Near invariance of quasi-energy spectrum of Floquet Hamiltonians

We begin by discussing more specifically the type of spectral localization we have in mind. First, fix a general quasi-momentum ${\bf k}_{\star}$ and energy $E_{\star}$, and consider initial data which is “$\varepsilon-$ spectrally localized” with respect to $H^{0}$– a Bloch wave-packet of band-width $\varepsilon$ (see Proposition 4.7). The (unforced) evolution of such data by $U^{0}(t)=e^{-iH^{0}t}$, on large finite time-scales, has the structure of a slowly-varying spatial and temporal modulation of Bloch modes with energy $E_{\star}$ and quasi-momentum ${\bf k}_{\star}$. The slow evolution is described by an effective Hamiltonian, which is determined by the local character of the band structure (energy dispersion curves and eigenspaces) near $({\bf k}_{\star},E_{\star})$. The effective Hamiltonian captures the transport and spreading dynamics of such data.

We choose the forcing time-scale of our time-dependent Hamiltonian, $H^{\varepsilon}(t)$, so that there is a non-trivial interplay between the unforced transport dynamics and the effect of time-dependent forcing. ¹¹1Such balancing corresponds to what is typically done in experiments. For example, a wavepacket excitation is designed by a choice of a laser frequency and pulse band-width, and balanced with forcing to measure effects on experimentally accessible spatial and temporal scales. The parameter $a\geq 1$ in (1.1) is chosen to achieve this balance of effects. For example, the choice $a=1$ in (1.1) is appropriate for the dynamics where our Bloch wavepacket is concentrated near a point where the dispersion is locally linear, thus allowing for simple transport or conical diffraction, for example. In Section 5 we discuss a number of examples.

A key to our analysis is the space ${\rm BL}_{\varepsilon}$ of band-limited wavepackets (Definition 4.1): the space of Fourier band-limited envelope modulations of Bloch modes (${\bf k}_{\star}$ pseudo-periodic eigenstates) of $H^{0}$ with energy $E_{\star}$. ${\rm BL}_{\varepsilon}$ states are good approximations to $\varepsilon-$ spectrally localized a Floquet-Bloch wavepackets of band-width $\varepsilon$ (Proposition 4.7, also [39]).

Let $\Pi^{\varepsilon}$ denote the spectral measure associated with unitary operator $M^{\varepsilon}$ (see Section 2); let $\mathcal{P}_{0}^{\varepsilon}$ be the projection onto the subspace of $L^{2}(\mathbb{R}^{n})$, which is the space of functions which are $\varepsilon-$ localized, with respect to $H^{0}$, in quasi-momentum and energy about $({\bf k}_{\star},E_{\star})$. In particular, note that $\mathcal{P}_{0}^{\varepsilon}M_{0}^{\varepsilon}=M_{0}^{\varepsilon}\mathcal{P}_{0}^{\varepsilon}$. Our main near-invariance results are:

1. (1)

   Theorem 4.4 [Near invariance on ${\rm range}(\mathcal{P}_{0}^{\varepsilon})$] Let $\mathcal{I}\subset S^{1}$ denote any arc such that the spectrum of $M^{\varepsilon}_{0}\circ\mathcal{P}_{0}^{\varepsilon}$ is contained in $\mathcal{I}$. Then, for $\varepsilon>0$ sufficiently small,

   $$\Pi^{\varepsilon}\left[\mathcal{I}\right]\circ\mathcal{P}_{0}^{\varepsilon}=\mathcal{P}_{0}^{\varepsilon}+\mathcal{O}_{B(L^{2}(\mathbb{R}^{n}))}(\varepsilon^{n+1})$$

   or equivalently $\Pi^{\varepsilon}\left[S^{1}\setminus\mathcal{I}\right]\circ\mathcal{P}_{0}^{\varepsilon}=\mathcal{O}_{B(L^{2}(\mathbb{R}^{n}))}(\varepsilon^{n+1})\,.$ This provides an answer to Question 2: when states in the range of $\mathcal{P}_{0}^{\varepsilon}$ evolve under the forced monodromy operator, $M^{\varepsilon}$, the resulting state has very small projection onto quasi-energies far from their quasi-energies under $M^{\varepsilon}_{0}=\exp(-iH_{0}T_{\rm per}^{\varepsilon})$. Equivalently, denoting the spectral measure of $M^{\varepsilon}_{0}$ by $\Pi^{\varepsilon}_{0}$,

   $$\left(\Pi^{\varepsilon}[\mathcal{I}]-\Pi^{\varepsilon}_{0}[\mathcal{I}]\right)\circ\mathcal{P}_{0}^{\varepsilon}=\mathcal{O}_{B(L^{2}(\mathbb{R}^{n}))}(\varepsilon^{n+1})\,.$$

In the present article, we give a novel perspective on the $L^{2}(\mathbb{R}^{n})$ spectrum of parametrically (periodically) forced Schrödinger equations. Here, we draw connections between our result and other approaches to similar problems.

In Section 2 we provide necessary background on Floquet-Bloch theory of periodic Hamiltonians, the spectral theorem for unitary operators, and introduce relevant notation. Section 3 presents a brief intuitive introduction to effective dynamics and homogenization of periodic Hamiltonians. The main results of this article are presented in Section 4. In Section 5 we demonstrate how these results apply to a number of specific periodic Hamiltonians, $H^{0}$ and the associated $H^{\varepsilon}$, of physical interest. The proofs of the main results are presented in Section 6. Finally, a formal derivation of an effective transport equation (see Section 5.1) is presented in Section 7.

AS would like to thank P. Kuchment and V. Rom-Kedar, whose questions inspired this research. This research was supported in part by National Science Foundation grants DMS-1620418, DMS-1908657 and DMS-1937254 (MIW) as well as Simons Foundation Math + X Investigator Award #376319 (MIW, AS) and the AMS-Simons Travel Grant (AS).

We consider Hamiltonians $H^{0}=-\Delta+V$, where $V$ is periodic with respect to a lattice $\Lambda=\mathbb{Z}{\bf v}_{1}\oplus\cdots\oplus\mathbb{Z}{\bf v}_{n}$, where $\{{\bf v}_{1},\dots,{\bf v}_{n}\}$ is a linearly independent set of vectors in $\mathbb{R}^{n}$. ²²2We believe our analysis can be extended to general classes of elliptic operators (scalar Hamiltonians and systems) $H^{0}$, whose coefficients are periodic with respect to a lattice; see [29]. Since $H^{0}$ commutes with lattice translations, the Hamiltonian $H^{0}$ admits a fiber-decomposition $H^{0}=\int^{\oplus}_{\mathcal{B}}H_{\bf k}^{0}\,d{\bf k}$, where $H^{0}_{\bf k}$ denotes the operator $H^{0}$ acting in the space $L^{2}_{\bf k}$, consisting of $L^{2}_{\rm loc}$ functions which are ${\bf k}-$ pseudoperiodic, i.e. $\psi\in L^{2}_{\bf k}$ if $\psi({\bf x}+{\bf v})=e^{i{\bf k}\cdot{\bf v}}\psi({\bf x})$ a.e. in ${\bf x}\in\mathbb{R}^{2}$. The set $\mathcal{B}$ is the Brillouin zone, a choice of fundamental cell in $(\mathbb{R}_{\bf x}^{n})^{*}=\mathbb{R}^{n}_{\bf k}$. For each quasimomentum, ${\bf k}\in\mathcal{B}$, $H_{{\bf k}}^{0}$ is self-adjoint and compact resolvent. Hence, for each ${\bf k}$, $H_{{\bf k}}^{0}$ has a real sequence of discrete eigenvalues of finite multiplicity

$$E_{1}({\bf k})\leq E_{2}({\bf k})\leq\dots\leq E_{b}({\bf k})\leq\dots\,,$$

tending to infinity. The corresponding $L^{2}_{\bf k}$ eigenfunctions, denoted by $\Phi_{b}({\bf x};{\bf k})$:

$$H^{0}\Phi_{b}({\bf x};{\bf k})=E_{b}({\bf k})\Phi_{b}({\bf x};{\bf k})\,,\qquad{\bf x}\mapsto\Phi_{b}({\bf x},{\bf k})\in L^{2}_{\bf k}\,,$$

or equivalently ${\bf x}\mapsto e^{-i{\bf k}\cdot{\bf x}}\Phi_{b}({\bf x},{\bf k})\in L^{2}(\mathbb{R}^{n}/\Lambda)$. These eigenfunctions may be chosen to form an orthonormal basis for $L^{2}_{\bf k}$. Each function ${\bf k}\mapsto E_{b}({\bf k})$ is continuous and piecewise analytic [28, Theorem 5.5], and hence Lipschitz continuous.

Each image, $E_{b}(\mathcal{B})$, is a subinterval of $\mathbb{R}$ called the $b^{\rm th}$ spectral band. The graphs of ${\bf k}\mapsto E_{b}({\bf k})$ are called dispersion surfaces. The collection of all pairs $({\bf k},E_{b}({\bf k}))$ and corresponding normalized $L^{2}_{\bf k}$ eigenfunctions, $\Phi_{b}({\bf x};{\bf k})$, is called the band structure of $H^{0}$. Finally, the family of Floquet-Bloch modes $\cup_{{\bf k}\in\mathcal{B}}\{\Phi_{b}(\cdot,{\bf k})\}_{b\geq 1}$ is complete in $L^{2}(\mathbb{R}^{n})$; for any $f\in L^{2}(\mathbb{R}^{n})$,

$$f({\bf x})=\frac{1}{{\rm vol}(\mathcal{B})}\sum_{b\geq 1}\int_{\mathcal{B}}\left\langle\Phi_{b}(\cdot,{\bf k}),f\right\rangle_{L^{2}(\mathbb{R}^{n})}\Phi_{b}({\bf x},{\bf k})d{\bf k}\,,$$

where the sum is interpreted as a convergence of partial sums in $L^{2}(\mathbb{R}^{n})$. For simplicity, we will assume henceforward that ${\rm vol}(\mathcal{B})=1$.

In (1.3a) we introduced the (unitary in $L^{2}(\mathbb{R}^{n})$) evolution $U^{\varepsilon}(t)$ associated with the dynamics (1.1). In this section we give a brief outline of a construction of $U^{\varepsilon}(t)$ and then discuss the spectral measure of the associated monodromy operator. Under very general assumptions on $H^{0}$ and the operators $\{W(t,\cdot)\}$, a unitary propagator can be shown to exist using semigroup methods, see [34, Section 5] and [37, Section X.12].

From $U^{0}(t)=e^{-iH^{0}t}$ and $U^{\varepsilon}(t)$, we obtain the unitary monodromy operators $M_{0}^{\varepsilon}$ and $M^{\varepsilon}$ which, by the spectral theorem, are equipped with associated spectral (projection-valued) measures $\Pi^{\varepsilon}_{0}$ and $\Pi^{\varepsilon}$, respectively. For completeness, we review the definition and properties of a spectral measure and the spectral theorem for unitary operators. We refer the reader to [7, 17, 38, 40] for details.

Let $\mathcal{H}$ be a Hilbert space, let $X$ be a set, and $\Sigma$ a $\sigma$-algebra in $X$. A map $\Pi:\Sigma\to B(\mathcal{H})$, the Banach space of bounded linear operators on $\mathcal{H}$, is called a projection-valued measure if the following properties hold:

1. (1)

   $\Pi(I)$ is an orthogonal projection for every $I\in\Sigma$.

2. (2)

   $\Pi(\emptyset)=0$ and $\Pi(X)={\rm Id}$.

3. (3)

   If $\{I_{j}\}_{j\geq 1}\subset\Sigma$ are disjoint then

   $$\Pi\left(\bigcup\limits_{j\geq 1}I_{j}\right)v=\sum\limits_{j\geq 1}\Pi(I_{j})v\,,\qquad v\in\mathcal{H}\,.$$

4. (4)

   $\Pi(I_{1}\cap I_{2})=\Pi(I_{1})\Pi(I_{2})$ for all $I_{1},I_{2}\in\Sigma$.

1. (1)

   We adopt the convention of considering all $\mathbb{C}^{N}$ vectors as column vectors. If $a,b\in\mathbb{C}^{N}$, $a^{\top}b=a\cdot b$.

2. (2)

   Fourier transform on $L^{2}(\mathbb{R}^{n})$: For a function, $f$, defined on $\mathbb{R}^{n}$, define its Fourier transform as

   $$\hat{f}(\xi)=\mathcal{F}[f](\xi)=\hat{f}(\xi)\equiv\frac{1}{(2\pi)^{n}}\int\limits_{\mathbb{R}^{n}}f(x)e^{-i\xi\cdot x}\,dx.\,$$

   Furthermore, introduce

   $$\check{g}(x)\equiv\frac{1}{(2\pi)^{n}}\int\limits_{\mathbb{R}^{n}}g(\xi)e^{ix\cdot\xi}\,d\xi.$$

   The mappings $f\mapsto\hat{f}$ and $g\mapsto\check{g}$ map Schwartz class, $\mathcal{S}(\mathbb{R}^{n})$, to itself and for all $f\in\mathcal{S}(\mathbb{R}^{n})$, we have the Plancherel identity $\|f\|_{L^{2}(\mathbb{R}^{n})}=\|\hat{f}\|_{L^{2}(\mathbb{R}^{n})}$ and the inversion formula $\left(\hat{f}\right)^{\check{}}=f$. Hence, $\check{f}=\mathcal{F}^{-1}f$ on $\mathcal{S}(\mathbb{R}^{n})$. By density, both mappings extend to bounded linear transformations on $L^{2}(\mathbb{R}^{n})$ which satisfy the Plancherel identity, and the inversion formula.

3. (3)

   Pauli matrices are given by $\sigma_{0}={\rm I}$, and

   $$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\,,\qquad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\ i&0\end{array}\right)\,,\qquad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right)\,.$$

4. (4)

   For a vector ${\bf v}=(v_{1},v_{2})\in\mathbb{C}^{2}$, we write $(v_{1},v_{2})\cdot(\sigma_{1},\sigma_{2})=v_{1}\sigma_{1}+v_{2}\sigma_{2}$.

Our first assumption concerns the character of the energy band structure near $({\bf k}_{\star},E_{\star})$; in particular if $({\bf k}_{\star},E_{\star})$ is a degeneracy, then this degeneracy is isolated:

With Question 2 in mind and assuming spectral separation as defined in Hypothesis 1, we define a projection, $\mathcal{P}_{0}^{\varepsilon}$, associated with a subspace of $L^{2}(\mathbb{R}^{n})$ consisting of states, which are superpositions of modes whose quasimomenta and energy are near $({\bf k}_{\star},E_{\star})$:

(4.2)

$$\mathcal{P}_{0}^{\varepsilon}\equiv\int\limits_{|{\bf k}-{\bf k}_{\star}|<\varepsilon}{\rm Proj}\left(\left|H^{0}_{{\bf k}}-E_{\star}\right|<L\varepsilon\right)\,d{\bf k}\,,$$

where $L>0$ is fixed.

We next present two additional assumptions concerning the underlying wave-packet dynamics. Let $({\bf k}_{\star},E_{\star})$ satisfy the spectral separation Hypothesis 1 with parameters $N\geq 1$ and $b\geq 1$. Denote the vector of degenerate Floquet-Bloch modes:

$$\Phi_{\star}({\bf x})\equiv\left(\begin{array}[]{c}\Phi_{b_{\star}}({\bf x};{\bf k}_{\star})\\ \vdots\\ \Phi_{{b_{\star}}+N-1}({\bf x};{\bf k}_{\star})\end{array}\right)\ .$$

We next introduce the subspace of $L^{2}(\mathbb{R}^{n})$, consisting of Fourier band-limited wave-packets, which are modulations $\Phi_{\star}$.

The effective evolution operator, $U_{\rm eff}^{\varepsilon}(t)$, naturally gives rise to an

(4.5)

$$\textrm{{\it Effective monodromy operator defined on ${\rm BL}_{\varepsilon}$}:}\quad M_{\rm eff}^{\varepsilon}\equiv U_{\rm eff}^{\varepsilon}(T_{\rm per}^{\varepsilon});$$

for $\psi_{0}=\alpha_{0}^{\top}(\varepsilon{\bf x})\Phi({\bf x})\in{\rm BL}_{\varepsilon}$,

$$(M_{\rm eff}^{\varepsilon}\psi_{0})({\bf x})=\mathscr{U}_{\rm eff}(T;-i\nabla)[\alpha_{0}](\varepsilon{\bf x})\cdot\Phi_{\star}({\bf x})\ .$$

Since the monodromy operator $M^{\varepsilon}$ is unitary (see (1.3)), $M^{\varepsilon}$ has a spectral representation as an integral with respect to a projection-valued spectral measure, $\Pi^{\varepsilon}$, which is supported on the unit circle; see Sec. 2.2. We now state our main theorem, which addresses Question 2.

Denote by $(a,b)$ the arc $\{e^{-iy}~{}~{}|~{}~{}y\in(a,b)\}\subseteq S^{1}$.

Theorem 4.4 is a near-invariance (or stability) result for a spectral subspace associated with $H^{0}$, the range of $\mathcal{P}_{0}^{\varepsilon}$, under the perturbed dynamics $H^{\varepsilon}(t)$. Indeed, let $\underline{A}=0$; Then, under Hypothesis 1, the non-driven monodromy operator $M^{\varepsilon}_{0}$, restricted to the range of $\mathcal{P}_{0}^{\varepsilon}$, is given by:

$$M^{\varepsilon}_{0}\mathcal{P}_{0}^{\varepsilon}u({\bf x})=\sum_{b=b_{\star}}^{b_{\star}+N-1}\int\limits_{|{\bf k}-{\bf k}_{\star}|<\varepsilon}\left\langle\Phi_{{b_{\star}}}(\cdot;{\bf k}),u\rangle\Phi_{{b_{\star}}}({\bf x};{\bf k})\right\rangle e^{-iE_{b}({\bf k})T_{\rm per}^{\varepsilon}}\,d{\bf k}\,.$$

Since the dispersion surfaces are Lipschitz continuous, there is a constant $C>0$ such that for $|{\bf k}-{\bf k}_{\star}|\lesssim\varepsilon$ and for all $b_{\star}\leq b\leq b_{\star}+N-1$ we have

$$|E_{b}({\bf k})|\leq C\cdot|{\bf k}-{\bf k}_{\star}|+\mathcal{O}(\varepsilon^{2})=\mathcal{O}(\varepsilon)\,.$$

Denoting the spectral measure of $M^{\varepsilon}_{0}$ by $\Pi^{\varepsilon}_{0}$, we have that for a fixed $g$ and sufficiently small $\varepsilon>0$, by inspecting the explicitly expression above,

(4.9)

$$\Pi_{0}^{\varepsilon}\left[(-g,g)\right]\circ\mathcal{P}_{0}^{\varepsilon}=\mathcal{P}_{0}^{\varepsilon}\quad{\rm and}\quad\Pi_{0}^{\varepsilon}\left[S^{1}\setminus(-g,g)\right]\circ\mathcal{P}_{0}^{\varepsilon}=0\,.$$

As discussed in Remark 1.1, it is non-trivial that a form of (4.9) persists for time-periodic forcing $\underline{A}\neq 0$ in (1.1), due to the formally order-one cumulative effect of a perturbation of size $\varepsilon^{a}$ on the time-scale $T_{\rm per}^{\varepsilon}\sim\varepsilon^{-a}$.

As a step toward the proof of Theorem 4.4, we first prove its analog, Theorem 4.6, a strict invariance property for functions in ${\rm BL}_{\varepsilon}$ (see (4.3)), a closed subspace of $L^{2}(\mathbb{R}^{n})$. Since ${\rm BL}_{\varepsilon}$ approximates the range of $\mathcal{P}_{0}^{\varepsilon}$ (Proposition 4.7), we can then use Theorem 4.6 to prove Theorem 4.4, which concerns the range of $\mathcal{P}_{0}^{\varepsilon}$.

${\rm BL}_{\varepsilon}$ is a very natural space with which to study the effects of time-dependent forcing. In fact, the proof of Theorem 4.4, follows from its analog for the space ${\rm BL}_{\varepsilon}$:

Theorem 4.6 is proved in Section 6. Here, we first use it to give a proof of the main result, Theorem 4.4 (concerning $\mathcal{P}_{0}^{\varepsilon}$).