf{B}}$ and $\widetilde{\gamma}_{0,\mathbf{B}}$ are introduced in (1.6) and (1.7) respectively and that $\delta_{0}\in]0,1[$ and $\widehat{\nu}_{0}>0$ will be introduced in (4.2) and in (4.5) (there is no need in this subsection to know more about them) we will find that

\widehat{\gamma}_{0,\mathbf{B}}=2^{-2/3}\widehat{\nu}_{0}\delta_{0}^{1/3}C(\tau,\delta_{0})\,,

(3.4)

and for $\tau\leq\tau_{0}$ , the equality,

\{x\in\Gamma~{}|~{}\widetilde{\gamma}_{0,\mathbf{B}}(x)=2^{-2/3}\widehat{\nu}_{0}\delta_{0}^{1/3}\}=\{(0,\pm 1,0)\}\,,

(3.5)

where $\tau_{0}$ is a constant and $C(\tau,\delta_{0})$ is explicitly computed (see Proposition 3.4).

The inward normal of $\Omega=\{x\in\mathbb{R}^{3}~{}|~{}|x|<1\}$ along $\partial\Omega$ is

\mathbf{N}(x)=-x\quad(|x|=1)\,.

(3.6)

The restriction of the magnetic field $\mathbf{B}$ to the boundary is then tangent to $\partial\Omega$ on the following set

\Gamma=\{x\in\partial\Omega~{}|~{}x\cdot{\bf A}(x)=0\}\,.

(3.7)

3.2 $\Gamma$ is a regular curve

For $|x|=1$ , the equation $x\cdot{\bf A}(x)=0$ reads as follows

x_{1}\cos(\tau x_{3})+x_{2}\sin(\tau x_{3})=0\,.

(3.8)

Proposition 3.1.

The set $\Gamma$ introduced in (3.7) is a $C^{\infty}$ regular curve.

Proof.

The proof follows by constructing an atlas on $\Gamma$ ,

\{\big{(}\mathbf{c}_{i},U:=(-1,1)\big{)},~{}1\leq i\leq 4\}

which turns $\Gamma$ to a $C^{\infty}$ regular curve.

Let us introduce the charts $(\mathbf{c}_{1},U)$ and $(\mathbf{c}_{2},U)$ which cover $\Gamma\setminus\{(0,0,\pm 1)\}$ . These charts are obtained by expressing $x_{1}$ and $x_{2}$ in (3.8) in terms of $x_{3}\in(-1,1)$ , provided that $(x_{1},x_{2},x_{3})\not=(0,0,\pm 1)$ . We write for $\alpha\in]-\pi,\pi]$

x_{1}=\sqrt{1-x_{3}^{2}}\,\cos\alpha,\quad x_{2}=\sqrt{1-x_{3}^{2}}\,\sin\alpha\,.

(3.9)

Then (3.8) becomes, for $x_{3}^{2}<1$ ,

\cos(\tau x_{3}-\alpha)=0

(3.10)

which in turn yields

\alpha=\tau x_{3}-\frac{\pi}{2}+k\pi,\quad k\in\mathbb{Z}\,.

In this way, we get two branches of $\Gamma$ parameterized by $x_{3}$ and defined as follows

x_{3}\in(-1,1)\mapsto\mathbf{c}_{1}(x_{3}):=\left(\begin{array}[]{c}x_{1}=\sqrt{1-x_{3}^{2}}\,\sin(\tau x_{3})\\ x_{2}=-\sqrt{1-x_{3}^{2}}\,\cos(\tau x_{3})\\ x_{3}\end{array}\right)

and

x_{3}\in(-1,1)\mapsto\mathbf{c}_{2}(x_{3}):=\left(\begin{array}[]{c}x_{1}=-\sqrt{1-x_{3}^{2}}\sin(\tau x_{3})\\ x_{2}=\sqrt{1-x_{3}^{2}}\cos(\tau x_{3})\\ x_{3}\end{array}\right)\,.

Both of the foregoing branches represent regular curves. Furthermore, $\mathbf{c}_{1}$ and $\mathbf{c}_{2}$ can be extended by continuity to the interval $[-1,1]$ , yielding a continuous representation of all $\Gamma$ .

Now we introduce the charts $(\mathbf{c}_{3},U)$ and $\mathbf{c}_{4},U)$ that cover the points $(0,0,\pm 1)$ . In a neighborhood of $(x_{1},x_{2},x_{3})=(0,0,\pm 1)$ , we parameterize a branch of $\Gamma$ with respect to $\rho:=\sqrt{x_{1}^{2}+x_{2}^{2}}$ as follows

x_{1}=\rho\cos\alpha\,,~{}x_{2}=\rho\sin\alpha\,,~{}x_{3}=\sqrt{1-\rho^{2}}\,.

With this in hand, (3.10) continues to hold for $x_{3}\not=0$ and we can write again $\alpha=\tau x_{3}-\frac{\pi}{2}+k\pi$ for some $k\in\mathbb{Z}$ . Consequently, we get two regular branches of $\Gamma$ defined as follows

\rho\in(-1,1)\mapsto\mathbf{c}_{3}(\rho):=\left(\begin{array}[]{rl}x_{1}&=\rho\sin(\tau\sqrt{1-\rho^{2}})\\ x_{2}&=-\rho\cos(\tau\sqrt{1-\rho^{2}})\\ x_{3}&=\sqrt{1-\rho^{2}}\end{array}\right)

and

\rho\in(-1,1)\mapsto\mathbf{c}_{4}(\rho):=\left(\begin{array}[]{rl}x_{1}&=-\rho\sin(\tau\sqrt{1-\rho^{2}})\\ x_{2}&=\rho\cos(\tau\sqrt{1-\rho^{2}})\\ x_{3}&=\sqrt{1-\rho^{2}}\end{array}\right)\,.

3.3 Explicit formulas in adapted coordinates

Note that $\mathbf{c}:=\mathbf{c}_{1}$ and $\mathbf{c}_{2}$ parameterize all of $\Gamma\setminus\{(0,0,\pm 1)\}$ . By symmetry considerations, we will compute, on $\mathbf{c}\big{(}\,(-1,1)\,\big{)}$ only,

|d^{T}(\mathbf{B}\cdot\mathbf{N})|=\tau|d^{T}(\mathbf{A}\cdot\mathbf{N})|\quad{\rm and}\quad|\mathbf{B}\cdot\mathbf{T}|=\tau|\mathbf{A}\cdot\mathbf{T}|\,.

(3.11)

First we note that $\mathbf{N}=-x$ on $\partial\Omega$ and introduce the arc-length parameter

s(x_{3})=\int_{0}^{x_{3}}|\mathbf{c}^{\prime}(\tilde{x}_{3})|d\tilde{x}_{3}

of $x_{3}\mapsto\mathbf{c}(x_{3})$ , which satisfies

s^{\prime}(x_{3})=|\mathbf{c}^{\prime}(x_{3})|=\sqrt{\frac{1+\tau^{2}(1-x_{3}^{2})^{2}}{1-x_{3}^{2}}}\,.

(3.12)

Clearly, $x_{3}\in(-1,1)$ can be expressed in terms of the arc-length parameter as $x_{3}=x_{3}(s)$ with

m(x_{3}):=\frac{dx_{3}}{ds}(s(x_{3}))=\sqrt{\frac{1-x_{3}^{2}}{1+\tau^{2}(1-x_{3}^{2})^{2}}}\,.

(3.13)

The arc-length parameterization is now given by

\gamma(s):=\mathbf{c}(x_{3}(s))\,,

(3.14)

and consequently, with $\mathbf{c}=\mathbf{c}_{1}$ , we have

\mathbf{N}(\gamma(s))=-\mathbf{\gamma}(s)=\left(\begin{array}[]{c}-\sqrt{1-x_{3}^{2}}\sin(\tau x_{3})\\ \sqrt{1-x_{3}^{2}}\cos(\tau x_{3})\\ -x_{3}\end{array}\right)~{}{\rm with~{}}x_{3}=x_{3}(s)\,,

(3.15)

and

	$\displaystyle\mathbf{T}(\gamma(s))$	$\displaystyle=\frac{d}{ds}\gamma(s)=\left(\begin{array}[]{c}T_{1}\\ T_{2}\\ T_{3}\end{array}\right)$
		$\displaystyle=m(x_{3})\left(\begin{array}[]{c}-\frac{x_{3}\sin(\tau x_{3})}{\sqrt{1-x_{3}^{2}}}+\tau\sqrt{1-x_{3}^{2}}\cos(\tau x_{3})\\ \frac{x_{3}\cos(\tau x_{3})}{\sqrt{1-x_{3}^{2}}}+\tau\sqrt{1-x_{3}^{2}}\sin(\tau x_{3})\\ 1\end{array}\right)\,.$

We also introduce the normal vector to $\Gamma$ on $\gamma(s)$ ,

	$\displaystyle\mathbf{V}(\gamma(s))$	$\displaystyle=\mathbf{T}(\gamma(s))\times\mathbf{N}(\gamma(s))=\left(\begin{array}[]{c}V_{1}\\ V_{2}\\ V_{3}\end{array}\right)$
		$\displaystyle=m(x_{3})\left(\begin{array}[]{c}-\frac{x_{3}^{2}\cos(\tau x_{3})}{\sqrt{1-x_{3}^{2}}}-\tau x_{3}\sqrt{1-x_{3}^{2}}\sin(\tau x_{3})-\sqrt{1-x_{3}^{2}}\cos(\tau x_{3})\\ -\frac{x_{3}^{2}\sin(\tau x_{3})}{\sqrt{1-x_{3}^{2}}}+\tau x_{3}\sqrt{1-x_{3}^{2}}\cos(\tau x_{3})-\sqrt{1-x_{3}^{2}}\sin(\tau x_{3})\\ \tau(1-x_{3}^{2})\end{array}\right)\,.$

We are now ready to prove that our magnetic field $\mathbf{B}$ verifies the condition (C2) appearing in Assumption 1.2

Proposition 3.2.

Let $\mathbf{B}$ be the magnetic field introduced in (3.2). For all $x\in\Gamma$ , we have

|\mathbf{B}(x)\cdot\mathbf{T}(x)|=\frac{\tau(1-x_{3}^{2})}{\sqrt{1+\tau^{2}(1-x_{3}^{2})^{2}}}\,.

In particular, $\mathbf{B}$ satisfies the condition $\rm(C2)$ .

Proof.

It is straightforward to compute

|\mathbf{A}(x)\cdot\mathbf{T}(x)|=\frac{1}{\tau}(|\cos(\tau x_{3})T_{1}+\sin(\tau x_{3})T_{2}|)=\frac{1-x_{3}^{2}}{\sqrt{1+\tau^{2}(1-x_{3}^{2})^{2}}}\,,

(3.16)

which holds for all $-1\leq x_{3}<1$ and $x=\mathbf{c}(x_{3})$ . Similarly, we can compute $|\mathbf{A}(x)\cdot\mathbf{T}(x)|$ for all $x=\mathbf{c}_{2}(x_{3})\in\Gamma$ , and get that (3.16) holds globally on $\Gamma$ , since $\Gamma$ is a regular curve. Finally, $\mathbf{B}(x)$ is orthogonal to $\mathbf{T}(x)$ if and only if $x_{3}^{2}=1$ , thereby (C2) holds.

Our next task is to show that our magnetic field satisfies the condition (C1) in Assumption 1.1.

Proposition 3.3.

Let $\mathbf{B}$ be the magnetic field introduced in (3.2). For all $x\in\Gamma$ , we have

\kappa_{n,{\mathbf{B}}}(x)=\sqrt{1+\tau^{2}(1-x_{3}(s)^{2})^{2}}\,.

(3.17)

In particular, $\mathbf{B}$ satisfies the condition $\rm(C1)$ .

Proof.

By Proposition 3.1, $\Gamma$ is a regular curve. So all we need to verify that $\mathbf{B}$ satisfies $\rm(C1)$ , is to derive (3.17) and observe that it yields $\kappa_{n,{\mathbf{B}}}(x)\not=0$ everywhere on $\Gamma$ .

Consider $x=\mathbf{c}_{1}(x_{3})$ with $x_{3}=x_{3}(s)$ , i.e. $x=\gamma(s)$ . At the point $\gamma(s)$ , the geodesic $\Lambda_{\gamma(s)}$ normal to the curve $\Gamma$ is the great circle (of center $0$ and radius $1$ ) in the $(\mathbf{V}(\gamma(s)),\mathbf{N}(\gamma(s))$ plane. A point $P=P(r,s)$ on $\Lambda_{\gamma(s)}$ can be described by the corresponding vector $\mathbf{p}(r,s)=\vbox{\halign{#\cr\tiny\rightarrowfill\cr\nointerlineskip\vskip 1.0pt\cr$OP\mskip 2.0mu$\cr}}$ as follows

\mathbf{p}(r,s)=-\cos r\,\mathbf{N}(\gamma(s))-\sin r\,\mathbf{V}(\gamma(s))\,,

where $r$ is the angle between $\mathbf{p}$ and $-\mathbf{N}$ ; hence $r$ is an arc-length length parameter of $\Lambda_{\gamma(s)}$ , and for $r=0$ , $p(r,s)=\gamma(s)$ .

Refer to caption — Figure 1: The curve $\Gamma$ and the geodesic $\Lambda_{\gamma(s)}$ passing through $\gamma(s)$ .

Now, we can introduce the coordinates $(r,s,t)$ in a neighborhood of $\gamma(s_{0})$ as follows (see Fig. 1)

x(r,s,t)=-(\cos r+t)\mathbf{N}(\gamma(s))-\sin r\mathbf{V}(\gamma(s))\,.

(3.18)

For $x=\gamma(s)$ , we would like to compute $\kappa_{n,\mathbf{B}}(x)=|d^{T}(\mathbf{B}\cdot\mathbf{N})|$ . We will show that $\kappa_{n,\mathbf{B}}(x)=\partial_{r}(\mathbf{B}\cdot\mathbf{N})|_{r=t=0}$ and end up with the computation of $\big{|}\partial_{r}(\mathbf{B}\cdot\mathbf{N})|_{r=t=0}\,.$

Notice that, by (3.15), we have

\begin{array}[]{ll}x_{3}(r,s,t)&=-(\cos r+t)\mathbf{N}_{3}(\gamma(s))-\sin r\mathbf{V}_{3}(\gamma(s))\\ &=(\cos r+t)x_{3}(s)-\sin r\,m(x_{3}(s))\tau(1-x_{3}(s)^{2})\,,\end{array}

and we observe that by (3.18),

\frac{\partial x}{\partial r}\,\big{|}_{r=t=0}=-V(\gamma(s))\,.

(3.19)

In particular we have

\frac{\partial x_{3}}{\partial r}\,\big{|}_{r=t=0}=-\tau(1-x_{3}^{2}(s))m(x_{3}(s))\,.

Now, using (3.13) and (3.15), we get from (3.1) that

\frac{\partial{\bf A}}{\partial r}\cdot\mathbf{N}\,\big{|}_{r=t=0}=-\frac{\tau(1-x_{3}^{2})^{2}}{\sqrt{1+\tau^{2}(1-x_{3}^{2})^{2}}}\,.

(3.20)

Moreover, by (3.19) we have

\frac{\partial}{\partial r}\mathbf{N}(x(r,s,t))\,\big{|}_{r=t=0}=\mathbf{V}(\gamma(s))

and

	$\displaystyle\mathbf{A}\cdot\frac{\partial}{\partial r}\mathbf{N}(x(r,s,t))\,\big{\|}_{r=t=0}$	$\displaystyle=\frac{1}{\tau}\cos(\tau x_{3}(s))V_{1}+\frac{1}{\tau}\sin(\tau x_{3}(s))V_{2}$		(3.21)
		$\displaystyle=-\frac{1}{\tau\sqrt{1+\tau^{2}(1-x_{3}^{2})^{2}}}\,.$		(3.21)

Summing up, we deduce from (3.20) and (3.21) that

\big{|}\partial_{r}(\mathbf{A}\cdot\mathbf{N})\,|_{r=t=0}\big{|}=\frac{1}{\tau}\sqrt{1+\tau^{2}(1-x_{3}^{2})^{2}}\,.

(3.22)

We also observe that $\partial_{s}(\mathbf{A}\cdot\mathbf{N})\,|_{r=t=0}=0$ and we get

|d^{T}(\mathbf{A}\cdot\mathbf{N})|(\gamma(s))=\frac{1}{\tau}\sqrt{1+\tau^{2}(1-x_{3}^{2})^{2}}\,,

(3.23)

on each branch (including the end points). Inserting this into (3.11), we get the identity in (3.17).

We return to the function in (1.7) and can give its expression in coordinates. We deduce from (3.16) and (3.17):

\widetilde{\gamma}_{0,\mathbf{B}}(x)=2^{-2/3}\widehat{\nu}_{0}\delta_{0}^{1/3}\left(1+\tau^{2}(1-x_{3}^{2})^{2}\right)^{1/3}\left(1-(1-\delta_{0})\frac{\tau(1-x_{3}^{2})}{\sqrt{1+\tau^{2}(1-x_{3}^{2})^{2}}}\right)^{1/3}

for all $x=(\pm\sqrt{1-x_{3}^{2}}\sin(\tau x_{3}),\mp\sqrt{1-x_{3}^{2}}\cos(\tau x_{3}),x_{3})$ with $-1\leq x_{3}\leq 1$ .

Consequently, we can compute the quantity appearing in the two terms asymptotics by computing $\inf_{x\in\Gamma}\widetilde{\gamma}_{0,\mathbf{B}}(x)$ and determining where the infimum is attained.

Proposition 3.4.

Let

\tau_{0}=\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{\delta_{0}+\delta_{0}(1-\delta_{0})}}-1\right)^{1/2}\,.

The following holds:

If $0<\tau\leq\tau_{0}$ , then

\inf_{x\in\Gamma}\widetilde{\gamma}_{0,\mathbf{B}}(x)=2^{-2/3}\widehat{\nu}_{0}\delta_{0}^{1/3}(1+\tau^{2})^{1/3}\Big{(}1-(1-\delta_{0})\frac{\tau^{1/3}}{(1+\tau^{2})^{1/6}}\Big{)}=\widetilde{\gamma}_{0,\mathbf{B}}(0,\pm 1,0)\,.

If $\tau>\tau_{0}$ , then

\inf_{x\in\Gamma}\widetilde{\gamma}_{0,\mathbf{B}}(x)=2^{-2/3}\widehat{\nu}_{0}\delta_{0}^{1/3}(1+\tau_{0}^{2})^{1/3}\Big{(}1-(1-\delta_{0})\frac{\tau_{0}^{1/3}}{(1+\tau_{0}^{2})^{1/6}}\Big{)}\,,

and the minimum is attained on the points

\Big{(}\,\pm\sqrt{\frac{\tau_{0}}{\tau}}\sin\tau\sqrt{1-\frac{\tau_{0}}{\tau}},\mp\sqrt{\frac{\tau_{0}}{\tau}}\cos\tau\sqrt{1-\frac{\tau_{0}}{\tau}},\sqrt{1-\frac{\tau_{0}}{\tau}}\,\Big{)}

and

\Big{(}\,\pm\sqrt{\frac{\tau_{0}}{\tau}}\sin\tau\sqrt{1-\frac{\tau_{0}}{\tau}},\pm\sqrt{\frac{\tau_{0}}{\tau}}\cos\tau\sqrt{1-\frac{\tau_{0}}{\tau}},-\sqrt{1-\frac{\tau_{0}}{\tau}}\,\Big{)}\,.

Remark 3.5.

In the case where $\Omega=B(0,1)$ is the unit ball and the magnetic field is constant, $\mathbf{B}=(0,0,1)$ , we have $\Gamma=\{x_{1}^{2}+x_{2}^{2}=1,~{}x_{3}=0\}$ and $\widetilde{\gamma}_{0,\mathbf{B}}(x)$ is constant on $\Gamma$ . Proposition 3.4 shows a quite different phenomenon when only the intensity of $\mathbf{B}$ is constant, $|\mathbf{B}|=1$ . In fact, $\widetilde{\gamma}_{0,\mathbf{B}}(x)$ is no more constant along $\Gamma$ and may have two symmetric minimum points, $(0,\pm 1,0)$ , which is the signature of an interesting double well tunnel effect [HeSj] related to the magnetic geometry of the problem.

Proof of Proposition 3.4.

Let us introduce $v=\tau(1-x_{3}^{2})\in[0,\tau]$ and $\mu_{0}=1-\delta_{0}\in(0,1)$ . Then

\widetilde{\gamma}_{0,\mathbf{B}}(x)=2^{-2/3}\widehat{\nu}_{0}\delta_{0}^{1/3}\big{(}f(v)\big{)}^{1/3}

where

f(v)=1+v^{2}-\mu_{0}v\sqrt{1+v^{2}}\,.

We have to minimize $f(v)$ on $[0,\tau]$ . Notice that

f^{\prime}(v)=2v-\mu_{0}\frac{1+2v^{2}}{\sqrt{1+v^{2}}}\,,

and the equation $f^{\prime}(v)=0$ has a unique positive solution, which is the solution of

v^{4}+v^{2}=\frac{\mu_{0}^{2}}{4(1-\mu_{0}^{2})}\,.

This solution is given by

\tau_{0}=\frac{1}{\sqrt{2}}\frac{\mu_{0}}{\sqrt{1+\sqrt{1-\mu_{0}^{2}}}\sqrt{1-\mu_{0}^{2}}}

and observe that $f^{\prime}(v)<0$ for $0<v<\tau_{0}$ and $f^{\prime}(v)>0$ for $v>\tau_{0}$ . Then, for $\tau\leq\tau_{0}\,$ ,

\min_{v\in[0,\tau]}f(v)=f(\tau)\,,

while for $\tau>\tau_{0}\,$ ,

\min_{v\in[0,\tau]}f(v)=f(\tau_{0})\,.

4 1D Models

The aim of this section is to recall the now standard properties of two important models.

4.1 The de Gennes model

We refer to [DaHe, HelMo2] for the proof of these now standard properties which are presented below. For $\xi\in\mathbb{R}$ , we consider the harmonic oscillator on $\mathbb{R}_{+}$ :

H(\xi):=D_{t}^{2}+(t-\xi)^{2}\,,

(4.1)

with Neumann boundary condition at $0$ . We denote by $\mu(\xi)$ its lowest eigenvalue. $\xi\mapsto\mu(\xi)$ admits a unique minimum at a point $\xi_{0}$ which in addition is non-degenerate. This leads to introduce the spectral constants, $\Theta_{0}$ and $\delta_{0}$ :

\Theta_{0}=\inf_{\xi\in\mathbb{R}}\mu(\xi)=\mu(\xi_{0}),\quad\delta_{0}=\mu^{\prime\prime}(\xi_{0})\,,

(4.2)

where $\xi_{0}=\sqrt{\Theta_{0}}$ .
Moreover $\frac{1}{2}<\Theta_{0}<1$ and that $0<\delta_{0}<1$ . $\Theta_{0}$ is called the de Gennes constant.
If $\varphi_{0}\in L^{2}(\mathbb{R}_{+})$ denotes the positive and normalized ground state of $H(\xi_{0})$ ,

\int_{\mathbb{R}_{+}}(t-\xi_{0})|\varphi_{0}(t)|^{2}dt=0\,,

(4.3)

which amounts to saying, via the Feynman-Hellmann formula, that $\mu^{\prime}(\xi_{0})=0$ . We also introduce the regularized resolvent $\mathcal{R}_{0}\in\mathcal{L}(L^{2}(\mathbb{R}_{+}))$ as follows

\mathcal{R}_{0}u=\begin{cases}\big{(}H(\xi_{0})-\Theta_{0}\big{)}^{-1}u&{\rm if}~{}u\perp\varphi_{0}\\ 0&{\rm if}~{}u\parallel\varphi_{0}\end{cases}\,.

(4.4)

4.2 The Montgomery model

Here we refer to [HelMo1] and [PanKw]. In Theorem 1.3, the constant $\widehat{\nu}_{0}>0$ is related to the Montgomery model [Mon] whose spectral analysis has a long story including recently (see [HeLe] and references therein). For $\rho\in\mathbb{R}$ , we introduce, in $L^{2}(\mathbb{R})$ , the operator

S(\rho)=D_{r}^{2}+(r^{2}-\rho)^{2}\,,

and denote its lowest eigenvalue by $\mu^{\rm Mon}(\rho)$ . Then

\widehat{\nu}_{0}:=\inf_{\rho\in\mathbb{R}}\mu^{\rm Mon}(\rho)=\mu^{\rm Mon}(\rho_{0})\,,

(4.5)

where $\rho_{0}\in\mathbb{R}$ is the unique minimum of $\mu^{\rm Mon}$ , which has been later shown to be non degenerate [HeKo]. Finally, the normalized positive ground state $\psi_{0}\in L^{2}(\mathbb{R})$ of $S(\rho_{0})$ belongs to the Schwartz space $S(\mathbb{R})$ and is an even function.

5 Model operator for non-uniform magnetic fields

Given real parameters $\eta,\zeta,\gamma$ and $\theta$ , we consider the operator

\begin{array}[]{ll}P^{h,\eta,\zeta}_{0;\gamma,\theta}&:=(hD_{r}-\sin\theta\,t-\cos\theta\,(\eta s+\zeta r)t)^{2}\\ &\quad+(hD_{s}+\cos\theta\,t-\sin\theta\,(\eta s+\zeta r)\,t+\gamma\frac{r^{2}}{2})^{2}\\ &\quad+h^{2}D_{t}^{2}\;,\end{array}

(5.1)

on $\mathbb{R}^{2}\times\mathbb{R}^{+}$ (actually in a neighborhood of $(0,0,0)$ ). Let us fix a positive constant $M$ . We assume that

\eta,\zeta,\gamma\in[-M,M]\,.

(5.2)

We note, when $\eta=\zeta=0$ , we recover the model studied in [HelMo4, Sec. 11]. Our aim is to compare this situation with that when $\eta=\zeta=0$ . Our main result on this model is Proposition 5.5 below, which is useful in our derivation of the lower bound matching with the asymptotics in Theorem 1.4. The lower bound in this proposition is uniform with respect to the various parameters appearing in (5.1) provided (5.2) holds and $h$ is sufficiently small.

Let us look at this model more carefully. We proceed essentially like in the case $\eta=\zeta=0$ . We do the following scaling

r=h^{\frac{1}{3}}\hat{r}\;,\;s=h^{\frac{1}{3}}\hat{s}\;,\;t=h^{\frac{1}{2}}\hat{t}\;.

(5.3)

After division by $h$ , this leads to (forgetting the hats)

\begin{array}[]{l}P_{1;\gamma,\theta}^{h,\eta,\zeta}:=\left(h^{\frac{1}{6}}D_{r}-\sin\theta\,t-h^{\frac{1}{3}}\cos\theta\,t(\eta s+\zeta r)\right)^{2}\\ \quad\quad+\left(h^{\frac{1}{6}}D_{s}+\cos\theta\,t+h^{\frac{1}{6}}\gamma\frac{r^{2}}{2}-h^{\frac{1}{3}}\sin\theta\,t(\eta s+\zeta r)\right)^{2}+D_{t}^{2}\end{array}

(5.4)

on $\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{+}$ .
Hence we have

\sigma(P_{0;\gamma,\theta}^{h,\eta,\zeta})=h\,\sigma(P_{1;\gamma,\theta}^{h,\eta,\zeta}).

Unlike the case where $\eta=\zeta=0$ , we can no more perform a partial Fourier transform in the $s$ -variable. But we can rewrite this operator as in the following lemma.

Lemma 5.1.

It holds,

P_{1;\gamma,\theta}^{h,\eta,\zeta}=D_{t}^{2}+(t-h^{\frac{1}{6}}L_{1,\gamma,\theta})^{2}+h^{\frac{1}{3}}(L_{2,\gamma,\theta}^{h,\eta,\zeta})^{2}\;,

(5.5)

where

\begin{array}[]{ll}L_{1;\gamma,\theta}&=\sin\theta D_{r}-\cos\theta\left(\frac{\gamma}{2}r^{2}+D_{s}\right)\;,\\ L^{h,\eta,\zeta}_{2;\gamma,\theta}&:=\cos\theta D_{r}+\sin\theta\left(\frac{\gamma}{2}r^{2}+D_{s}\right)-h^{\frac{1}{6}}(\zeta r+\eta s)t\,.\end{array}

(5.6)

Note that to compare with the case considered in [HelMo4] ( $\eta=\zeta=0$ ) we can write

L_{2;\gamma,\theta}^{h,\eta,\zeta}=L_{2;\gamma,\theta}-h^{\frac{1}{6}}(\zeta r+\eta s)t\,,

(5.7)

where $L_{2;\gamma,\theta}:=L_{2;\gamma,\theta}^{0,0,0}$ .

Proof of Lemma 5.1.

Let $P_{1;\gamma,\theta}^{h}:=P_{1;\gamma,\theta}^{h,0,0}$ . Then (see [HelMo4, Eq. (11.4)])

P_{1;\gamma,\theta}^{h}=D_{t}^{2}+(t-h^{\frac{1}{6}}L_{1,\gamma,\theta})^{2}+h^{\frac{1}{3}}(L_{2,\gamma,\theta})^{2}\,.

With $p=(\eta s+\zeta r)t$ , we have

P_{1;\gamma,\theta}^{h,\eta,\zeta}=P_{1;\gamma,\theta}^{h}+h^{\frac{1}{3}}\big{[}-2(h^{\frac{1}{6}}p)L_{2;\gamma,\theta}-h^{\frac{1}{6}}\big{(}\cos\theta(D_{r}p)+\sin\theta(D_{s}p)\big{)}+(h^{\frac{1}{6}}p)^{2}\big{]}\,.

Finally, we observe by (5.7),

(L_{2;\gamma,\theta}^{h,\eta,\zeta})^{2}=(L_{2,\gamma,\theta})^{2}-2(h^{\frac{1}{6}}p)L_{2;\gamma,\theta}-h^{\frac{1}{6}}\big{(}\cos\theta(D_{r}p)+\sin\theta(D_{s}p)\big{)}+(h^{\frac{1}{6}}p)^{2}\,.

When $\eta=\zeta=0$ , this is the operator studied in [HelMo4], modulo a Fourier transformation with respect to the $s$ variable. Let us recall the following important result [HelMo4, Lem. 13.4] corresponding to the case $(\eta,\zeta)=(0,0)$ .

Proposition 5.2 (Helffer-Morame).

For any $C_{0}>0$ , $\delta\in]0,\frac{1}{3}[$ and $M>0$ , there exist positive constants $C$ and $h_{0}$ such that, for all $\theta\in\mathbb{R}$ , $|\gamma|\leq M$ , and $h\in]0,h_{0}]$ , we have, for any $u\in C_{0}^{\infty}(]-C_{0}h^{\delta},C_{0}h^{\delta}[\times\mathbb{R}\times\overline{\mathbb{R}_{+}})$

\langle P_{0;\gamma,\theta}^{h,0,0}u\,,\,u\rangle\geq\big{(}h\Theta_{0}+h^{\frac{4}{3}}c^{\rm conj}(\gamma,\theta)-C(h^{\frac{11}{8}}+h^{\delta+\frac{13}{12}})\big{)}\;\|u\|^{2}\;,

(5.8)

where

c^{\rm conj}(\gamma,\theta):=\Big{(}\frac{1}{2}\Big{)}^{\frac{2}{3}}\delta_{0}^{\frac{1}{3}}|\gamma|^{\frac{2}{3}}(\delta_{0}\sin^{2}\theta+\cos^{2}\theta)^{\frac{1}{3}}\,{\hat{\nu}_{0}},

and $P^{h,0,0}_{0;\gamma,\theta}$ is the operator introduced in (5.1).

Remark 5.3.

The underlying estimate in Proposition 5.2 is in fact

\langle P_{1;\gamma,\theta}^{h,0,0}u\,,\,u\rangle\geq\big{(}\Theta_{0}+h^{\frac{1}{3}}c^{\rm conj}(\gamma,\theta)-C(h^{\frac{3}{8}}+h^{\delta+\frac{1}{12}})\big{)}\;\|u\|^{2}.

We can not directly compare $P_{1;\gamma,\theta}^{h,\eta,\zeta}$ and $P_{1;\gamma,\theta}^{h,0,0}$ but this can be done by introducing a small perturbation of $P_{1;\gamma,\theta}^{h,0,0}$ whose spectrum is just lifted. To achieve this goal we introduce for $\tau>0$

P_{1;\gamma,\theta,\tau}^{h}:=D_{t}^{2}+(t-h^{\frac{1}{6}}L_{1,\gamma,\theta})^{2}+(1-h^{\tau})h^{\frac{1}{3}}(L_{2,\gamma,\theta})^{2},

where we have modified the coefficient of $(L_{2,\gamma,\theta})^{2}$ by $\epsilon=h^{1/3+\tau}$ . Heuristically this leads to a maximal shift of the bottom of the spectrum by $\mathcal{O}(h^{1/3+\tau})$ . More precisely, we show by a slight variation of the argument in [HelMo4, Lem. 13.3]

Proposition 5.4.

For all $\tau\in\,]0,1[$ , for any $C_{0}>0$ , $\delta\in]0,\frac{1}{3}[$ and $M>0$ , there exist positive constants $C$ and $h_{0}$ such that, for all $\theta\in\mathbb{R}$ , $|\gamma|\leq M$ , and $h\in]0,h_{0}]$ , we have, for any $u\in C_{0}^{\infty}(]-C_{0}h^{\delta},C_{0}h^{\delta}[\times\mathbb{R}\times\overline{\mathbb{R}_{+}})$

\langle P_{1,\gamma,\theta,\tau}^{h}u\,,\,u\rangle\geq\big{(}\Theta_{0}+h^{\frac{1}{3}}c^{\rm conj}(\gamma,\theta)-C(h^{\tau+\frac{1}{3}}+h^{\frac{3}{8}}+h^{\delta+\frac{1}{12}})\big{)}\;\|u\|^{2}\,.

(5.9)

Note that the estimate in Proposition 5.2 holds without constraint on the support of the function in $s$ . This will not be the case for $(\eta,\zeta)\not=0$ .
We now compare $\langle P_{1;\gamma,\theta}^{h,\eta,\zeta}u,u\rangle$ and $\langle P_{1;\gamma,\theta,\tau}^{h}u,u\rangle$ when

u\in C_{0}^{\infty}\big{(}\,]-C_{0}h^{\delta-\frac{1}{3}},C_{0}h^{\delta-\frac{1}{3}}[\times\,]-C_{0}h^{\delta-\frac{1}{3}},C_{0}h^{\delta-\frac{1}{3}}[\times\overline{\mathbb{R}_{+}}\,\big{)}.

and $\eta,\zeta$ satisfies (5.2).

Let us fix

\delta\in\,]\frac{1}{4},\frac{1}{3}[\mbox{ and }\tau\in]0,\frac{1}{6}[\,.

(5.10)

The estimates below hold uniformly with respect to $u$ , $\theta\in\mathbb{R}$ and $\eta,\zeta,\gamma$ satisfying (5.2).
Comparing $L_{2,\gamma,\theta}^{h,\eta,\zeta}$ and $L_{2,\gamma,\theta}$ in (5.7), we find¹¹1We use $2ab\leq\varepsilon a^{2}+\varepsilon^{-1}b^{2}$ with $\varepsilon=h^{\tau}$ , $a=\|L_{2,\gamma,\theta}^{h,\eta,\zeta}u\|$ and $b=\|L_{2,\gamma,\theta}u\|^{2}$ ., for all $\tau>0$ ,

\langle(L_{2,\gamma,\theta}^{h,\eta,\zeta})^{2}u,u\rangle=\|L_{2,\gamma,\theta}^{h,\eta,\zeta}u\|^{2}\geq(1-h^{\tau})\|L_{2,\gamma,\theta}u\|^{2}+(1-h^{-\tau})\|(L_{2,\gamma,\theta}^{h,\eta,\zeta}-L_{2,\gamma,\theta})u\|^{2}\,.

Consequently,

\langle P_{1;\gamma,\theta}^{h,\eta,\zeta}u,u\rangle\geq\langle\big{(}D_{t}^{2}+(t-h^{\frac{1}{6}}L_{1;\gamma,\theta})^{2}+(1-h^{\tau})h^{\frac{1}{3}}(L_{2;\gamma,\theta})^{2}\big{)}u,u\rangle\\ -h^{\frac{1}{3}-\tau}||(L_{2;\gamma,\theta}^{h,\eta,\zeta}-L_{2;\gamma,\theta})u||^{2}\;.

(5.11)

This implies (see (5.7) and the condition on the support of $u$ ),

\langle P_{1;\gamma,\theta}^{h,\eta,\zeta}u,u\rangle\geq\langle P_{1;\gamma,\theta,\tau}^{h}u,u\rangle-C(\eta^{2}+\zeta^{2})h^{2\delta-\tau}||tu||^{2}\;,

(5.12)

where we used (see (5.7))

L_{2;\gamma,\theta}^{h,\eta,\zeta}-L_{2;\gamma,\theta}=h^{1/6}t\,\mathcal{O}((|s|+|r|))=t\,\mathcal{O}(h^{\delta-\frac{1}{6}})

in the support of $u$ .
By (5.9) and (5.12) we have

\langle P^{h,\eta,\zeta}_{1;\gamma,\theta}u,u\rangle\geq\big{(}\Theta_{0}+c^{\rm conj}(\gamma,\theta)h^{1/3}-C(h^{\frac{3}{8}}+h^{\delta+\frac{1}{12}}+h^{\tau})\big{)}\|u\|^{2}\\ -C(\eta^{2}+\zeta^{2})h^{2\delta-\tau}\|tu\|^{2}\;.

(5.13)

Note that by (5.10) we have

h^{\frac{3}{8}}+h^{\delta+\frac{1}{12}}+h^{\tau+\frac{1}{3}}+h^{2\delta-\tau}=\mathcal{O}(h^{\frac{1}{3}+\varsigma})\,,

for some $\varsigma=\varsigma(\delta,\tau)>0$ .
Consequently, there exist $C$ , $\varsigma>0$ and $h_{0}$ such that, $\forall h\in]0,h_{0}]$ ,

\langle P_{1;\gamma,\theta,\tau}^{h,\eta,\zeta}u,u\rangle\geq\big{(}\Theta_{0}+c^{\rm conj}(\gamma,\theta)h^{1/3}-Ch^{\frac{1}{3}+\varsigma}\big{)}\|u\|^{2}-Ch^{\frac{1}{3}+\varsigma}\|tu\|^{2}\,,

(5.14)

for any $u\in C_{0}^{\infty}(\,]-Ch^{\delta-\frac{1}{3}},Ch^{\delta-\frac{1}{3}}[^{2}\times\overline{\mathbb{R}_{+}}\,)$ .
By coming back to the initial coordinates, we get the following generalization of Proposition 5.2.

Proposition 5.5.

Let $C_{0},M>0$ and $\delta\in]\frac{1}{4},\frac{1}{3}[$ be given. There exist positive constants $C$ , $h_{0}$ , and $\varsigma$ , such that, for all $h\in]0,h_{0}]$ , $\theta\in\mathbb{R}$ and $\gamma,\eta,\zeta\in[-M,M]$ , we have, for any $u\in C_{0}^{\infty}(]-C_{0}h^{\delta},C_{0}h^{\delta}[^{\,2}\,\times\,\overline{\mathbb{R}^{+}})$ ,

\langle P_{0,\gamma,\theta}^{h,\eta,\zeta}u\,,\,u\rangle\geq\big{(}h\Theta_{0}+h^{\frac{4}{3}}c^{\rm conj}(\gamma,\theta)-Ch^{\frac{4}{3}+\varsigma}\big{)}\|u\|^{2}-Ch^{\frac{1}{3}+\varsigma}\|tu\|^{2}.

(5.15)

Note here that the last term will be small when considering localized states satisfying (6.6).

6 Localization of bound states

We recall that the bound states of the operator $P^{h}_{\bf A}$ in (1.3) are localized on the boundary near the curve where the magnetic field is tangent to the boundary $\partial\Omega$ . The localization is related with the analysis of a family of model operators in the half-space [LuPa5].

Consider $\mathbb{R}_{+}^{3}:=\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\,|\,x_{1}>0\}$ and the Neumann realization in $\mathbb{R}_{+}^{3}$ of the operator,

H(\nu)=D_{x_{1}}^{2}+D_{x_{2}}^{2}+(D_{x_{3}}+x_{1}\cos\nu-x_{2}\sin\nu)^{2}\,,

where $\nu\in[-\frac{\pi}{2},\frac{\pi}{2}]$ .
More precisely, $H(\nu)$ is self-adjoint in $L^{2}(\mathbb{R}_{+}^{3})$ with the following domain

{\rm Dom}\big{(}H(\nu)\big{)}=\{u\in L^{2}(\mathcal{\mathbb{R}}^{3}_{+})\,|\,H(\nu)u\in L^{2}(\mathcal{\mathbb{R}}^{3}_{+}),~{}\partial_{x_{1}}u|_{x_{1}=0}=0\}\,.

We denote by

\sigma(\nu)=\inf_{\nu\in[-\frac{\pi}{2},\frac{\pi}{2}]}{\rm spec}\,\big{(}H(\nu)\big{)}\,.

(6.1)

We gather some properties of the lowest eigenvalue $\sigma(\nu)$ (see [LuPa5], [HelMo2], and [HelMo4, Sec. 3.3]):

Proposition 6.1.

The following properties hold for the lowest eigenvalue $\sigma(\nu)$ of $H(\nu)$ :

•

For all $\nu\in[-\frac{\pi}{2},\frac{\pi}{2}]$ , $\sigma(-\nu)=\sigma(\nu)$ .
•

$[0,\frac{\pi}{2}]\ni\nu\mapsto\sigma(\nu)$ is monotone increasing and $\sigma(0)=\Theta_{0}$ .
•

$\sigma(\nu)\geq\Theta_{0}\cos^{2}\nu+\sin^{2}\nu$ .
•

As $\nu\to 0$ , $\sigma(\nu)=\Theta_{0}+\sqrt{\delta_{0}}\,|\nu|+\mathcal{O}(\nu^{2})$ .

Here we recall that $\Theta_{0}$ and $\delta_{0}$ are introduced in (4.2).

Let us return to the magnetic field in (1.1). Recall that, for $x\in\Omega$ , $p(x)\in\partial\Omega$ satisfies ${\rm dist}(x,\partial\Omega)={\rm dist}(x,p(x))$ , and it is uniquely defined when $x$ is sufficiently close to the boundary. For all $x\in\overline{\Omega}$ , we introduce $\nu(x)\in[-\frac{\pi}{2},\frac{\pi}{2}]$ by

(\mathbf{B}\cdot\mathbf{N})(p(x))=\sin\nu(x)\,.

(6.2)

Hence $\nu(x)=0$ implies that $\mathbf{B}(p(x))$ is tangent to $\partial\Omega$ at $p(x)$ , in other words that $x$ belongs to $\Gamma$ (see (1.4)). Now we recall the following lower bound related to the operator $P_{\bf A}^{h}$ established in [HelMo4, Thm. 4.3]:

Proposition 6.2.

Under Assumption (1.1), there exist constants $C,h_{0}>0$ such that, for all $h\in(0,h_{0}]$ and $u\in H^{1}(\Omega)$ , we have

\int_{\Omega}|(h\nabla-i{\bf A})u|^{2}dx\geq\int_{\Omega}(hW_{h}(x)-Ch^{5/4})|u(x)|^{2}dx\,,

where

W_{h}(x)=\begin{cases}1&{\rm if~{}}{\rm dist}(x,\partial\Omega)\geq 2h^{3/8}\\ \,\sigma(\nu(x))&{\rm if~{}}{\rm dist}(x,\partial\Omega)\leq 2h^{3/8}\end{cases}\,.

If additionally $u\in H^{1}_{0}(\Omega)$ , we have for some positive constant $C_{0}$ the stronger lower bound

\int_{\Omega}|(h\nabla-i{\bf A})u|^{2}dx\geq(h-C_{0}h^{5/4})\int_{\Omega}|u|^{2}\,dx\,.

Combining the lower bound in Proposition 6.2 with the following leading term expansion of the lowest eigenvalue (see [HelMo4, Thm. 4.4])

\lambda_{1}^{N}({\bf A},h)=\Theta_{0}h+o(h)\,,

(6.3)

we get decay estimates for the ground states. Let us recall these localization estimates (see [FoHe2, Sec. 9.4] for details).

Proposition 6.3.

Given $M>0$ , there exists a positive constant $\alpha$ such that, if $u_{h}$ is a normalized bound state of $\mathcal{P}_{h}$ with eigenvalue $\lambda(h)\leq Mh$ , then as $h\to 0_{+}$ ,

\int_{\Omega}\Big{(}|u_{h}(x)|^{2}+h^{-1}|(h\nabla-i{\bf A})u_{h}|^{2}\Big{)}\exp\Big{(}\frac{\alpha\,{\rm dist}(x,\partial\Omega)}{h^{1/2}}\Big{)}dx=\mathcal{O}(1)\,.

(6.4)

Furthermore, there exist constants $\alpha_{1},\epsilon_{0}>0$ such that, as $h\to 0_{+}$ ,

\int_{\{{\rm dist}(x,\partial\Omega)<\epsilon_{0}\}}\Big{(}|u_{h}(x)|^{2}+h^{-1}|(h\nabla-i{\bf A})u_{h}|^{2}\Big{)}\exp\Big{(}\frac{\alpha_{1}\,d_{\Gamma}(x)}{h^{1/4}}\Big{)}dx=\mathcal{O}(1)\,,

where

d_{\Gamma}(x)={\rm dist}_{\partial\Omega}(p(x),\Gamma)\,,

(6.5)

and ${\rm dist}_{\partial\Omega}$ is the geodesic distance on $\partial\Omega$ .

Hence we have two levels of localization, first a strong one near $\partial\Omega$ and then an additional but weaker one near $\Gamma$ . Along the proof of Theorem 1.4, we will only use (6.4) and generalizations/consequences of it, as explained in the below remark.

Remark 6.4 (Applications of Proposition 6.3).

Let $u_{h}$ be a normalized ground state of $P_{\bf A}^{h}$ .

1.

By (6.3), the hypothesis in Proposition 6.3 holds, hence the ground state $u_{h}$ satisfies (6.4) and (6.5).

Pick an arbitrary point $x_{0}\in\Gamma$ . In the coordinates introduced in (2.7), where $t(x)={\rm dist}(x,\partial\Omega)$ , $r(x)=d_{\Gamma}(x)$ and $u_{h}(x)=\tilde{u}_{h}(r,s,t)$ (see (2.19)), we deduce from (6.4) the following weaker, but quite useful estimates. For any $n\geq 0$ ,

\int_{\tilde{V}_{0}}t^{n}|\tilde{u}_{h}|^{2}dsdrdt=\mathcal{O}(h^{n/2}),

(6.6)

and

\int_{\tilde{V}_{0}}t^{n}|(h\nabla_{r,s,t}-\tilde{\bf A})\tilde{u}_{h}|^{2}drdsdt=\mathcal{O}(h^{1+\frac{n}{2}})\,,\\

(6.7)

where $\tilde{V}_{0}:=\tilde{V}_{x_{0}}$ and $\tilde{\bf A}$ are introduced in (2.18) and (2.22) respectively.

7 Estimating the quadratic form

7.1 A comparison estimate

We fix $\delta$ and $\epsilon_{2}$ satisfying

\frac{5}{18}<\delta<\frac{1}{3}\mbox{ and }0<\epsilon_{2}<1\,.

(7.1)

We also fix $R_{0}>0$ , $h_{0}>0$ , $x_{0}\in\Gamma$ and introduce for $h\in(0,h_{0}]$ the set

Q_{h}(x_{0},R_{0},\delta,\epsilon_{2})\\ =\{x\in\Omega\,:\,|r(x)-r_{0}|\leq R_{0}h^{\delta},~{}|s(x)-s_{0}|\leq R_{0}h^{\delta},~{}0<t(x)<\epsilon_{2}\}

(7.2)

where $(r(x),s(x),t(x))$ are introduced in (2.2) and, since $x_{0}\in\Gamma$ ,

y^{0}:=(r_{0},s_{0},t_{0}):=(r(x_{0}),s(x_{0}),t(x_{0}))=(0,s(x_{0}),0)\,.

(7.3)

For simplicity, we omit most of the time the reference to $\delta$ and $\epsilon_{2}$ .

Let $\tilde{\bf A}=(\tilde{A}_{1}^{(2)},\tilde{A}_{2}^{(2)},\tilde{A}_{3}^{(2)})$ be the magnetic potential associated with ${\bf A}$ via (2.22), with $y=(y_{1},y_{2},y_{3})=(r,s,t)$ (see (2.7)). We introduce the following magnetic potential

\tilde{\bf A}^{(2)}(y)=\sum_{|\beta|\leq 2}\frac{\partial^{\beta}\tilde{\bf A}}{\partial y^{\beta}}(y^{0})\frac{(y-y^{0})^{\beta}}{\beta!},

(7.4)

which is the quadratic Taylor expansion of $\tilde{\bf A}$ at $y^{0}$ . We introduce the quadratic form associated with the magnetic potential $\tilde{\bf A}^{(2)}$ as follows

q_{\tilde{\bf A}^{(2)}}^{h}(u)=\int_{\tilde{Q}_{h}(x_{0},R_{0})}(1-r\kappa_{g}(x_{0}))\Big{(}|(hD_{t}-\tilde{A}_{3}^{(2)})u|^{2}\\ +(1+2r\kappa_{g}(x_{0}))|(hD_{s}-\tilde{A}_{2}^{(2)})u|^{2}+|(hD_{r}-\tilde{A}_{1}^{(2)})u|^{2}\Big{)}drdsdt\,,

where

\tilde{Q}_{h}(x_{0},R_{0},\delta,\epsilon_{2})=\{(r,s,t)~{}:~{}\max(|r|,|s-s_{0}|)<R_{0}h^{\delta},~{}0<t<\epsilon_{2}\}\,,

(7.5)

and (see (2.11))

\kappa_{g}(x_{0})\mbox{ is the geodesic curvature of }\Gamma\mbox{ at }x_{0}\,.

The next lemma compares the quadratic forms $u\mapsto q_{\tilde{\bf A}^{(2)}}^{h}(u)$ and $u\mapsto q_{\bf A}^{h}(u)$ introduced in (2.21). The errors that will arise are controlled by the following energy

M_{h}(u)=\sum_{n=0}^{6}h^{-n/2}\int_{\Omega}t(x)^{n}\Big{(}|u|^{2}+h^{-1}|(h\nabla-i{\bf A})u|^{2}\Big{)}dx\,,

(7.6)

where $t(x)={\rm dist}(x,\partial\Omega)$ . Notice that,

$\displaystyle{\rm(a)}\quad\int_{\Omega}\|u\|^{2}dx$	$\displaystyle\leq M_{h}(u)\,,$	(7.7)
$\displaystyle{\rm(b)}\quad\int_{\Omega}\|(h\nabla-i{\bf A})u\|^{2}dx$	$\displaystyle\leq M_{h}(u)h\,,$
$\displaystyle{\rm(c)}\quad\int_{\Omega}t(x)^{n}\big{(}\|u\|^{2}+h^{-1}\|(h\nabla-i{\bf A})u\|^{2}\big{)}dx$	$\displaystyle\leq M_{h}(u)h^{n/2}\quad(1\leq n\leq 6)\,.$

Lemma 7.1.

There exist constants $C,h_{0},\varsigma_{0}>0$ such that, for all $h\in(0,h_{0}]$ and $u\in H^{1}(\Omega)$ satisfying ${\rm supp\,}\,u\subset Q_{h}(x_{0},R_{0})$ , we have

(1-C\,h^{2\delta})q_{\tilde{\bf A}^{(2)}}^{h}(u)-CM_{h}(u)h^{\frac{4}{3}+\varsigma_{0}}\leq q_{\bf A}^{h}(u)\leq(1+C\,h^{2\delta})q_{\tilde{\bf A}^{(2)}}^{h}(u)+CM_{h}(u)h^{\frac{4}{3}+\varsigma_{0}}\,.

(7.8)

Proof.

Let us recall two useful estimates whose proof does not require that the magnetic field ${\rm curl}\,{\bf A}$ is constant (see [HelMo4, Lem. 10.1]):

q_{\bf A}^{h}(u)\geq(1-Ch^{2\delta})q_{\tilde{\bf A}^{(2)}}^{h}(u)-C\|t^{1/2}(hD_{x}-{\bf A})u\|^{2}\\ -C\big{(}q_{\tilde{\bf A}^{(2)}}^{h}(u)\big{)}^{1/2}\|(h^{3\delta}+h^{2\delta}t+h^{\delta}t^{2}+t^{3})u\|\\ -C\|(h^{3\delta}+h^{2\delta}t+h^{\delta}t^{2}+t^{3})u\|^{2}\,,

(7.9)

and

q_{\bf A}^{h}(u)\leq(1+Ch^{2\delta})q_{\tilde{\bf A}^{(2)}}^{h}(u)+C\|t^{1/2}(hD_{x}-{\bf A})u\|^{2}\\ +C\big{(}q_{\tilde{\bf A}^{(2)}}^{h}(u)\big{)}^{1/2}\|(h^{3\delta}+h^{2\delta}t+h^{\delta}t^{2}+t^{3})u\|\\ +C\|(h^{3\delta}+h^{2\delta}t+h^{\delta}t^{2}+t^{3})u\|^{2}\,.

(7.10)

In the sequel we use the notation $\mathcal{O}(c_{h}h^{\rho+})$ in the following manner

f_{h}=\mathcal{O}(c_{h}h^{\rho+})\mbox{ if and only if }\exists\,\epsilon>0\mbox{ s.t. }f_{h}=\mathcal{O}(c_{h}h^{\rho+\epsilon}).

(7.11)

Since we have assumed (7.1), we have

\min(6\delta\,,\,2\delta+1\,,\,3\delta+\frac{1}{2}\,,\,2-2\delta)>\frac{4}{3}\,.

We can now estimate the error terms appearing in (7.9) and (7.10). We deduce from (7.7) (a) that

\|h^{3\delta}u\|^{2}=\mathcal{O}(M_{h}h^{6\delta})=\mathcal{O}(M_{h}h^{\frac{4}{3}+})\,,

where we write $M_{h}$ instead of $M_{h}(u)$ for the sake of simplicity.

Using again (7.7) with $n=1$ , $n=2$ , $n=4$ and $n=6$ , we get

\begin{array}[]{rl}\|t^{1/2}(hD_{x}-{\bf A})u\|^{2}&=\mathcal{O}(M_{h}h^{\frac{5}{4}}),\\ \|h^{2\delta}tu\|^{2}&=\mathcal{O}(M_{h}h^{4\delta+1})\,,\\ \|h^{\delta}t^{2}u\|^{2}&=\mathcal{O}(M_{h}h^{2\delta+2})\,,\\ \|t^{3}u\|^{2}&=\mathcal{O}(M_{h}h^{3})\,.\end{array}

Consequently,

\|t^{1/2}(hD_{x}-{\bf A})u\|^{2}+\|(h^{3\delta}+h^{2\delta}t+h^{\delta}t^{2}+t^{3})u\|^{2}=\mathcal{O}(M_{h}h^{\frac{4}{3}+})\,.

(7.12)

Notice that $|\tilde{\bf A}-\tilde{\bf A}^{(2)}|=\mathcal{O}(h^{3\delta})+\mathcal{O}(t^{3})$ in $\tilde{Q}_{h}(x_{0},R_{0})$ . By the triangle inequality and (2.21)

q_{\tilde{\bf A}^{(2)}}^{h}(u)\leq C\,\big{(}q_{{\bf A}}^{h}(u)+\|t^{3}u\|^{2}\big{)}\,.

So by using (7.7) we get

q_{\tilde{\bf A}^{(2)}}^{h}(u)=\mathcal{O}(M_{h}h)\,.

Consequently, the foregoing estimate and (7.12) yield,

\big{(}q_{\tilde{\bf A}^{(2)}}^{h}(u)\big{)}^{1/2}\|(h^{3\delta}+h^{2\delta}t+h^{\delta}t^{2}+t^{3})u\|=\mathcal{O}(M_{h}h)\,.

This finishes the proof of (7.8).

7.2 Normal form

Recall that we have fixed an arbitrary point $x_{0}\in\Gamma$ and denoted its coordinates, in the $(r,s,t)$ -frame, by $(0,s_{0},0)$ . Let us also recall that the magnetic field $\mathbf{B}(x_{0})$ can be expressed by (2.5).

Performing an appropriate gauge transformation on the set $\tilde{Q}_{h}(x_{0},R_{0})$ introduced in (7.5), will yield a convenient normal form of the magnetic potential $\tilde{{\bf A}}^{(2)}$ introduced in (7.4).

Lemma 7.2.

There exist positive constants $C$ and $\widehat{C}$ , and for all $x_{0}\in\Gamma$ , there exist $\check{\kappa},\zeta\in[-\widehat{C},\widehat{C}]$ and a smooth function $\check{p}$ on a neighborhood of $\tilde{Q}_{h}(x_{0},R_{0},\delta,\epsilon_{2})$ , such that,

|\tilde{\bf A}^{(2)}(r,s,t)-{\bf A}^{00}(r,s,t)+\nabla\check{p}(r,s,t)|\leq C\,\big{(}r^{3}+t^{2}+|s-s_{0}|^{3}\big{)}\,,

where

{\bf A}^{00}(r,s,t)=\Big{(}ta_{1}(r,s),ta_{2}(r,s)+\frac{1}{2}\kappa_{n,\mathbf{B}}(x_{0})r^{2},0\Big{)},

$\kappa_{n,\mathbf{B}}(x_{0})$ is introduced in (1.5), and

	$\displaystyle a_{1}(r,s)$	$\displaystyle=\sin\theta(s_{0})+\big{(}\zeta r+\check{\kappa}(s-s_{0})\big{)}\cos\theta(s_{0})\,,$
	$\displaystyle a_{2}(r,s)$	$\displaystyle=-\cos\theta(s_{0})+r\kappa_{g}(x_{0})\cos\theta(s_{0})+(\zeta r+\check{\kappa}(s-s_{0}))\sin\theta(s_{0})\,.$

Here $\theta(s_{0})$ is the angle introduced in (2.6) with $x=x_{0}$ .

This lemma is an extension of Lemma 9.1 in [HelMo4] to the case when the magnetic field is not necessarily constant. In the constant magnetic field case we have $\zeta=0$ and $\check{\kappa}=\kappa_{g}(x_{0})$ , where $\kappa_{g}$ is the geodesic curvature introduced in (2.11). Note that we do not try at the moment to explicitly compute $\check{\kappa}$ and $\zeta$ in the general case. We plan indeed to show that the result on the lowest eigenvalue is independent of $\check{\kappa}$ and $\zeta$ .

Proof of Lemma 7.2.

Our goal is to determine the Taylor expansion up to order $1$ of the magnetic field vector and corresponding magnetic field $2$ -form in the variables $(r,s,t)$ , the Taylor expansion being computed at $t=r=0$ and $s=s_{0}$ . Up to a translation, we assume that $s_{0}=0$ .
Writing the magnetic vector field in (1.2) as

\mathbf{B}=\widetilde{b}_{1}\partial_{r}+\widetilde{b}_{2}\partial_{s}+\widetilde{b}_{3}\partial_{t}\;,

(7.13)

the Taylor expansion of order $1$ at $(0,0,0)$ takes the form

\begin{array}[]{ll}\widetilde{b}_{1}(r,s,t)&=\cos\theta+\gamma_{1}r+\delta_{1}s+\sigma_{1}t+\mathcal{O}(r^{2}+s^{2}+t^{2})\;,\\ \widetilde{b}_{2}(r,s,t)&=\sin\theta+\gamma_{2}r+\delta_{2}s+\sigma_{2}t+\mathcal{O}(r^{2}+s^{2}+t^{2})\;,\\ \widetilde{b}_{3}(r,s,t)&=\gamma_{3}r+\sigma_{3}t+\mathcal{O}(r^{2}+s^{2}+t^{2})\;.\end{array}

(7.14)

where $\theta=\theta(s_{0})$ and where we used (2.4)-(2.5). Here we have used that by definition of the coordinate $r$ , the function $(r,s)\mapsto\widetilde{b}_{3}(r,s,0)$ vanishes exactly at order $1$ on $r=0$ . Note that $\gamma_{3}$ is $\kappa_{n,\mathbf{B}}(x_{0})$ , introduced in (1.5).

We now express that on $t=0$ the norm of $\mathbf{B}$ should be one. In fact

|\mathbf{B}|^{2}=\sum\limits_{1\leq i,j\leq 1}g_{ij}\tilde{b}_{i}\tilde{b}_{j}+\tilde{b}_{3}^{2}

(7.15)

where the coefficients $g_{ij}$ can be computed by (2.8), (2.9) and (2.12).
For $t=0$ , this reads

(\widetilde{b}_{1}(r,s,0)^{2}+\alpha(r,s)(\widetilde{b}_{2}(r,s,0))^{2}+(\widetilde{b}_{3}(r,s,0))^{2}=1\;,

(7.16)

where $\alpha(r,s)$ is introduced in (2.12) and satisfies (2.13). We expand the last formula around $t=r=s=0$ . This leads, by taking $t=0$ and considering the coefficients of $r$ and $s$ , to the two identities

\gamma_{1}\cos\theta+\gamma_{2}\sin\theta-\kappa_{g}(x_{0})\sin^{2}\theta=0\;,

and

\delta_{1}\cos\theta+\delta_{2}\sin\theta=0\;.

So it is natural to introduce the new parameters $\hat{\kappa}$ and $\zeta$ as follows

\check{\kappa}=-\delta_{1}\sin\theta+\delta_{2}\cos\theta,\quad\zeta=-\gamma_{1}\sin\theta+(\gamma_{2}-\kappa_{g}(x_{0})\sin\theta)\cos\theta\,.

(7.17)

So we observe that

\delta_{1}=-\check{\kappa}\sin\theta\;,\delta_{2}=\check{\kappa}\cos\theta\,,

and

\gamma_{1}=-\zeta\sin\theta\;,\;\gamma_{2}=\zeta\cos\theta+\kappa_{g}(x_{0})\,\sin\theta\;.

Hence our “normal” form becomes

\widetilde{b}_{j}(r,s,t)=\widetilde{b}_{j}^{0}(r,s,t)+{\cal O}(r^{2}+s^{2}+t^{2})

with

\begin{array}[]{ll}\widetilde{b}_{1}^{0}(r,s,t)&=\cos\theta-(\zeta\,r+\check{\kappa}s)\sin\theta+\sigma_{1}t\;,\\ \widetilde{b}_{2}(r,s,t)&=\sin\theta+(\zeta\,r+\check{\kappa}\,s)\cos\theta+\kappa_{g}(x_{0})r\sin\theta+\sigma_{2}t\;,\\ \widetilde{b}_{3}(r,s,t)&\equiv\gamma_{3}\,r+\sigma_{3}t\;,\end{array}

(7.18)

with

\gamma_{3}={\kappa_{n,{\bf B}}(x_{0})}=\partial_{r}\langle\mathbf{B}\,|\,N\rangle\;.

(7.19)

Now consider $\widetilde{\mathbf{B}}={\rm curl}\,_{(r,s,t)}\tilde{\bf A}$ . We have $\widetilde{\mathbf{B}}=|g|^{1/2}(\tilde{b}_{1},\tilde{b}_{2},\tilde{b}_{3})$ (see [HelMo4, Eq. (5.13)]), where $g$ is introduced in (2.9). So we obtain by (2.15),

\widetilde{\mathbf{B}}_{ij}(r,s,t)=\widetilde{\mathbf{B}}_{ij}^{0}(r,s,t)+{\cal O}(r^{2}+s^{2}+t^{2})

with

\begin{array}[]{ll}\widetilde{\mathbf{B}}_{23}^{0}(r,s,t)&=(1-\kappa_{g}(x_{0})r)\cos\theta-(\zeta\,r+\check{\kappa}s)\sin\theta+\sigma_{1}t\;,\\ \widetilde{\mathbf{B}}_{31}^{0}(r,s,t)&=\sin\theta+(\zeta\,r+\check{\kappa}\,s)\cos\theta+\sigma_{2}t\;,\\ \widetilde{\mathbf{B}}_{12}^{0}(r,s,t)&\equiv\gamma_{3}\,r+\sigma_{3}t\;.\end{array}

(7.20)

Notice that the condition ${\rm div}_{(r,s,t)}\widetilde{\mathbf{B}}=0$ reads (at $r=t=0$ and $s=0$ ) as follows

\sigma_{3}=\big{(}\kappa_{g}(x_{0})-\check{\kappa}\big{)}\cos\theta+\zeta\sin\theta\,.

We have now to choose a suitable corresponding magnetic potential to $\widetilde{\mathbf{B}}^{0}$ . We find

\tilde{{\bf A}}^{00}(r,s,t)=\left(\begin{array}[]{c}\tilde{A}_{1}^{00}\\ \tilde{A}_{2}^{00}\\ \tilde{A}_{3}^{00}\end{array}\right)=\left(\begin{array}[]{c}ta_{1}(r,s)+\frac{\sigma_{2}}{2}t^{2}\vskip 6.0pt plus 2.0pt minus 2.0pt\\ ta_{2}(r,s)+\frac{1}{2}\gamma_{3}r^{2}-\frac{\sigma_{1}}{2}t^{2}\vskip 6.0pt plus 2.0pt minus 2.0pt\\ 0\end{array}\right)={\bf A}^{00}(r,s,t)+\mathcal{O}(t^{2})\,,

(7.21)

with

a_{1}(r,s)=\sin\theta+(\zeta r+\check{\kappa}s)\cos\theta\;,

(7.22)

a_{2}(r,s)=-(1-\kappa_{g}(x_{0})r)\cos\theta+(\zeta r+\check{\kappa}s)\sin\theta\;.

(7.23)

Moreover ${\rm curl\,}\tilde{\bf A}^{(2)}=\widetilde{\mathbf{B}}^{0}$ in the simply connected domain $\tilde{Q}_{h}(x_{0},R_{0},\delta,\epsilon_{2})$ , so we can find a function $\check{p}$ such that $\tilde{\bf A}^{(2)}=\tilde{\bf A}^{00}-\nabla\check{p}$ .

Finally, $\gamma_{j}(s):=\frac{\partial\tilde{b}_{j}}{\partial r}(0,s,0)$ and $\delta_{j}(s):=\frac{\partial\tilde{b}_{j}}{\partial s}(0,s,0)$ are bounded functions. Setting $M_{j}=\sup\big{(}|\gamma_{j}(s)|+|\delta_{j}(s)|\big{)}$ and $M=\max(M_{1},M_{2})$ , we get from (7.17) that

|\check{\kappa}|\leq 2M\mbox{ and }|\zeta|\leq 2M+\|\kappa_{g}\|_{\infty}\,.

7.3 A second comparison estimate

We use the magnetic potential in Lemma 7.2 to approximate the quadratic form, as we did in Lemma 7.1. In particular, we approximate the metric by a flat one. Let us introduce the quadratic form corresponding to the magnetic potential in Lemma 7.1 (see [HelMo4, Lem. 10.2]):

q_{{\bf A}^{00}}^{h}(v)=\int_{\tilde{Q}_{h}(x_{0},R_{0})}\Big{(}|hD_{t}v|^{2}+(1+2r\kappa_{g}(x_{0}))|(hD_{s}-A^{00}_{2})v|^{2}\\ +|(hD_{r}-A_{1}^{00})v|^{2}\Big{)}drdsdt,

(7.24)

where $v\in H^{1}(\tilde{Q}_{h}(x_{0},R_{0}))$ and $\tilde{Q}_{h}(x_{0},R_{0})=\tilde{Q}_{h}(x_{0},R_{0},\delta,\epsilon_{2})$ is the set introduced in (7.5).

We can obtain a further approximation of the quadratic form for functions obeying the conditions in (7.7).

Lemma 7.3 (Helffer-Morame).

There exist positive constants $C,h_{0},\varsigma_{0}$ such that, for all $h\in(0,h_{0}]$ and $u\in H^{1}(\Omega)$ s.t. ${\rm supp\,}\,u\subset Q_{h}(x_{0},R_{0},\delta,\epsilon_{2})$ , we have

q_{{\bf A}^{00}}^{h}(\tilde{u})-CM_{h}(u)h^{\frac{4}{3}+\varsigma_{0}}\leq q_{\bf A}^{h}(u)\leq q_{{\bf A}^{00}}^{h}(\tilde{u})+CM_{h}(u)h^{\frac{4}{3}+\varsigma_{0}}\,,

where $M_{h}(u)$ is introduced in (7.6) and

\tilde{u}=(1-r\kappa_{g}(x_{0}))^{1/2}u\,e^{-i\check{p}/h}.

Proof.

We have the following two estimates from [HelMo4, Lem. 10.2] (whose proof does not require that the magnetic field ${\rm curl}\,{\bf A}$ is constant)

q_{\bf A}^{h}(u)\geq q_{{\bf A}^{00}}^{h}(\tilde{u})-C\|t^{1/2}(hD_{x}-{\bf A})u\|^{2}\\ -C\big{(}q_{{\bf A}^{00}}^{h}(\tilde{u})\big{)}^{1/2}\|(h^{3\delta}+h+h^{2\delta}t+t^{2})u\|\\ -C\|(h^{3\delta}+h+h^{2\delta}t+t^{2})u\|^{2}\,,

and

q_{\bf A}^{h}(u)\leq q_{{\bf A}^{00}}^{h}(\tilde{u})+C\|t^{1/2}(hD_{x}-{\bf A})u\|^{2}\\ +C\big{(}q_{{\bf A}^{00}}^{h}(\tilde{u})\big{)}^{1/2}\|(h^{3\delta}+h+h^{2\delta}t+t^{2})u\|\\ +C\|(h^{3\delta}+h+h^{2\delta}t+t^{2})u\|^{2}\,.

We can then estimate the remainder terms, using (7.7), as we did in the proof of Lemma 7.1. The only term that was not present satisfies

\|t^{2}u\|^{2}\leq M_{h}(u)h^{2}\,,

where we used (7.7) (c) with $n=4$ .

7.4 An estimate away from the curve $\Gamma$

Let us now look at the quadratic form, $q_{\bf A}^{h}(u)$ , when $u$ is supported away from $\Gamma$ . We start with a rough lower bound.

Lemma 7.4.

Given $c>0$ , $\epsilon_{2}\in(0,1)$ and $\rho\in(0,\frac{1}{4})$ , there exist positive constants $h_{0},\tilde{c}$ such that, if $u\in H^{1}(\Omega)$ satisfies

{\rm supp\,}\,u\subset\{x\in\Omega~{}:~{}{\rm dist}(x,\partial\Omega)<\epsilon_{2}\,,~{}d_{\Gamma}(x)\geq c\,h^{\rho}\},

where $d_{\Gamma}(x)={\rm dist}_{\partial\Omega}(p(x),\Gamma)$ is introduced in (6.5), then

q_{\bf A}^{h}(u)\geq(\Theta_{0}+\tilde{c}\,h^{\rho})h\int_{\Omega}|u|^{2}dx\,.

Proof.

If we verify that, for a given constant $c>0$ ,

d_{\Gamma}(x)\geq ch^{\rho}\implies\exists\,c^{\prime}>0,~{}|\nu(x)|\geq c^{\prime}h^{\rho}\,,

(7.25)

then the proof follows from Proposition 6.2, by using that $h^{5/4}=o(h^{1+\rho})$ and the lower bound from Proposition 6.1,

\sigma(\nu)\geq\Theta_{0}+\frac{\sqrt{\delta_{0}}}{2}|\nu|\,,

in a neighborhood of $0$ .

Let us denote by $m_{*}=\min_{x\in\Gamma}\kappa_{n,\mathbf{B}}(x)$ , then $m_{*}>0$ by Assumption 1.1, and (7.25) holds with $c^{\prime}=m_{*}c/2$ . In fact, if $|\nu(x)|\leq c^{\prime}h^{\rho}$ , we get by (6.2)

|\mathbf{B}\cdot\mathbf{N}(p(x))|\leq c^{\prime}h^{\rho}\,,

and it follows from (2.5) that (recall that $d_{\Gamma}(x)=|r|$ , see Sec. 2)

m_{*}d_{\Gamma}(x)\leq c^{\prime}h^{\rho}=m_{*}\frac{c}{2}h^{\rho}\,.

The next proposition is an improvement of Proposition 7.4 since it allows for the support of $u$ to be closer to the curve $\Gamma$ .

Proposition 7.5.

Given $c>0$ , $\epsilon_{2}\in(0,1)$ and $\delta\in[\frac{1}{4},\frac{1}{3})$ , there exist positive constants $h_{0},c_{*},C,\varsigma_{0}$ such that, if $u\in H^{1}(\Omega)$ satisfies

{\rm supp\,}\,u\subset\{x\in\Omega~{}:~{}{\rm dist}(x,\partial\Omega)<\epsilon_{2},~{}d_{\Gamma}(x)\geq c\,h^{\delta}\},

(7.26)

where $d_{\Gamma}(x)={\rm dist}_{\partial\Omega}(p(x),\Gamma)$ is introduced in (6.5), then

q_{\bf A}^{h}(u)\geq(\Theta_{0}+c_{*}h^{\delta})h\int_{\Omega}|u|^{2}dx-CM_{h}(u)h^{\frac{4}{3}+\varsigma_{0}}\,,

where $M_{h}(u)$ is introduced in (7.6).

Proof.

Step 1. Let us fix constants $c,R_{0}>0$ , $\epsilon_{2}\in(0,1)$ , $\delta\in[\frac{1}{4},\frac{1}{3})$ and $\rho\in(0,\frac{1}{4})$ . We assume that ${\rm supp\,}\,u\subset Q_{h}(x_{0}^{*},R_{0},\delta,\epsilon_{2})$ where $x_{0}^{*}\in\partial\Omega$ with boundary coordinates $(r_{0},s_{0},t_{0}=0)$ satisfies (for $h$ small enough) $c\,h^{\delta}\leq|r_{0}|=d_{\Gamma}(x_{0}^{*})\leq 2c\,h^{\rho}$ and $Q_{h}(x_{0}^{*},R_{0},\delta,\epsilon_{2})$ is introduced in (7.2).

We denote by $\tilde{Q}_{h}(x_{0}^{*})=\tilde{Q}_{h}(x_{0}^{*},R_{0},\delta,\epsilon_{2})$ the neighborhood associated with $Q_{h}(x_{0}^{*},R_{0},\delta,\epsilon_{2})$ by (7.5). By a translation, we may assume that $s_{0}=0$ .

Consider the magnetic potential $\tilde{\bf A}^{(2)}$ introduced in (7.4). We modify the coordinates $(r,s,t)$ so that, locally near $(r_{0},0,0)$ , the metric $G$ in (2.12) is diagonal²²2We consider the curve $\Gamma_{h}$ defined by $s\mapsto\Phi_{x_{0}}^{-1}(r_{0},s,0)$ , where $x_{0}=\gamma(x_{0}^{*})$ and $\Phi_{x_{0}}$ is the coordinate transformation introduced in (2.2). We parameterization $\Gamma_{h}$ by arc-length $s\mapsto\gamma_{h}(s)$ and define the adapted coordinates by considering the normal geodesic to $\Gamma_{h}$ passing through $x_{0}^{*}$ . with

\alpha(r_{0},s)=1\quad{\rm and}\quad\frac{\partial\alpha}{\partial r}(r_{0},s)=-2\kappa_{g}(\gamma(s))+\mathcal{O}(h^{\rho})\,.

(7.27)

By Taylor’s formula

\alpha(r,s)=1-2\kappa_{g}(\gamma(s))(r-r_{0})+\mathcal{O}(h^{\rho}(r-r_{0}))+\mathcal{O}((r-r_{0})^{2})\,.

In $\tilde{Q}_{h}(x_{0}^{*})$ , we write

|\kappa_{g}(\gamma(s))-\kappa_{g}(x_{0}^{*})|\leq Ch^{\delta},

\begin{aligned} \alpha(r,s)=1-2\kappa_{g}(x_{0}^{*})(r-r_{0})-Ch^{\delta+\rho}\end{aligned}\,,

and

hD_{y}-\tilde{\bf A}=(hD_{y}-\tilde{\bf A}^{(2)})-(\tilde{\bf A}-\tilde{\bf A}^{(2)})\,.

So we get, as in Lemma 7.1, the existence of $C^{\prime},\varsigma_{0}>0$ such that

q_{\bf A}^{h}(u)\geq(1-Ch^{\delta+\rho})q_{\tilde{\bf A}^{(2)}}^{h,x_{0}^{*}}(u)-C^{\prime}M_{h}(u)h^{\frac{4}{3}+\varsigma_{0}}\,,

where

q_{\tilde{\bf A}^{(2)}}^{h,x_{0}^{*}}(u)=\int_{\tilde{Q}_{h}(x_{0}^{*})}(1-(r-r_{0})\kappa_{g}(x_{0}^{*}))\Big{(}|(hD_{t}-\tilde{A}_{3}^{(2)})u|^{2}\\ +(1+2(r-r_{0})\kappa_{g}(x_{0}^{*}))|(hD_{s}-\tilde{A}_{2}^{(2)})u|^{2}+|(hD_{r}-\tilde{A}_{1}^{(2)})u|^{2}\Big{)}drdsdt\,.

Performing a change of variables

(r,s)\mapsto\big{(}(r-r_{0})\cos\omega-s\sin\omega,(r-r_{0})\sin\omega+s\cos\omega\big{)}

which amounts to a rotation in the $(r,s)$ -plane (centered at $(r_{0},0)$ ), we may assume that the second component of $\widetilde{\mathbf{B}}={\rm curl}_{(r,s,t)}\tilde{\bf A}=(\tilde{B}_{23},\tilde{B}_{31},\tilde{B}_{12})$ vanishes at $(r_{0},0,0)$ , by choosing $\omega$ so that

\tilde{B}_{31}(x_{0}^{*})\cos\omega+\tilde{B}_{23}(x_{0}^{*})\sin\omega=0\,.

At the same time, this rotation leaves $|\mathbf{B}|$ and the measure $drds$ invariant. Then performing a gauge transformation (see [HelMo4, Sec. 16.3]), we may assume that

\tilde{\bf A}^{(2)}(r,s,t)=\tilde{\bf A}^{(2,0)}(r,s,t)+\mathcal{O}(|r-r_{0}|t+|s|t+t^{2})

where

\tilde{\bf A}^{(2,0)}(r,s,t):=\left(\begin{array}[]{l}\tilde{c}_{1}^{0}\,s^{2}\\ \tilde{B}_{23}^{(0)}t+\tilde{B}_{12}^{(0)}(r-r_{0})+\tilde{c}_{2}^{0}\,(r-r_{0})^{2}\\ 0\end{array}\right)\,.

Here

\widetilde{\mathbf{B}}^{(0)}:=\widetilde{\mathbf{B}}(r_{0},0,0)=(\tilde{B}_{23}^{(0)},\tilde{B}_{31}^{(0)}=0,\tilde{B}_{12}^{(0)})

and $\tilde{c}_{1}^{0},\tilde{c}_{2}^{0}$ are constants.
Similarly to the proof of Lemma 7.1, by writing

hD_{y}-\tilde{\bf A}^{(2)}=hD_{y}-\tilde{\bf A}^{(2,0)}-(\tilde{\bf A}^{(2)}-\tilde{\bf A}^{(2,0)})

and

\|(hD_{y}-\tilde{\bf A}^{(2,0)})u\|\leq\|(hD_{y}-\tilde{\bf A})u\|+\|(\tilde{\bf A}-\tilde{\bf A}^{(2)})u\|+\|(\tilde{\bf A}^{(2)}-\tilde{\bf A}^{(2,0)})u\|\,,

we get

q_{\tilde{\bf A}^{(2)}}^{h,x_{0}^{*}}(u)\geq(1-Ch^{2\delta})q_{\tilde{\bf A}^{(2,0)}}^{h,x_{0}^{*}}(u)-C^{\prime\prime}M_{h}(u)h^{\frac{4}{3}+\varsigma_{0}}\,.

Thus we are left with finding a lower bound of $q_{\tilde{\bf A}^{(2,0)}}^{h,x_{0}^{*}}(u)$ .
Note that, since $|\mathbf{B}|=1$ and by (7.27), the metric satisfies $|g|=1$ on $x_{0}^{*}$ , we have by (7.15), $|\tilde{B}_{23}^{(0)}|^{2}+|\tilde{B}_{12}^{(0)}|^{2}=1$ .
Moreover, since $\mathbf{B}\cdot\mathbf{N}$ vanishes linearly on $\Gamma=\{r=0\}$ , there exist $C_{1}>0$ and $C_{2}>0$ such that

\frac{1}{C_{1}}|r_{0}|\leq|\tilde{B}_{12}^{(0)}|^{2}\leq C_{2}\,|r_{0}|\,,\quad|\,|\tilde{B}_{23}^{(0)}|-1\,|\leq C_{2}r_{0}^{2},\quad|\tilde{c}_{1}^{0}|+|\tilde{c}_{2}^{0}|\leq C_{2}\,.

The previous estimates yield a lower bound of $q_{\tilde{\bf A}^{(2,0)}}^{h}(u)$ by comparing with a model operator (after rescaling the variables $\tilde{r}=h^{1/3}(r-r_{0})$ , $\tilde{s}=h^{1/3}s$ and $\tilde{t}=h^{1/2}t$ ). In fact, by [HelMo4, Lemma 16.1], there exists $c_{1}>0$ such that,

q_{\tilde{\bf A}^{(2,0)}}^{h,x_{0}^{*}}(u)\geq(\Theta_{0}+c_{1}|r_{0}|)h\int_{\Omega}|u|^{2}dx\,.

Note that, we can use Lemma 16.1 of [HelMo4] under our assumptions on the support of $u$ .
Step 2. We can reduce to the setting of Step 1 and Lemma 7.4 by means of a partition of unity. In fact, consider an $h$ -dependent partition of unity $\chi_{1}^{2}+\chi_{2}^{2}=1$ on $\{{\rm dist}(x,\partial\Omega)<\epsilon_{2}\}$ such that

{\rm supp\,}\chi_{1}\subset\{d_{\Gamma}(x)\geq\frac{c}{2}h^{\rho}\},\quad{\rm supp\,}\chi_{2}\subset\{d_{\Gamma}(x)\leq ch^{\rho}\},\quad\sum_{i=1}^{2}|\nabla\chi_{i}|^{2}=\mathcal{O}(h^{-2\rho})\,.

If $u\in H^{1}(\Omega)$ satisfies (7.26), then

q_{\bf A}^{h}(u)=\sum_{i=1}^{2}\Big{(}q_{\bf A}^{h}(\chi_{i}u)-h^{2}\|\,|\nabla\chi_{i}|u\,\|^{2}\Big{)}\,,

where

	$\displaystyle q_{\bf A}^{h}(\chi_{1}u)\geq(\Theta_{0}+\tilde{c}\,h^{\rho})h\int_{\Omega}\|\chi_{1}u\|^{2}dx\quad{\rm by~{}Proposition~{}\ref{lem:10.1*f}}\,,$
	$\displaystyle q_{\bf A}^{h}(\chi_{2}u)\geq(1-Ch^{\delta+\rho})(\Theta_{0}+c_{1}h^{\delta})h\int_{\Omega}\|\chi_{2}u\|^{2}dx-M_{h}(u)h^{\frac{4}{3}+\varsigma_{0}}\quad{\rm by~{}Step~{}1}\,,$
	$\displaystyle\sum_{i=1}^{2}h^{2}\\|\,\|\nabla\chi_{i}\|u\,\\|^{2}=\mathcal{O}(h^{2-2\rho})=o(h^{1+\delta})\,,$

where in the last step we used that $0<\rho<\frac{1}{4}$ and $\frac{1}{4}<\delta<\frac{1}{3}$ .

8 Lower bound

8.1 Another model

The model in (5.1) corresponds to the quadratic form in (7.24) when $\kappa_{g}(x_{0})=0$ . However, when $\kappa_{g}(x_{0})\not=0$ , the situation is similar to [HelMo4, Sec. 15]. The model compatible with (7.24) can still be reduced to the one in (5.1) with appropriate choices of the parameters $\eta,\zeta,\gamma$ (see (8.22)).

8.1.1 A new model quadratic form

Let us fix a boundary point $x_{0}\in\Gamma$ and denote the model quadratic form near $x_{0}$ by

u\mapsto q_{m}(u):=q_{{\bf A}^{00}}(u)

(8.1)

where $q_{{\bf A}^{00}}$ is given in (7.24), $u\in H^{1}(\tilde{Q}_{h}(x_{0},R_{0}))$ and $\tilde{Q}_{h}(x_{0},R_{0})=\tilde{Q}_{h}(x_{0},R_{0},\delta,\epsilon_{2})$ is the set introduced in (7.5). Furthermore, we assume that that the metric is flat at $x_{0}$ and the coordinates of $x_{0}$ in the $(r,s,t)$ frame are $(0,s_{0}=0,0)$ , after performing a translation with respect to the $s$ variable.

Following the proof of [HelMo4, Lem. 15.1], we are led to the analysis of the model quadratic form (see Lemma 8.1)

q^{h}_{m,0}(u)=\int_{\tilde{Q}_{h}(x_{0},R_{0})}\Big{(}h^{2}|D_{t}u|^{2}+|tu-L^{h}_{1}u|^{2}+|L^{h}_{2}u|^{2}\Big{)}\;drdsdt\;,

(8.2)

where

	$\displaystyle L^{h}_{1}$	$\displaystyle=a_{1}\,hD_{r}+a^{0}_{2}\,hD_{s}-\frac{1}{2}\,\cos\theta\,\kappa_{n,\bf B}(x_{0})\,r^{2}\;,$		(8.3)
	$\displaystyle L^{h}_{2}$	$\displaystyle=a^{1}_{2}\,hD_{r}+a^{1}_{1}\,hD_{s}+\frac{1}{2}\,\sin\theta\,\kappa_{n,\bf B}(x_{0})\,r^{2}\;,$		(8.3)

and, with $\theta=\theta(s_{0})$ the angle defined by (2.6), we introduce the following functions

\begin{array}[]{ll}a_{1}(r,s)&=\sin\theta+\cos\theta(\zeta r+\check{\kappa}s)\;,\\ a_{2}(r,s)&=-\cos\theta+\kappa_{g}(x_{0})\cos\theta r+\sin\theta(\zeta r+\check{\kappa}s)\;,\\ a_{2}^{0}(r,s)&=-\cos\theta-\kappa_{g}(x_{0})\cos\theta r+\sin\theta(\zeta r+\check{\kappa}s)\;,\\ a_{2}^{1}(r,s)&=\cos\theta-\sin\theta(\zeta r+\check{\kappa}s)\;,\\ a_{1}^{1}(r,s)&=\sin\theta+\sin\theta\kappa_{g}(x_{0})r+\cos\theta(\zeta r+\check{\kappa}s)\;,\\ \alpha(r)&=1+2\kappa_{g}(x_{0})r\;.\end{array}

(8.4)

We will consider the form $q_{m,0}$ on the following class of functions

\mathcal{D}_{0}=\{u\in H^{1}(\Omega^{h})~{}:~{}u|_{(\partial\mathcal{Q}^{h})\times\,]0,h^{\delta}[}=0,~{}u|_{\mathcal{Q}^{h}\times\{h^{\delta}\}}=0\}

(8.5)

where

\Omega_{h}=\mathcal{Q}^{h}\times\,]0,h^{\delta}[\,,\quad\mathcal{Q}^{h}=]-R_{0}h^{\delta},R_{0}h^{\delta}[^{2}\,.

The precise relation between the model quadratic forms in (8.1) and (8.2) is given in the following lemma.

Lemma 8.1.

For any $\delta\in(\frac{5}{18},\frac{1}{3})$ and $\tau_{1}>0$ , there exists $C>0$ such that, for any $u\in\mathcal{D}_{0}$ and $h\in(0,1)$ ,

(1+Ch^{2\delta})q^{h}_{m}(u)\geq(1-Ch^{\tau_{1}})q^{h}_{m,0}(u)-C\big{(}\|(h^{2\delta}+h^{\tau_{1}})tu\|^{2}+h^{6\delta-\tau_{1}}\|u\|^{2}\big{)}\,.

Proof.

The proof follows that of Lemma 15.1 in [HelMo4] with some adjustments in the formulas (15.9), (15.16) and (15.17) in [HelMo4].
We have indeed

|1-(a_{1})^{2}-\alpha(a_{2})^{2}|\leq Ch^{2\delta}\;,

(8.6)

where we used that $\alpha(r)^{1/2}=1+\kappa_{g}(x_{0})r+\mathcal{O}(h^{2\delta})$ on the support of $u$ , which follows by (8.4).

We also observe that :

|\alpha a_{2}-a^{0}_{2}|+|\alpha^{1/2}a_{2}+a^{1}_{2}|+|\alpha^{1/2}a_{1}-a^{1}_{1}|\leq C(r^{2}+s^{2})

(8.7)

and

|\alpha^{1/2}a_{1}-\sin\theta|+|\alpha a_{2}+\cos\theta|\leq C(r^{2}+s^{2})^{1/2}\;.

(8.8)

Later on, we will choose $\delta$ and $\tau_{1}$ in a convenient way (see Remark 8.3).

8.1.2 Linearizing change of variable

In order to reduce to the case $\kappa_{g}=0$ and eliminate the slightly variable coefficients of $D_{r}$ and $D_{s}$ in (7.24), we argue as [HelMo4a, Sec. 15.2] by performing a change of variables. The argument does not work in our case in the same way as [HelMo4a, Sec. 15.2], but it leads to the fact that for our lower bound the only relevant parameters are $\eta:=\check{\kappa}-\kappa_{g}$ and $\zeta$ (see (7.24)).

The below computations are essentially the same as in [HelMo4, Sec. 15.2] but we have to do them carefully in order to capture the correct $\eta$ and $\zeta$ appearing in (5.1).

Let us follow, what this change of variable was doing. We introduce

\kappa:=\kappa_{g}(x_{0}).

(8.9)

Let us make the change of variables $(r,s)=\Phi_{\kappa}(p,q)$ with

\begin{array}[]{l}r=\sin\theta\,p+\cos\theta\,q-\frac{\kappa}{2}[-\cos\theta\,p+\sin\theta\,q]^{2}\;,\\ s=-\cos\theta\,p+\sin\theta\,q-\frac{\kappa}{2}[\sin(2\theta)\,(p^{2}-q^{2})+2\cos(2\theta)\,pq]\;,\end{array}

(8.10)

where $\theta=\theta(s_{0})$ is the angle defined by (2.6).

The map $\Phi_{\kappa}$ is a perturbation of a rotation and, by the local inversion theorem, it is easily seen as a local diffeomorphism sending a fixed neighborhood of $(0,0)$ onto another neighborhood of $(0,0)$ .

Then, for $h$ small enough, $\mathcal{Q}^{h}:=]-R_{0}h^{\delta},R_{0}h^{\delta}[^{2}$ is transformed by $\Phi_{\kappa}^{-1}$ to the set $\mathcal{Q}^{h}_{0}$ satisfying :

\mathcal{Q}^{h}_{0}=\Phi_{\kappa}^{-1}(\mathcal{Q}^{h})\;\subset\,\;]-R_{0}^{\prime}h^{\delta}\;,\;R_{0}^{\prime}h^{\delta}[\;\times\;]-R_{0}^{\prime}h^{\delta}\;,\;R_{0}^{\prime}h^{\delta}[\;.

(8.11)

Let us write

D_{p}=c_{11}D_{r}+c_{12}D_{s},\ \ \ D_{q}=c_{21}D_{r}+c_{22}D_{s}\;,

(8.12)

We can express the functions $c_{ij}$ in terms of the $(p,q)$ variables, by using (8.10). In fact, we introduce $c_{ij}(r,s)=\check{c}_{ij}(p,q)$ , and observe that

	$\displaystyle\check{c}_{11}(p,q)$	$\displaystyle=\frac{\partial r}{\partial p}=\sin\theta+\kappa\cos\theta\,(-\cos\theta\,p+\sin\theta\,q)\;;$
	$\displaystyle\check{c}_{12}(p,q)$	$\displaystyle=\frac{\partial s}{\partial p}=-\cos\theta-\kappa\,(\sin(2\theta)\,p+\cos(2\theta)\,q)\;;$
	$\displaystyle\check{c}_{21}(p,q)$	$\displaystyle=\frac{\partial r}{\partial q}=\cos\theta-\kappa\sin\theta\,(-\cos\theta\,p+\sin\theta\,q)\;;$
	$\displaystyle\check{c}_{22}(p,q)$	$\displaystyle=\frac{\partial s}{\partial q}=\sin\theta-\kappa\,(-\sin(2\theta)\,q+\cos(2\theta)\,p)\;.$

Then we return back to the $(r,s)$ variables, by using (8.10). Noticing that, as $(p,q)\to(0,0)$ ,

r=\sin\theta\,p+\cos\theta\,q+{\cal O}(p^{2}+q^{2})\,,\quad s=-\cos\theta\,p+\sin\theta\,q+\mathcal{O}(p^{2}+q^{2})\,,

(8.13)

we get

$\displaystyle c_{11}(r,s)$	$\displaystyle=\sin\theta+\kappa\cos\theta\,s+{\cal O}(r^{2}+s^{2})\;;$	(8.14)
$\displaystyle c_{12}(r,s)$	$\displaystyle=-\cos\theta-\kappa(\cos\theta\,r-\sin\theta\,s)+{\cal O}(r^{2}+s^{2})\;;$
$\displaystyle c_{21}(r,s)$	$\displaystyle=\cos\theta-\kappa\sin\theta\,s+{\cal O}(r^{2}+s^{2})\;;$
$\displaystyle c_{22}(r,s)$	$\displaystyle=\sin\theta+\kappa(\sin\theta\,r+\cos\theta\,s)+{\cal O}(r^{2}+s^{2})\;.$

Let us now control the measure in the change of variable. By an easy computation, we get :

dr\,ds=\check{\alpha}_{1}dpdq\,,\quad\check{\alpha}_{1}(p,q)=1+\kappa(\sin\theta\,p+\cos\theta\,q)+\mathcal{O}(p^{2}+q^{2})\,.

By using (8.13), $\alpha_{1}(r,s)=\check{\alpha}_{1}(p,q)$ satisfies

|\alpha_{1}-1-\kappa r|\leq C(r^{2}+s^{2})\;,

(8.15)

where $r=r(p,q)$ is defined in (8.10).

Similarly to Lemma 8.1 we get also that one can go from the control of $q_{m,0}^{h}(u)$ to the control of the new quadratic form³³3We express $L_{1}^{h}$ and $L_{2}^{h}$ (see (8.3)) in terms of the $(p,q)$ variables introduced in (8.10) and neglect the terms of order $\mathcal{O}(r^{2}+s^{2})=\mathcal{O}(p^{2}+q^{2})$ .

q^{h}_{m,1}(u)=\int_{\Omega^{h}_{0}}\Big{(}h^{2}|D_{t}u|^{2}+|tu-M^{h}_{1}u|^{2}+|M^{h}_{2}u|^{2}\Big{)}\check{\alpha}_{1}\,dpdqdt\;,

(8.16)

with

\Omega^{h}_{0}:=\mathcal{Q}^{h}_{0}\times]0,h^{\delta}[\;,

and

	$\displaystyle M^{h}_{1}$	$\displaystyle=hD_{p}+h\left((\check{\kappa}-\kappa)s+\zeta r\right)D_{q}-\frac{1}{2}\,\cos\theta\,\kappa_{n,\bf B}(x_{0})(\sin\theta\,p+\cos\theta\,q)^{2}\;,$		(8.17)
	$\displaystyle M^{h}_{2}$	$\displaystyle=hD_{q}-h\left((\check{\kappa}-\kappa)s+\zeta r\right)D_{p}+\frac{1}{2}\,\sin\theta\kappa_{n,\bf B}(x_{0})(\sin\theta\,p\;+\;\cos\theta q)^{2}\;,$		(8.17)

where $(r,s)=(\sin\theta p+\cos\theta q,-\cos\theta p+\sin\theta q)$ .

More precisely, we have the following comparison lemma (see Lemma 8.1 and [HelMo4, Lem. 15.4]).

Lemma 8.2.

For any $\tau_{1}>0$ , there exists $C>0$ such that, for any $u\in\mathcal{D}_{0}$ ,

(1+Ch^{2\delta})q^{h}_{m,0}(u)\geq(1-Ch^{\tau_{1}})q^{h}_{m,1}(\tilde{u})-C\big{(}\|(h^{2\delta}+h^{\tau_{1}})tu\|^{2}+h^{6\delta-\tau_{1}}\|u\|^{2}\big{)}\,,

where $\tilde{u}=u\circ\Phi_{\kappa}^{-1}$ is associated with $u$ by the transformation $\Phi_{\kappa}$ .

By a unitary transformation, and after control of a commutator, we can reduce to a flat measure ( $dpdq$ instead of $\check{\alpha}_{1}dpdq$ ) and obtain the new quadratic form defined as follows

q^{h}_{m,2}(v)=\int_{\Omega^{h}_{0}}[h^{2}|D_{t}v|^{2}+|tv-M^{h}_{1}v|^{2}+|M^{h}_{2}v|^{2}]\;dpdqdt\;,

(8.18)

with $v$ associated to $u$ by $v=\check{\alpha}_{1}^{1/2}\tilde{u}$ . In fact, we have [HelMo4, Eq. (15.29)]

(1+Ch^{1/2})q_{m,1}(\tilde{u})+Ch^{3/2}\|u\|^{2}\geq q^{h}_{m,2}(v)\,.

(8.19)

Let us consider the new model associated with the quadratic form in (8.18). We first observe that the result depends only on $\check{\kappa}-\kappa$ and on $\zeta$ . The proof is moreover uniform with respect to these parameters. As a consequence, if $\Phi=\Phi_{\kappa}$ was the transformation introduced in (8.9), the inverse (for $\kappa=0$ ) $\Phi_{0}^{-1}$ , more explicitly the transformation $(p,q)\mapsto(\tilde{r}=\sin\theta p+\cos\theta q\;,\;\tilde{s}=-\cos\theta p+\sin\theta q)$ will bring us (in the new variables $(\tilde{r},\tilde{s},t)$ ) to the initial model with $\kappa_{g}$ replaced by $0$ , and $\check{\kappa}$ replaced by $\check{\kappa}-\kappa_{g}(x_{0})$ . This can also be done by explicit computations.

Doing the transformations backwards, we are led to a magnetic Laplacian computed with a trivial metric $\kappa_{g}=0$ but with a new magnetic potential

a_{1}(r,s)^{\rm new}=\sin\theta+\cos\theta(\zeta r+(\check{\kappa}-\kappa_{g}(x_{0}))s)\;,

(8.20)

and

a_{2}(r,s)^{\rm new}=-\cos\theta+\sin\theta(\zeta r+(\check{\kappa}-\kappa_{g}(x_{0}))s)\;.

(8.21)

So the new model is not as simple as in the uniform magnetic field case (where $\check{\kappa}=\kappa_{g}$ ) but it is the model in (5.1), which we have studied in the previous section with

\eta=\check{\kappa}-\kappa_{g}(x_{0}),\quad\gamma=\kappa_{n,\mathbf{B}}(x_{0})\;.

(8.22)

In fact, since $v$ is supported in $\Omega_{0}^{h}$ , we have,

q_{m,2}^{h}(v)=\langle P_{0;\gamma,\theta}^{h,\eta,\zeta}v,v\rangle,

(8.23)

where $P_{0;\gamma,\theta}^{h,\eta,\zeta}$ is the operator in (5.1).

Remark 8.3.

We will choose $\tau_{1}$ in such a manner that $\frac{1}{3}<\tau_{1}<6\delta-\frac{4}{3}$ . This choice is possible when $\delta$ satisfies $\frac{5}{18}<\delta<\frac{1}{3}$ .

8.1.3 Conclusion

We can now write a lower bound for the quadratic form $q^{h}_{{\bf A}^{00}}(u)$ in (7.24), assuming that $u\in H^{1}(\tilde{Q}_{h}(x_{0},R_{0}))$ and $\tilde{Q}_{h}(x_{0},R_{0})$ is the set introduced in (7.5). Let $\frac{5}{18}<\delta<\frac{1}{3}$ and $\frac{1}{3}<\tau_{1}<6\delta-\frac{4}{3}$ . Collecting Lemmas 8.1, 8.2, (8.19), (8.23) and Proposition 5.5, we get the existence of positive constants $C$ and $\varsigma_{0}$ , such that

q^{h}_{{\bf A}^{00}}(u)\geq\big{(}h\Theta_{0}+h^{\frac{4}{3}}c^{\rm conj}\big{(}\theta,\kappa_{n,\mathbf{B}}(x_{0})\big{)}-Ch^{\frac{4}{3}+\varsigma_{0}}\big{)}\|u\|^{2}\\ -Ch^{\frac{1}{3}+\varsigma_{0}}\|tu\|^{2}-C\|(h^{2\delta}+h^{\tau_{1}})tu\|^{2}

(8.24)

where $c^{\rm conj}(\gamma,\theta)$ is introduced in Proposition 5.2 with $\theta=\theta(s_{0})$ the angle in (2.6).

8.2 The general case

We return now to the proof of the asymptotics of the lowest eigenvalue, $\lambda_{1}^{N}({\bf A},h)$ , of the operator $P_{\bf A}^{h}$ in (1.3). Under Assumptions (C1)-(C2), we will prove the following lower bound:

\lambda_{1}^{N}({\bf A},h)\geq\Theta_{0}h+\widehat{\gamma}_{0,\mathbf{B}}h^{\frac{4}{3}}+{\mathcal{O}}(h^{\frac{4}{3}+\eta_{*}}),

(8.25)

for some constant $\eta_{*}>0$ , where $\widehat{\gamma}_{0,\mathbf{B}}$ is introduced in (1.6).

Let $u_{h}$ be a normalized ground state of $P_{\bf A}^{h}$ , i.e.

\lambda_{1}^{N}({\bf A},h)=q_{\bf A}^{h}(u_{h})=\|(h\nabla-i{\bf A})u_{h}\|^{2}\,.

Consider $\frac{5}{18}<\delta<\frac{1}{3}$ and the following neighborhood of the curve $\Gamma$ ,

\Gamma_{\delta}^{h}=\{x\in\Omega~{}:~{}{\rm dist}(x,\partial\Omega)<h^{\delta},~{}{\rm dist}_{\partial\Omega}(x,\Gamma)<h^{\delta/2}\}\,.

(8.26)

In terms of the $(r,s,t)$ coordinates introduced in Sec. 2.1,

\Gamma_{\delta}^{h}=\{0<t<h^{\delta},~{}h^{\delta/2}<r<h^{\delta/2}\}\,.

Let $\chi_{h}\in C_{c}^{\infty}(\Gamma_{\delta}^{h};[0,1])$ be a smooth function such that

\chi_{h}=1~{}{\rm on~{}}\Gamma_{\delta,0}^{h}=\{x\in\Omega~{}:~{}{\rm dist}(x,\partial\Omega)<\frac{1}{2}h^{\delta},~{}{\rm dist}_{\partial\Omega}(x,\Gamma)<\frac{1}{2}h^{\delta/2}\}

and

|\nabla\chi_{h}|=\mathcal{O}(h^{-\delta/2}).

We introduce the function

w_{h}=\chi_{h}u_{h}\,.

(8.27)

By Proposition 6.3, the eigenfunction $u_{h}$ is exponentially small outside $\Gamma_{\delta}^{h}$ , since by our choice of $\delta$ we have $h^{\delta/2}\gg h^{1/4}$ and $h^{\delta}\gg h^{1/2}$ . So we have

\lambda_{1}^{N}({\bf A},h)=q_{\bf A}^{h}(u_{h})=q_{\bf A}^{h}(w_{h})+\mathcal{O}(h^{\infty}),\quad\|u_{h}\|=\|w_{h}\|+\mathcal{O}(h^{\infty})\,.

(8.28)

Consider now a partition of unity of $\mathbb{R}^{3}$

\sum_{j\in\mathbb{Z}^{3}}|\chi_{j}|^{2}=1,\quad\sum_{j\in\mathbb{Z}^{3}}|\nabla\chi_{j}|^{2}<\infty,\quad{\rm supp\,}\chi_{j}\subset j+[-1,1]^{3}\,,

and introduce the following functions

w_{h,j}=\chi_{j,\delta}(x)w_{h}(x),\quad\chi_{j,\delta}(x)=\chi_{j}(h^{-\delta}x)\,.

(8.29)

We can decompose the quadratic form $q_{\bf A}^{h}(w_{h})$ as follows

q_{\bf A}^{h}(w_{h})=\sum_{j\in\mathcal{J}_{h}}q_{\bf A}^{h}(w_{h,j})+\mathcal{O}(h^{2-2\delta}),

(8.30)

where

\mathcal{J}_{h}=\{j\in\mathbb{Z}^{3}~{}:~{}{\rm supp\,}\chi_{j,\delta}\cap\Omega\not=\emptyset\}\,.

(8.31)

Let $C_{1}>0$ be a fixed constant that we will choose later to be sufficiently large. We will estimate the energy $q_{\bf A}^{h}(w_{h,j})$ when the support of $w_{h,j}$ is near the curve $\Gamma$ , or away from $\Gamma$ , independently. So we introduce the sets of indices

\begin{aligned} \mathcal{J}_{h}^{1}&=\{j\in\mathcal{J}_{h}~{}:~{}{\rm dist}\big{(}{\rm supp\,}\chi_{\gamma,\delta},\Gamma\big{)}\leq C_{1}h^{\delta}\}\\ \mathcal{J}_{h}^{2}&=\{j\in\mathcal{J}_{h}~{}:~{}{\rm dist}\big{(}{\rm supp\,}\chi_{\gamma,\delta},\Gamma\big{)}\geq C_{1}h^{\delta}\}\end{aligned}\,.

(8.32)

By Proposition 7.5,

\sum_{j\in\mathcal{J}_{h}^{2}}q_{\bf A}^{h}(w_{h,j})\geq\sum_{j\in\mathcal{J}_{h}^{2}}\Big{(}(\Theta_{0}h+c_{*}h^{1+\delta})\|w_{h,j}\|^{2}-Ch^{\frac{4}{3}+\varsigma_{0}}M_{h}(w_{j,h}\Big{)}\,,

(8.33)

where $M_{h}(w_{j,h})$ is introduced in (7.6). Notice that

M_{h}(w_{j,h})\leq\sum_{n=0}^{6}h^{-n/2}\int_{\Omega}t(x)^{n}\Big{(}|\chi_{j,h}u_{h}|^{2}+2h^{-1}|\chi_{j,h}(h\nabla-i{\bf A})u_{h}|^{2}\\ +2h|\nabla(\chi_{h}\chi_{j,h})|^{2}|u_{h}|^{2}\Big{)}dx\,.

Since $\sum|\chi_{j,h}|^{2}\leq 1$ and $\sum|\nabla(\chi_{j,h}\chi_{h})|^{2}=\mathcal{O}(h^{-2\delta})$ , Proposition 6.3 together with (6.6) and (6.7) yield

\sum_{j\in\mathcal{J}_{h}}M_{h}(w_{j,h})=\mathcal{O}(1)\,.

Consequently, we infer from (8.33),

\sum_{j\in\mathcal{J}_{h}^{2}}q_{\bf A}^{h}(w_{h,j})\geq(\Theta_{0}h+c_{*}h^{1+\delta})\Big{(}\sum_{j\in\mathcal{J}_{h}^{2}}\|w_{h,j}\|^{2}\Big{)}-C^{\prime}h^{\frac{4}{3}+\varsigma_{0}}\,.

(8.34)

For $j\in\mathcal{J}_{h}^{1}$ , we estimate $q_{\bf A}^{h}(w_{h,j})$ by collecting (8.24) and the estimates in Lemma 7.1 and 7.3. We start by picking $R_{0}>0$ and $x_{0}^{j}\in\Gamma$ , so that

{\rm supp\,}\,w_{h,j}\subset Q_{h}(x_{0}^{j})

where $Q_{h}(x_{0}^{j})$ is introduced in (7.2). Eventually, we find

\sum_{j\in\mathcal{J}_{h}^{1}}q_{\bf A}^{h}(w_{h,j})\geq\sum_{j\in\mathcal{J}_{h}^{1}}(\Theta_{0}h+h^{4/3}c^{\rm conj}(\theta_{j},\kappa_{n,\mathbf{B}}(x_{j}^{0})))\|w_{h,j}\|^{2}-Ch^{\frac{4}{3}+\varsigma_{*}}\,,

for some constant $\varsigma_{*}>0$ , where

\theta_{j}=\theta(s_{0}^{j})

and $(0,s_{0}^{j},0)$ denote the coordinates of $x_{0}^{j}$ in the $(r,s,t)$ -frame (see Sec. 2 and Eq. 2.3). Note that we used Proposition 6.3 to control the term $\sum_{j\in\mathcal{J}_{h}^{1}}\|tw_{h,j}\|^{2}$ appearing in (8.24); in fact $\sum_{j\in\mathcal{J}_{h}^{1}}\|tw_{h,j}\|^{2}=\mathcal{O}(h)$ .

Since $c^{\rm conj}(\theta_{j},\kappa_{n,\mathbf{B}}(x_{j}^{0}))$ is bounded from below by $\widehat{\gamma}_{0,\mathbf{B}}$ (see (1.6)), we get

\sum_{j\in\mathcal{J}_{h}^{1}}q_{\bf A}^{h}(w_{h,j})\geq(\Theta_{0}h+\widehat{\gamma}_{0,\mathbf{B}}h^{4/3})\sum_{j\in\mathcal{J}_{h}^{1}}\|w_{h,j}\|^{2}-Ch^{\frac{4}{3}+\varsigma_{*}}\,.

(8.35)

Inserting (8.34) and (8.35) into (8.30), and using (8.28), we deduce the lower bound in (8.25), since $\frac{5}{18}<\delta<\frac{1}{3}$ .

9 Upper bound

Fortunately, the same quasi-mode constructed in [HelMo4, Sec. 12] (see also [Pan6] for a different formulation) yields an upper bound of the lowest eigenvalue $\lambda_{1}({\bf A},h)$ matching with the asymptotics in Theorem 1.4. More precisely, under Assumptions (C1)-(C2), we will prove that:

\lambda_{1}^{N}({\bf A},h)\leq\Theta_{0}h+\widehat{\gamma}_{0,\mathbf{B}}h^{\frac{4}{3}}+{\mathcal{O}}(h^{\frac{4}{3}+\eta^{*}}),

(9.1)

for some constant $\eta^{*}>0$ , where $\widehat{\gamma}_{0,\mathbf{B}}$ is introduced in (1.6).

However, while computing the energy of the quasi-mode, we observe additional terms (not present in [HelMo4]) due to the non-homogeneity of the magnetic field. These terms are treated in Sec. 9.2.

9.1 The quasi-mode

The construction of the quasi-mode in [HelMo4] is quite lengthy and involves many auxiliary functions related to the de Gennes and Montgomery models (see (4.1) and (4.5)). We present here the definition of the quasi-mode along with a useful result from [HelMo4, Sec. 12].

9.1.1 Geometry and normal form

Select a point $x_{0}\in\partial\Omega$ such that the function in (1.7) satisfies

\widetilde{\gamma}_{0,\mathbf{B}}(x_{0})=\widehat{\gamma}_{0,\mathbf{B}}\,.

Let us assume that the coordinates of $x_{0}$ in the $(r,s,t)$ -frame are $(0,s_{0}=0,t_{0})$ . The normal form of the effective magnetic potential in Lemma 7.2 now becomes

{\bf A}^{00}=\left(\begin{array}[]{c}A^{00}_{1}\\ A^{00}_{2}\\ A^{00}_{3}\end{array}\right)=\left(\begin{array}[]{c}t\sin\theta+t(\zeta r+\check{\kappa}s)\cos\theta\\ -t\cos\theta+rt\kappa\cos+t(\zeta r+\check{\kappa}s)\sin\theta+\frac{1}{2}\gamma r^{2}\\ 0\end{array}\right)\,,

(9.2)

where

\theta=\theta(s_{0}),\quad\kappa=\kappa_{g}(s_{0}),\quad\gamma=\kappa_{n,\mathbf{B}}(x_{0})\,.

(9.3)

9.1.2 Structure of the quasi-mode

Consider two positive constants $C_{0}$ and $\delta$ such that $\frac{5}{18}<\delta<\frac{1}{3}$ . Let $\chi$ be a smooth even function, valued in $[0,1]$ , equal to $1$ on $[-\frac{1}{4},\frac{1}{4}]$ and supported in $[-\frac{1}{2},\frac{1}{2}]$ . We set

\chi_{h}(s)=c_{1}h^{-\delta/2}\chi(C_{0}^{-1}h^{-\delta}s)\,,

(9.4)

where $c_{1}=C_{0}^{-1/2}\left(\int_{\mathbb{R}}\chi(\sigma)^{2}d\sigma\right)^{1/2}$ , so that $\chi_{h}$ is normalized as follows,

\int_{\mathbb{R}}|\chi_{h}(s)|^{2}ds=1\,.

Our quasi-mode, $u$ , is supported in the set $Q_{h}(x_{0},R_{0},\delta,\epsilon_{2})$ introduced in (7.2) and is of the form

u=e^{i\check{p}/h}(1-r\kappa)^{-1/2}\tilde{u}

(9.5)

where $(r,s,t)\mapsto\check{p}(r,s,t)$ is the function from Lemma 7.2 and the function $\tilde{u}$ is of the form

\tilde{u}(r,s,t)=\exp\left(-\frac{i\rho\gamma s}{h^{1/3}}\right)\exp\left(i\frac{r\sin\theta-s\cos\theta}{h^{1/2}}\xi_{0}\right)\chi_{h}(s)v(r,t)\,,

(9.6)

where $\xi_{0}=\sqrt{\Theta_{0}}$ is given by (4.2), $\theta$ and $\kappa$ are introduced in (9.3).
The choice of $\rho$ and $v$ will be specified later⁴⁴4 $\rho$ is defined in (9.2). For the definition of $v$ , see (9.11), (9.12) and (9.13). so that, for some constants $C,\varsigma_{*}>0$ , we have [HelMo4, Eq. (12.8)]

q^{h}_{M^{00}}(v)\leq\Big{(}\Theta_{0}h+\widehat{\gamma}_{0,\mathbf{B}}h^{\frac{4}{3}}+Ch^{\frac{4}{3}+\varsigma_{*}}\Big{)}\|v\|_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}^{2}\,.

(9.7)

Here $q_{M^{00}}(v)$ arises while computing the quadratic form of the quasi-mode in (9.5). It is defined as follows [HelMo4, Eq. (12.9)],

q^{h}_{M^{00}}(v)=\int_{\mathbb{R}\times\mathbb{R}_{+}}\Big{(}|(hD_{r}-M_{1}^{00}v|^{2}+|M_{2}^{00}v|^{2}+|hD_{t}v|^{2}\Big{)}drdt\,,

(9.8)

where

	$\displaystyle M_{1}^{00}(r,t)$	$\displaystyle=\sin\theta(t-h^{1/2}\xi_{0})$		(9.9)
	$\displaystyle M_{2}^{00}(r,t)$	$\displaystyle=(1+2\kappa r)^{1/2}\Big{(}-\cos\theta(t-h^{1/2}\xi_{0})+\kappa\cos\theta rt-b\frac{\gamma}{2}(r^{2}-h^{2/3}\rho)\Big{)}\,.$		(9.9)

Notice that, by our normalization of $\chi_{h}$ , we have

\int_{\mathbb{R}^{2}\times\mathbb{R}_{+}}|\tilde{u}(r,s,t)|^{2}drdsdt=\|v\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}\,.

(9.10)

9.1.3 Definition of the auxiliary objects

Let us recall the definition of the function $v$ and the parameter $\rho$ given in [HelMo4, Sec. 12]. The function $v$ depends on $h$ and is selected in the following form (see [HelMo4, Eq. (12.14)])

v(r,t)=h^{-5/12}v_{0}(\hat{r},\hat{t})

(9.11)

where

(\hat{r},\hat{t})=(h^{-1/3}r,h^{-1/2}t)\,.

The function $v_{0}$ is selected as in [HelMo4, Eq. (12.28)]:

v_{0}(\hat{r},\hat{t})=\chi(C_{0}^{-1}h^{-\delta+\frac{1}{3}}\hat{r})\chi(C_{0}^{-1}h^{-\delta+\frac{1}{2}}\hat{t})w_{h}(\hat{r},\hat{t}),

(9.12)

In the sequel, we skip the hats from the notation. The function $w_{h}$ is defined as follows⁵⁵5 For the convenience of the reader, we will recall the heuristics behind the construction of $w_{h}$ in Subsection 9.1.4. [HelMo4, Eq. (12.22)]

w_{h}(r,t)=\varphi_{0}(t)\psi(r)+h^{1/6}\varphi_{1}(t)L_{1}^{0}(r,D_{r})\psi(r)+h^{1/3}\varphi_{2}(t)\big{(}L_{1}^{0}(r,D_{r})\big{)}^{2}\psi(r)\,,

(9.13)

where $\varphi_{0}$ is the positive normalized ground state of the harmonic oscillator in (4.1),

\varphi_{1}(t)=2\mathcal{R}_{0}\big{(}(t-\xi_{0})\varphi_{0}\big{)},\quad\varphi_{2}(t)=2\mathcal{R}_{0}\big{(}(t-\xi_{0})\varphi_{1}-\langle(t-\xi_{0})\varphi_{1},\varphi_{0}\rangle\varphi_{0}\big{)}

and $\mathcal{R}_{0}$ is the regularized resolvent introduced in (4.4). Notice that $\varphi_{0},\varphi_{1}$ and $\varphi_{2}$ are Schwartz functions (i.e. in $\mathcal{S}(\mathbb{R}_{+})$ , see [FH06, Appendix A]). The definition of $w_{h}$ involves the differential operator

L_{1}^{0}(r,D_{r})=\sin\theta D_{r}-\frac{1}{2}\cos\theta\gamma(r^{2}-\rho)

(9.14)

and a function $\psi\in\mathcal{S}(\mathbb{R})$ defined via the ground state $\psi_{0}$ of the Montgomery model in (4.5) and the following phase function

\varphi(r)=\gamma\alpha(\theta)\Big{(}\frac{r^{3}}{6}+\frac{\rho r}{2}\Big{)}\,,

where

\alpha(\theta)=\frac{\sin\theta\cos\theta(1-\delta_{0})}{\delta_{0}\sin^{2}\theta+\cos^{2}\theta}\,,

and $\delta_{0}$ the constant introduced in (4.2). We define now the function $\psi(r)$ as follows

\psi(r)=\Big{(}\frac{c}{d}\Big{)}^{-1/12}\exp\big{(}i\varphi(r)\big{)}\,\psi_{0}\Big{(}\Big{(}\frac{c}{d}\Big{)}^{-1/6}r\Big{)}

where

c=\cos^{2}\theta+\delta_{0}\sin^{2}\theta,\quad d=\frac{\delta_{0}^{2}\gamma^{2}}{\delta_{0}\sin^{2}\theta+\cos^{2}\theta}\,,

(9.15)

and we choose (see (4.5))

\rho=\Big{(}\frac{c}{d}\Big{)}^{1/3}\rho_{0}.

(9.16)

We conclude by mentioning some estimates which follow easily from the definitions of $v$ and $v_{0}$ in (9.11) and (9.12):

$\displaystyle\\|v\\|_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}^{2}$	$\displaystyle=1+\mathcal{O}(h^{1/6})\,,$	(9.17)
$\displaystyle\int_{\mathbb{R}\times\mathbb{R}_{+}}r^{k}t^{n}\|v\|^{2}drdt$	$\displaystyle=\mathcal{O}(h^{\frac{k}{3}+\frac{n}{2}})\quad(k,n\geq 0)\,,$
$\displaystyle\int_{\mathbb{R}\times\mathbb{R}_{+}}\|hD_{r}v\|^{2}drdt$	$\displaystyle=\mathcal{O}(h^{5/3}),\quad\int_{\mathbb{R}\times\mathbb{R}_{+}}\|hD_{t}v\|^{2}=\mathcal{O}(h)\,.$

9.1.4 Heuristics on the construction of $w_{h}$ .

Starting from the definition of the function $v$ in (9.12), the quadratic form in (9.7) becomes (after neglecting error terms in the magnetic potential)

q^{h}_{M^{00}}(v)\approx h\tilde{q}^{h}(w_{h})

where

\tilde{q}^{h}(w_{h}):=\int_{\mathbb{R}^{2}_{+}}\Big{(}|D_{t}w_{h}|^{2}+\big{|}\big{(}t-\xi_{0}-h^{1/6}L_{1}^{0}(r,D_{r})\big{)}w_{h}\big{|}^{2}+h^{1/3}|L_{2}^{0}(h,D_{r})w_{h}|^{2}\Big{)}drdt\,,

$L_{1}^{0}(r,D_{r})$ is introduced in (9.14) and

L_{2}^{0}=\cos\theta D_{r}+\frac{1}{2}\sin\theta\gamma(r^{2}-\rho)\,.

The construction of $w_{h}$ is based on minimizing

\int_{\mathbb{R}}\left(\int_{\mathbb{R}_{+}}\Big{(}|D_{t}w_{h}|^{2}+\big{|}\big{(}t-\xi_{0}-h^{1/6}L_{1}^{0}(r,D_{r})\big{)}w_{h}\big{|}^{2}\Big{)}dt\right)dr\,,

which amounts to finding the lowest eigenvalue of the operator

\mathcal{T}_{h}:=D_{t}^{2}+\big{(}t-\xi_{0}-h^{1/6}L_{1}^{0}(r,D_{r})\big{)}^{2}\,.

Writing

\mathcal{T}_{h}=D_{t}^{2}+(t-\xi_{0})^{2}-2h^{1/6}(t-\xi_{0})^{2}L_{1}^{0}(r,D_{r})+h^{1/3}\big{(}L_{1}^{0}(r,D_{r})\big{)}^{2}\,,

it is natural to search for $w_{h}$ in the form in (9.13) and satisfying

\mathcal{T}_{h}w_{h}-\left(\mu_{0}+\mu_{1}h^{1/6}L_{1}^{0}(r,D_{r})+\mu_{2}^{1/3}\big{(}L_{1}^{0}(r,D_{r})\big{)}^{2}\right)w_{h}\approx 0

in the following sense (after taking the coefficients of $h^{i/6}$ to be $0$ , for $i=0,1,2$ )

	$\displaystyle\big{(}D_{t}^{2}+(t-\xi_{0})^{2}-\mu_{0}\big{)}\varphi_{0}$	$\displaystyle=0$
	$\displaystyle\big{(}D_{t}^{2}+(t-\xi_{0})^{2}-\mu_{0}\big{)}\varphi_{1}$	$\displaystyle=\mu_{1}\varphi_{0}$
	$\displaystyle\big{(}D_{t}^{2}+(t-\xi_{0})^{2}-\mu_{0}\big{)}\varphi_{2}$	$\displaystyle=\mu_{2}\varphi_{0}+\mu_{1}\varphi_{1}$

Eventually, this leads to $\mu_{0}=\Theta_{0}$ , $\mu_{1}=0$ , $\mu_{2}=\frac{1}{2}\mu^{\prime\prime}(\xi_{0})$ and $\varphi_{0},\varphi_{1},\varphi_{2}$ as in (9.13).

9.2 Energy estimates

We will estimate the following energy arising from Lemma 7.3:

q^{h}_{{\bf A}^{00}}(\tilde{u})=\int_{\mathbb{R}^{2}\times\mathbb{R}_{+}}\Big{(}|(hD_{r}-A_{1}^{00})\tilde{u}|^{2}+(1+2\kappa r)|(hD_{s}-A_{2}^{00})\tilde{u}|^{2}+|hD_{t}\tilde{u}|^{2}\Big{)}drdsdt\,,

where $A_{1}^{00},A_{2}^{00}$ are introduced in (9.2).

Actually, $q^{h}_{{\bf A}^{00}}(\tilde{u})$ is bounded from above by $q_{M^{00}}(v)$ modulo error terms, where $q_{M^{00}}(v)$ and $v$ are introduced in (9.8) and (9.6) respectively. Due to the non-homogeneity of the magnetic field, the error terms involve a quantity⁶⁶6 This is $A(v)+B(v)$ appearing in (9.20), which would equals $0$ if the magnetic field were constant. introduced in (9.20) whose control has to be done carefully.

Due to the phase terms in the definition of $\tilde{u}$ in (9.6), we have

q^{h}_{{\bf A}^{00}}(\tilde{u})=\int_{\mathbb{R}^{2}\times\mathbb{R}_{+}}\Big{(}|hD_{t}\tilde{u}|^{2}+|(hD_{s}-A_{2,\rm new}^{00})\tilde{u}|^{2}+|(hD_{r}-A_{1,\rm new}^{00})\tilde{u}|^{2}\Big{)}drdsdt

where

\left(\begin{array}[]{c}A_{1,\rm new}^{00}\vskip 6.0pt plus 2.0pt minus 2.0pt\\ A_{2,\rm new}^{00}\end{array}\right)=\left(\begin{array}[]{c}M_{1,\zeta}^{00}\vskip 6.0pt plus 2.0pt minus 2.0pt\\ M_{2,\zeta}^{00}\end{array}\right)+\left(\begin{array}[]{c}\check{\kappa}\cos\theta st\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \check{\kappa}\sin\theta st\end{array}\right)\,,

and

$\displaystyle M_{1,\zeta}^{00}(r,t)$	$\displaystyle=\sin\theta(t-h^{1/2}\xi_{0})+\zeta\cos\theta rt$	(9.18)
$\displaystyle M_{2,\zeta}^{00}(r,t)$	$\displaystyle=(1+2\kappa r)^{1/2}\Big{(}-\cos\theta(t-h^{1/2}\xi_{0})$
	$\displaystyle\qquad+(\kappa\cos\theta+\zeta\sin\theta)rt-\frac{\gamma}{2}(r^{2}-h^{2/3}\rho)\Big{)}\,.$

Since the function $s\mapsto\chi_{h}(s)$ is even, we have

\langle(hD_{s}-\check{\kappa}\sin\theta st)u,M_{2,\zeta}^{00}u\rangle_{L^{2}(\mathbb{R}^{2}\times\mathbb{R}_{+})}=0

and

\langle\check{\kappa}\cos\theta st\,u,(hD_{r}-M_{1,\zeta}^{00})u\rangle_{L^{2}(\mathbb{R}^{2}\times\mathbb{R}_{+})}=0\,.

Moreover, we have the estimates

	$\displaystyle\\|(hD_{s}-\check{\kappa}\sin\theta st)\tilde{u}\\|_{L^{2}(\mathbb{R}^{2}\times\mathbb{R}_{+})}$	$\displaystyle\leq C\int_{\mathbb{R}\times\mathbb{R}_{+}}(h^{2-2\delta}+h^{2\delta}t^{2})\|v\|^{2}drdt$
		$\displaystyle=\mathcal{O}(h^{2-2\delta}+h^{2\delta+1})\,,$

and

\|\check{\kappa}\cos\theta st\,u\|_{L^{2}(\mathbb{R}^{2}\times\mathbb{R}_{+})}\leq Ch^{2\delta}\int_{\mathbb{R}\times\mathbb{R}_{+}}t^{2}|v|^{2}drdt=\mathcal{O}(h^{2\delta+1}).

Notice that we used (9.17) and also that $|s|\leq C_{0}h^{\delta}$ in the support of $\tilde{u}$ . Consequently, we get

	$\displaystyle q^{h}_{{\bf A}^{00}}(\tilde{u})$	$\displaystyle\leq\int_{\mathbb{R}\times\mathbb{R}_{+}}\Big{(}\|(hD_{r}-M_{1,\zeta}^{00})v\|^{2}+\|M_{2,\zeta}^{00}v\|^{2}+\|hD_{t}v\|^{2}\Big{)}drdt$		(9.19)
		$\displaystyle\qquad+\mathcal{O}\big{(}h^{2-2\delta}+h^{2\delta+1}\big{)}\,.$		(9.19)

Let us now reduce the computations to the potentials $M_{1}^{00}$ and $M_{2}^{00}$ in (9.9) which amount to $M_{1,\zeta}^{00}$ and $M_{2,\zeta}^{00}$ with $\zeta=0$ . A straightforward computation yields,

\|(hD_{r}-M_{1,\zeta}^{00})v\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}+\|M_{2,\zeta}^{00}v\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}\\ =\|(hD_{r}-M_{1}^{00})v\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}+\|M_{2}^{00}v\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}+\zeta\big{(}A(v)+B(v)\big{)},

(9.20)

where

	$\displaystyle A(v)$	$\displaystyle:=\zeta\cos^{2}\theta\,\\|rtv\\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}-2\cos\theta\,{\rm Re}\langle(hD_{r}-M_{1}^{00})v\,,\,rtv\rangle_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}$
	$\displaystyle B(v)$	$\displaystyle:=\zeta\sin^{2}\theta\,\\|rtv\\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}+2\sin\theta\,{\rm Re}\langle M_{2}^{00}v\,,\,rtv\rangle_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}$

and by (9.17)

\|rtv\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}=\mathcal{O}(h^{5/3}),\quad\langle hD_{r}v\,,\,rtv\rangle_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}=\mathcal{O}(h^{5/3})\,.

So, we end up with estimating

F(v):=\langle(\cos\theta M_{1}^{00}+\sin\theta M_{2}^{00})v\,,\,rtv\rangle_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}\,.

Notice that

\cos\theta M_{1}^{00}(r,t)+\sin\theta M_{2}^{00}(r,t)=\cos\theta\sin\theta\big{(}1-(1+2\kappa r)^{2}\big{)}(t-h^{1/2}\xi_{0})\\ +(1+2\kappa r)^{2}\cos\theta\Big{(}\kappa\cos\theta rt-\frac{\gamma}{2}\Big{(}r^{2}-h^{2/3}\rho\Big{)}\Big{)}\,.

By expanding

(1+2\kappa r)^{1/2}=1+\kappa r+\mathcal{O}(r^{2})\quad(r\to 0)\,,

we observe that, for $|r|\leq r_{0}$ and $r_{0}$ sufficiently small,

|\cos\theta M_{1}^{00}(r,t)+\sin\theta M_{2}^{00}(r,t)|\leq C(r^{2}+t^{2}+h^{2/3})\,,

so we get by (9.17) and the Cauchy-Schwarz inequality that

F(v)=\mathcal{O}(h^{3/2})\,.

Therefore, $A(v)+B(v)=\mathcal{O}(h^{3/2})$ and we deduce from (9.20) and (9.19) that

q^{h}_{{\bf A}^{00}}(\tilde{u})\leq q^{h}_{M^{00}}(v)+\mathcal{O}\big{(}h^{2-2\delta}+h^{2\delta+1}+h^{3/2}\big{)}\,,

(9.21)

where $q^{h}_{M^{00}}(v)$ is introduced in (9.8).

9.3 Conclusion

Collecting (9.21) and (9.7), we get

q^{h}_{{\bf A}^{00}}(\tilde{u})\leq(\Theta_{0}h+\widehat{\gamma}_{0,\mathbf{B}}h^{\frac{4}{3}}+Ch^{\frac{4}{3}+\eta}\Big{)}\|v\|_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}^{2}+R_{h}(v)

where

R_{h}(v)=\mathcal{O}\big{(}h^{2-2\delta}+h^{2\delta+1}+h^{3/2}\big{)}=\mathcal{O}(h^{\frac{4}{3}+\hat{\eta}})

for some $\hat{\eta}>0$ , thanks to the condition $\frac{5}{18}<\delta<\frac{1}{3}$ .

We insert this into Lemma 7.3 with $u$ given in (9.5). Notice that $u$ satisfies (7.7) with $M_{h}(u)=\mathcal{O}(1)$ . So by Lemma 7.3 and (9.10), we get for some $\eta_{*}>0$

q^{h}_{{\bf A}}(u)\leq(\Theta_{0}h+\widehat{\gamma}_{0,\mathbf{B}}h^{\frac{4}{3}}+Ch^{\frac{4}{3}+\eta_{*}}\Big{)}\|v\|_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}^{2}\,.

Comparing (9.10) and (9.5), we get by (2.20),

\|u\|^{2}=\big{(}1+\mathcal{O}(h^{2\delta})\big{)}\|v\|^{2}_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}\,.

(9.22)

Applying the min-max principle, and noticing that $1+2\delta>\frac{4}{3}$ for $\frac{5}{18}<\delta<\frac{1}{3}$ , we finish the proof of (9.1).

Acknowledgments

Preliminary discussions of the first author on this problem with Xingbin Pan more than twelve years ago are acknowledged. This work was initiated while the second author visited the Laboratoire de Mathématiques Jean Leray (LMJL) at Nantes Université in 2021. The authors would like to thank the support from the Fédération de recherche Mathématiques des Pays de Loire and Nantes Université.

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$\displaystyle\\|v\\|_{L^{2}(\mathbb{R}\times\mathbb{R}_{+})}^{2}$	$\displaystyle=1+\mathcal{O}(h^{1/6})\,,$	(9.17)
$\displaystyle\int_{\mathbb{R}\times\mathbb{R}_{+}}r^{k}t^{n}\|v\|^{2}drdt$	$\displaystyle=\mathcal{O}(h^{\frac{k}{3}+\frac{n}{2}})\quad(k,n\geq 0)\,,$
$\displaystyle\int_{\mathbb{R}\times\mathbb{R}_{+}}\|hD_{r}v\|^{2}drdt$	$\displaystyle=\mathcal{O}(h^{5/3}),\quad\int_{\mathbb{R}\times\mathbb{R}_{+}}\|hD_{t}v\|^{2}=\mathcal{O}(h)\,.$

Helical magnetic fields and semi-classical asymptotics of the lowest eigenvalue

Abstract

1 Main results

Assumption 1.1 (C1).

Assumption 1.2 (C2).

Theorem 1.3.

Theorem 1.4.

Discussion and applications

Organization and outline of the proof

2 Adapted coordinates

2.1 Description of the coordinates

2.2 The metric in the new coordinates

2.3 The operator and quadratic form

3 Helical magnetic fields

3.1 Preliminaries

3.2 Γ\Gamma is a regular curve

Proposition 3.1.

Proof.

3.3 Explicit formulas in adapted coordinates

Proposition 3.2.

Proof.

Proposition 3.3.

Proof.

Proposition 3.4.

Remark 3.5.

Proof of Proposition 3.4.

4 1D Models

4.1 The de Gennes model

4.2 The Montgomery model

5 Model operator for non-uniform magnetic fields

Lemma 5.1.

Proof of Lemma 5.1.

Proposition 5.2 (Helffer-Morame).

Remark 5.3.

Proposition 5.4.

Proposition 5.5.

6 Localization of bound states

Proposition 6.1.

Proposition 6.2.

Proposition 6.3.

Remark 6.4 (Applications of Proposition 6.3).

7 Estimating the quadratic form

7.1 A comparison estimate

Lemma 7.1.

Proof.

7.2 Normal form

Lemma 7.2.

Proof of Lemma 7.2.

7.3 A second comparison estimate

Lemma 7.3 (Helffer-Morame).

Proof.

7.4 An estimate away from the curve Γ\Gamma

Lemma 7.4.

Proof.

Proposition 7.5.

Proof.

8 Lower bound

8.1 Another model

8.1.1 A new model quadratic form

Lemma 8.1.

Proof.

8.1.2 Linearizing change of variable

Lemma 8.2.

Remark 8.3.

8.1.3 Conclusion

8.2 The general case

9 Upper bound

9.1 The quasi-mode

9.1.1 Geometry and normal form

9.1.2 Structure of the quasi-mode

9.1.3 Definition of the auxiliary objects

9.1.4 Heuristics on the construction of whw_{h}.

9.2 Energy estimates

9.3 Conclusion

Acknowledgments

References

3.2 $\Gamma$ is a regular curve

7.4 An estimate away from the curve $\Gamma$

9.1.4 Heuristics on the construction of $w_{h}$ .