Spectral localization estimates for abstract linear Schrödinger equations

Jingxuan Zhang Yau Mathematical Sciences Center
Tsinghua University
Haidian District
Beijing 100084, China Department of Mathematical Sciences
University of Copenhagen
Universitetsparken 5
2100 Copenhagen, Denmark [email protected]

Spectral localization estimates for abstract linear Schrödinger equations

Abstract.

We study the propagation properties of abstract linear Schrödinger equations of the form $i\partial_{t}\psi=H_{0}\psi+V(t)\psi$ , where $H_{0}$ is a self-adjoint operator and $V(t)$ a time-dependent potential. We present explicit sufficient conditions ensuring that if the initial state $\psi_{0}$ has spectral support in $(-\infty,0]$ with respect to a reference self-adjoint operator $\phi$ , then, for some $c>0$ independent of $\psi_{0}$ and all $t\neq 0$ , the solution $\psi_{t}$ remains spectrally supported in $(-\infty,c\left\lvert t\right\rvert]$ with respect to $\phi$ , up to an $O(\left\lvert t\right\rvert^{-n})$ remainder in norm. The main condition is that the multiple commutators of $H_{0}$ and $\phi$ are uniformly bounded in operator norm up to the $(n+1)$ -th order. We then apply the abstract theory to a class of nonlocal Schrödinger equations on $\mathbb{R}^{d}$ , proving that any solution with compactly supported initial state remains approximately supported, up to a polynomially suppressed tail in $L^{2}$ -norm, inside a linearly spreading region around the initial support for all $t\neq 0$ .

Key words and phrases:

Schrödinger equations; A priori estimates

2020 Mathematics Subject Classification:

35Q41 (primary); 35B40 , 35B45 ,81U90 , 37K06 (secondary)

1. Introduction

We consider the following abstract linear Schrödinger equation on a Hilbert space $\mathfrak{h}$ :

(1.1)

\displaystyle i\partial_{t}\psi=H(t)\psi,\quad H(t)=H_{0}+V(t).

Here $H_{0}$ is a densely defined, self-adjoint operator on $\mathfrak{h}$ and $V(t)$ is a time-dependent potential such that $H(t)$ admits bounded propagator on $\mathfrak{h}$ for all times.

Our aim in this paper is to control the spectral localization properties of states evolving according to (1.1). Specifically, fix a reference self-adjoint operator $\phi$ on $\mathfrak{h}$ and let $P_{a}$ be the spectral projection of $\phi$ onto the half-line $(a,\infty)$ . Under suitable conditions on the commutator between $\phi$ and the system Hamiltonian, we prove that any solution $\psi_{t},\,t\in\mathbb{R}$ to (1.1) with initial state $\psi_{0}$ satisfies, for some $c,\,C>0$ independent of $\psi_{0}$ and all $t\neq 0$ ,

(1.2)

\displaystyle\left\lVert P_{c\left\lvert t\right\rvert}\psi_{t}\right\rVert_{\mathfrak{h}}\leq C\left(\left\lVert P_{0}\psi_{0}\right\rVert_{\mathfrak{h}}+\left\lvert t\right\rvert^{-n}\left\lVert\psi_{0}\right\rVert_{\mathfrak{h}}\right).

Estimate (1.2) ensures that if the initial state $\psi_{0}$ has spectral support in $(-\infty,0]$ w.r.t. the reference operator $\phi$ , then the corresponding solution $\psi_{t}$ remains spectrally localized in $(-\infty,c\left\lvert t\right\rvert]$ w.r.t. $\phi$ up to a polynomially decaying remainder in norm. See Theorem 2.1 for precise statement and the subsequent remarks for generalizations. The physical significance is discussed in Section 2.2.

We then apply the general theory (1.2) to study spacetime localization properties for some concrete models of nonlocal dispersive equations. We work in $\mathfrak{h}=L^{2}(\mathbb{R}^{d})$ and, for $X\subset\mathbb{R}^{d}$ , $a\geq 0$ , denote by ${\bf 1}_{X}$ the characteristic function of $X$ ,

X_{a}:=\left\{x\in\mathbb{R}^{d}:d_{X}(x)\leq a\right\},\quad\text{and}\quad X_{a}^{\mathrm{c}}:=\mathbb{R}^{d}\setminus X_{a}.

We choose the reference operator $\phi=d_{X}$ , the multiplication operator by the distance functions $d_{X}(x):=\inf_{y\in X}\left\lvert y-x\right\rvert$ , so that $P_{a}$ amounts to the multiplication operator by $\mathbf{1}_{X_{a}^{\mathrm{c}}}$ . The Hamiltonians $H(t)=H_{0}+V(t)$ are given by a class of nonlocal non-autonomous operators on $L^{2}(\mathbb{R}^{d})$ , with $H_{0}$ satisfying suitable finite moment bounds and potential $V(t)$ given by arbitrary uniformly bounded multiplication operators. Under condition (4.4) on the kernel of $H_{0}$ , we prove that if the wave function $\psi_{t}$ evolving according to such a nonlocal Schrödinger equation is supported in $X$ at $t=0$ , then there holds the dispersive estimate

(1.3)

\displaystyle\sup_{t\neq 0}\left\lvert t\right\rvert^{n}\int_{X_{c\left\lvert t\right\rvert}^{\mathrm{c}}}\left\lvert\psi_{t}\right\rvert^{2}\leq C\left\lVert\psi_{0}\right\rVert_{L^{2}}.

From here we conclude the Strichartz-type estimates

(1.4)

\displaystyle\left\lVert\mathbf{1}_{X_{c\left\lvert t\right\rvert}^{\mathrm{c}}}\psi_{t}\right\rVert_{L^{p}_{t}(L^{2}_{x})}\leq C\left\lVert\psi_{0}\right\rVert_{L^{2}},

for all $p>1/n$ and $\psi_{0}$ with $\operatorname{supp}\psi_{0}\subset X$ . Furthermore, by Markov’s inequality and (1.3), we have

(1.5)

\displaystyle\sup_{t\neq 0}\mu\bigl{(}\bigl{\{}{{x\in X_{c\left\lvert t\right\rvert}^{\mathrm{c}}}:\left\lvert\psi_{t}\right\rvert^{2}\geq\left\lvert t\right\rvert^{-n}}\}\bigr{)}\leq C\left\lVert\psi_{0}\right\rVert_{L^{2}}.

Localization estimates (1.3)–(1.5) impose direct constraints on the size of the probability tails. See Section 4.1 for the precise results.

The novelty of our results in this paper is twofold. Firstly, we identify explicit sufficient conditions ensuring spectral localization estimate (1.2) for abstract Schrödinger equations (see conditions (C1)–(C2) below). In particular, the system Hamiltonian $H(t)$ does not enter the conditions directly, but only through its commutators with the reference operator $\phi$ appearing in (1.2). Secondly, comparing to recent results [MR4254070, MR4604685] where similar propagation estimates as (1.2) are obtained for the standard Schrödinger equation and the Hartree equation, our results hold in an abstract Hilbert space setting and complement existent results in the case where the Hamiltonian involves nonlocal operators and the reference operator $\phi$ is not given by specific forms.

1.1. Organization of the paper

In Section 2, we present our main results, illustrate the key technical steps, and discuss the methodology in the context of relevant literature. In Section 3, we furnish the proofs of the main results, Theorems 2.2–2.4, in the general setting.

In Section 4, we illustrate applications to a large class of nonlocal Schrödinger equations on $\mathbb{R}^{d}$ . The models under consideration have the favourable property that they satisfy the main condition laid out in Section 2 with any multiplication operator $\phi$ by functions in the homogeneous Sobolev space $\dot{W}^{1,\infty}$ (i.e., weakly differentiable with $L^{\infty}$ -gradient). These could be viewed as typical models for nonlocal Hamiltonians, as the fractional Laplacian $H=(-\Delta)^{1/2}$ does enjoy the same property, owning to the theory of Calderón commutators.

In Section 5, we complete the proof of some technical estimates needed to establish certain expansion formulae in Section 3. In the appendix, we prove certain commutator bounds for the nonlocal Hamiltonians studied in Section 4.

Notation

We denote by ${\bf 1}$ the identity operator, $\mathcal{D}(A)$ the domain of an operator $A$ , and $\|\cdot\|$ the norm of operators on $\mathfrak{h}$ and sometimes that of vectors in $\mathfrak{h}$ . $\mathcal{B}(\mathfrak{h})$ denotes the space of bounded operators on $\mathfrak{h}$ . We make no distinction in our notation between a function $f\in\mathfrak{h}$ and the associated multiplication operator $\psi(x)\mapsto f(x)\psi(x)$ on $\mathfrak{h}$ .

The commutator $[A,B]$ of two operators $A$ and $B$ is first defined as a quadratic form on $\mathcal{D}(A)\cap\mathcal{D}(B)$ (always assumed to be dense in $\mathfrak{h}$ ) and then extended to an operator. Similarly, the multiple commutators of $A$ and $B$ are defined recursively by $\operatorname{ad}^{0}_{B}(A)=A$ and $\operatorname{ad}^{p}_{B}(A)=[\operatorname{ad}^{p-1}_{B}(A),B]$ for $p=1,2,\ldots$ .

2. Setup and main result

Let $\mathfrak{h}$ be a complex Hilbert space equipped with inner product $\left\langle\cdot,\,\cdot\right\rangle$ and induced norm $\left\lVert\cdot\right\rVert$ . We consider a dynamical system described by (1.1), with $H(t)$ defined on a common dense domain $\mathcal{D}=\mathcal{D}(H_{0})$ for all times. Furthermore we assume (1.1) is globally well-posed on $\mathfrak{h}$ . By standard perturbation theory, a simple sufficient condition is that $H_{0}$ is a densely defined self-adjoint operator on $\mathfrak{h}$ and $V(t)$ is uniformly bounded for all $t$ .

We will mainly work in the Heisenberg picture and study the Heisenberg evolution, $\alpha_{t}$ , for differentiable families $A(t)\in\mathcal{B}(\mathfrak{h})$ , characterized by the duality relation

(2.1)

\displaystyle\left\langle\psi_{0},\,\alpha_{t}(A(t))\psi_{0}\right\rangle=\left\langle\psi_{t},\,A(t)\psi_{t}\right\rangle,

where $\psi_{t},\,t\in\mathbb{R}$ is the unique global solution to the Schrödinger equation (1.1) with $\psi_{t}|_{t=0}=\psi_{0}\in\mathcal{D}$ .

2.1. Main result

Throughout the paper, we fix a reference self-adjoint operator $\phi$ on $\mathfrak{h}$ , such that $\mathcal{D}(H_{0})\cap\mathcal{D}(\phi)$ is dense in $\mathfrak{h}$ . Our main assumptions are then stated in terms of commutators between the reference operator $\phi$ and the system Hamiltonian in (1.1).

First, for some integer $n\geq 1$ , we assume that $\phi$ ‘almost commutes’ with the potential, in the sense that $[\phi,V(t)]$ extends to bounded operators for all times and satisfies, for some fixed $C_{V}>0$ ,

(C1)

\displaystyle\left\lVert G\right\rVert_{L^{1}(\mathbb{R})}\leq C_{V}\quad\text{ where }\quad G(t):=\left\lVert[\phi,V(t)]\right\rVert.

Second, we assume that the multiple commutators $\operatorname{ad}^{p}_{\phi}(H_{0}),\,p=1,\ldots,n+1$ , all extend to bounded operators on $\mathfrak{h}$ ; namely, there exist $\kappa_{1},\ldots,\kappa_{n+1}>0$ such that

(C2)

\displaystyle\left\lVert\operatorname{ad}^{p}_{\phi}(H_{0})\right\rVert\leq\kappa_{p}\qquad(p=1,\ldots,n+1).

To state our main result, let

(2.2)

\displaystyle\kappa\equiv\kappa_{1},

so that, by (C1), $\kappa$ amounts to the norm of the generalized group velocity operator $i[H,\phi]$ . Recall that $P_{a},\,a\in\mathbb{R}$ is the spectral projection of $\phi$ onto $(a,\infty)$ , explicitly,

(2.3)

\displaystyle P_{a}\equiv{\bf 1}_{(a,\infty)}(\phi).

The main result of this paper is the following:

Theorem 2.1 (Spectral localization).

Suppose (C1)–(C2) hold for some $n\geq 1$ . Then, for any $c>\kappa$ , there exists $C=C(n,c)>0$ such that for all $t\neq 0$ , the following operator inequality holds on $\mathfrak{h}$ :

(2.4)

\boxed{\alpha_{t}\left(P_{{c\left\lvert t\right\rvert}}\right)\leq P_{0}+C(P_{0}+C_{V})\left\lvert t\right\rvert^{-1}+C\left\lvert t\right\rvert^{-n}.}

This theorem is proved at the end of Section 2.3

Remark 1.

Let $C_{V}=0$ in (C1). Then by the duality relation (2.1), it is readily verifiable that (2.4) is equivalent to the localization estimate (1.2) in the Schrödinger picture.

Remark 2.

For $[H_{0},\phi]=0$ , we have $\kappa=0$ and for large $t$ , the propagation estimate (2.4) affirms the fact that $\phi$ is conserved along the evolution (1.1).

Remark 3.

Replacing $\phi\to-\phi$ in (2.3) and setting $P_{a}^{-}:={\bf 1}_{(-\infty,a)}$ , we conclude from (2.4) that

(2.5)

{\alpha_{t}\left(P_{{-c\left\lvert t\right\rvert}}^{-}\right)\leq P_{0}^{-}+C(P_{0}+C_{V})\left\lvert t\right\rvert^{-1}+C\left\lvert t\right\rvert^{-n}}.

Similarly, shifting the reference operator $\phi\to\phi-b$ yields

(2.6)

{\alpha_{t}\left(P_{{b+c\left\lvert t\right\rvert}}\right)\leq P_{b}+C(P_{b}+C_{V})\left\lvert t\right\rvert^{-1}+C\left\lvert t\right\rvert^{-n}},\quad b\in\mathbb{R}.

Note that both ways of modification above have no bearing on conditions (C1)–(C2).

Remark 4.

In [MR1720738], Hunziker, Sigal, and Soffer proved a corresponding ‘minimal escape velocity’ bound which complements estimate (2.4): Suppose $C_{V}=0$ in (C1) and condition (C2) holds for some $n\geq 1$ . Assume further $i[H_{0},\phi]\geq\theta$ for some $\theta>0$ . Then, by [MR1720738, eq. (1.11)], for any $0<\vartheta<\theta$ and $a\in\mathbb{R}$ , there exists $C>0$ s.th. for all $t>0$ ,

(2.7)

\alpha_{t}\left(P_{{a+\vartheta{t}}}^{-}\right)\leq{P_{{a}}^{-}+C{t}^{-n}}.

Combining (2.7) and (2.6) with $C_{V}=0$ , we obtain a refined two-sided control on the propagation of spectral supports of evolving states: Fix $a>b$ and assume the initial state $\psi_{0}\in\operatorname{Ran}{\bf 1}_{(a,b)}(\phi)$ . Then, for any $\vartheta<\theta\leq\kappa<c$ , the solution $\psi_{t},\,t>0$ generated by $\psi_{0}$ remains localized in the linearly expanding ‘annular regions’ $\operatorname{Ran}{\bf 1}_{(a+\vartheta t,b+ct)}(\phi)$ , up to an $O(t^{-n})$ remainder in norm.

2.2. Literature review and discussion

The question of locality versus nonlocality has always been a fundamental issue in the study of physical phenomena and the corresponding mathematical models. On $\mathfrak{h}=L^{2}(\mathbb{R}^{d})$ , our results yield, through (1.3)–(1.5), spacetime localization estimates for the propagation of information in a range of non-relativistic quantum mechanical models. Establishing such estimates is a delicate matter, because the dispersive structure of the governing evolution equations generally leads to an apparent lack of locality in the restrictive sense. For instance, suppose one defines, as usual, the maximal propagation speed as the infimum of all $c$ ’s such that states initially supported in $X\subset\mathbb{R}^{d}$ at time $t=0$ remains supported, for all times $t>0$ , in the light cone $X_{ct}$ . Then, even for the simplest example of a free particle evolving according to the Schrödinger equation $i\partial_{t}\psi=-\Delta\psi$ , infinite speed of propagation can be observed by examining the Fourier transform of the solution and using the superlinear growth of the dispersion relation. This general idea also leads to infinite speed of propagation, with the usual definition above, for typical $1$ -body quantum evolutions described by a large class of dispersive equations [MR1264524, MR1443322].

Historically, for quantum evolutions described by the standard Schrödinger equations with Hamiltonians $H=-\Delta+V$ , where the potential $V$ is sufficiently regular, one approach to recover an appropriate sense of locality is to introduce an energy cutoff adapted to the spectrum of $H$ on the initial state, and then show that the probability of finding the (microlocalized) state in the classically forbidden region vanishes asymptotically in time. Thus, localization properties of evolving states are reformulated in terms of propagation estimates for certain time-dependent observables identifying the spacetime support of states at time $t$ , and finite speed of propagation is established in terms of the resulting propagation estimates. This idea is also the starting point of the present paper.

More precisely, V. Enss proved in his seminal works [En1, En2] that if a particle with unit mass is initially localized in a ball $X$ at $t=0$ and has energy below $c^{2}/2$ , then the probability, $p(t)$ , of finding the particle at time $t>0$ in the classically forbidden region $X_{ct}^{\mathrm{c}}$ vanishes as a $L^{1}$ function, i.e., $\int p(t)\,dt<\infty$ . This way one obtains effective light cones (ELCs), viz., regions outside of which the probability of finding the particle vanishes asymptotically in time. Notice that the ELCs obtained this way are energy-dependent, in agreement with the physical intuition that a particle should move at a speed proportional to the square root of its energy.

The result of Enss was subsequently improved in [SigSof, Skib] and, more recently, [MR4254070, MR4604685], to Schrödinger equations with time-dependent Hamiltonians and the Hartree equation. Historically, such propagation properties have played crucial roles in scattering theory of Schrödinger operators, leading to important breakthroughs in the study of asymptotic completeness of $N$ -body problems by Enss [En3, En4], Skibsted [Skib2, Skib3], and Sigal-Soffer [SigSof1, SigSof2, SigSof3, SigSof4], among many others. For reviews of the development in scattering theory along this line, see [HunSig2, HunSig3].

Our approach to derive the main propagation estimate (1.2) is a generalization of the monotonicity method originated from the classical works of Sigal and Soffer’s [SigSof, SigSof1, SigSof2, SigSof3, SigSof4] in scattering theory, in which the authors proved that for general time-dependent Schrödinger equations on $\mathbb{R}^{d}$ , evolving states admit ELCs that spread out in space at a finite rate. The results of the seminal works [SigSof, SigSof2] were improved in [Skib, MR1335378, MR4254070, MR4604685] and extended to non-relativistic QED in [BoFauSig], open quantum systems in [MR4529865, MVBvNL], nonlinear equations in [MR4604685] and condensed matter physics in [FLS1, FLS2, LRSZ]. Our result is motivated and built upon the works mentioned above and more.

2.3. Key steps for proving Theorem 2.1

In this section, we illustrate the key steps in the proof of our main result, the operator inequality (2.4).

Our approach to study the localization properties of quantum evolutions, pioneered in [SigSof, Skib, MR1335378, MR4254070, BoFauSig, FLS1, FLS2], is based on approximate monotonicity formulae for certain propagation-identifying observables. As the starting point, we define, for any differentiable family $A(t)\in\mathcal{B}(\mathfrak{h})$ , the Heisenberg derivative

(2.8)

\displaystyle D_{(\cdot)}A(t):=\partial_{t}A(t)+i[(\cdot),A(t)].

It is readily verified that the evolution $\alpha_{t}$ , defined by relation (2.1), solves the Heisenberg equation

(2.9)

\displaystyle\partial_{t}(\alpha_{t}(A(t)))=\alpha_{t}(D_{H}(A(t))).

By the fundamental theorem of calculus and the fact that $\alpha_{0}(A(0))=A(0)$ , it follows that

(2.10)

\displaystyle\alpha_{t}(A(t))=A(0)+\int_{0}^{t}\alpha_{t}(D_{H}(A(r)))\,dr.

Our goal is to establish monotonicity estimates, based on (2.10), for the evolution of suitable time-dependent observables $A(t)$ . Roughly speaking, we design $A(t)$ so that it satisfies the following properties:

•

$\alpha_{t}(A(t))$ is constrained by a time-decaying envelope, with $\alpha(A(t))\lesssim A(0)+\left\lvert t\right\rvert^{-n}$ .
•

$A(t)$ is comparable with $P_{c\left\lvert t\right\rvert}$ , with $P_{c\left\lvert t\right\rvert}\leq A(t)$ for $t\neq 0$ while $A(0)\leq P_{0}$ .

Combining these, we arrive at the desired estimate (2.4).

Below we elaborate on the key steps in establishing these properties.

2.3.1. Construction of ASTLOs.

We fix a test light-cone slope $c>0$ and let $s>0$ be a large adiabatic parameter. For the reference operator $\phi$ entering (C1)–(C2), times $t\in\mathbb{R}$ , and smooth cutoff functions $\chi$ in a suitable class $\mathcal{X}\subset C^{\infty}\cap L^{\infty}(\mathbb{R},\mathbb{R}_{\geq 0})$ (see Figure 1 below), we define the adiabatic spectral localization observables, or ASTLOs (adopting the terminology of [FLS1, FLS2]), as

(ASTLO)

\mathcal{A}_{s}(t,\chi):=\chi\left(\frac{\phi-c\left\lvert t\right\rvert}{s}\right).

The precise construction of the class $\mathcal{X}$ is rather flexible and not so relevant to the present exposition; we defer it to (3.2) where the detailed proof is presented. Note that $\mathcal{A}_{s}(\cdot)$ is an operator-valued function defined by functional calculus of the self-adjoint operator $s^{-1}(\phi-c\left\lvert t\right\rvert)$ .

Figure 1. A typical function

\chi\in\mathcal{X}

compared with the characteristic function of

\mathbb{R}_{>0}

. Here

\delta>0

is a parameter entering the definition of

\mathcal{X}

through (RME) below. In essence,

\chi

is a smoothed-out version of

\mathbf{1}_{\mathbb{R}>0}

with derivative supported in

(0,\delta)

Remark 5.

Regarding the nomenclature of ‘ASTLO’, we say that $\mathcal{A}_{s}(t,\chi)$ is adiabatic since, for a test light-cone slope $c=O(1)$ and a large adiabatic parameter $s\gg 1$ , the velocity $\partial_{t}\mathcal{A}_{s}(t,\chi)=-cs^{-1}\mathcal{A}_{s}(t,\chi^{\prime})=O(s^{-1})$ varies at a slow scale. To see that $\mathcal{A}_{s}(t,\chi)$ identifies the spectral localization property of states, one can view $\mathcal{A}_{s}(\cdot,\chi)$ as a smoothed spectral cutoff function associated to $\phi$ and various half-lines (see Figure 1). Indeed, for functions $\chi$ in appropriate classes and suitably chosen $s$ , we can arrange to have

(2.11)		$\displaystyle\mathcal{A}_{s}(0,\chi)\leq$	$\displaystyle P_{0},$
(2.12)		$\displaystyle P_{c\left\lvert t\right\rvert}\leq$	$\displaystyle\mathcal{A}_{s}(t,\chi)\quad(t\neq 0).$

The detailed relations are formulated and proved in Proposition 3.4.

2.3.2. Differential inequality for ASTLOs.

By relation (2.9) and the almost-commuting assumption (C1), it is easy to verify that for some absolute constant $C>0$ ,

(H)

\displaystyle\partial_{t}\alpha_{t}(\mathcal{A}_{s}(t,\chi))\leq\alpha_{t}(D_{H_{0}}\mathcal{A}_{s}(t,\chi))+Cs^{-1}G(t).

See Lemma 5.5 and (2.3.4) for details. Further exploiting relation (H), we obtain

Theorem 2.2 (Recursive monotonicity estimate).

Suppose the evolution $\alpha_{t}$ satisfies identity (H), and condition (C2) holds for some $n\geq 1$ . Then, for any $c>\kappa$ and $\chi\in\mathcal{X}$ , there exists $C>0,\,\xi\in\mathcal{X}$ depending only on $n,\chi$ such that for $\delta:=c-\kappa>0$ and all $s>0,\,t\in\mathbb{R}$ :

(RME)

\displaystyle\partial_{t}\alpha_{t}\left(\mathcal{A}_{s}(t,\chi)\right)\leq

\displaystyle-\delta s^{-1}{\alpha_{t}\left(\mathcal{A}_{s}(t,\chi^{\prime})\right)}+Cs^{-2}{\alpha_{t}\left(\mathcal{A}_{s}\left(t,\xi^{\prime}\right)\right)}+Cs^{-1}\left(s^{-n}+G(t)\right).

This theorem is proved in Section 3.1.

Remark 6.

Note that for the free evolution with $V(t)\equiv 0$ , relation (H) follows with equality immediately from definitions (2.1) and (2.8). See discussions in Section 2.4 and concrete examples in Section 4 in which $V(t)$ is nontrivial.

Remark 7.

The differential inequality (RME) is ‘recursive monotone’ because the second term on the r.h.s. is of the same form as the leading term, and the latter is non-positive. Notice that (RME) (more precisely, the proof of it) is the only place where information of the evolution $\alpha_{t}$ is used. The only property we require of the underlying evolution is the differential identity (H), which asserts that the velocity $\partial_{t}\alpha_{t}(\mathcal{A}_{s})$ is approximately determined by the Heisenberg derivative $D_{H_{0}}\mathcal{A}_{s}$ corresponding to the autonomous part $H_{0}$ in the system Hamiltonian, up to a small time-decaying remainder due to the potential.

To prove Theorem 2.2, we use the commutator bounds (C2) to derive an expansion formula for $i[H_{0},\mathcal{A}_{s}]$ in terms of the bounded multiple commutators entering (C2). Combining this expansion with the explicit form of $\partial_{t}\mathcal{A}_{s}$ and (H) then yields (RME).

2.3.3. Monotonicity estimate for ASTLOs.

Since the evolution $\alpha_{t}$ is positivity-preserving, control over $\alpha_{t}(A_{s}(t,\chi))$ along $t$ yields the same over $\alpha_{t}(P_{c\left\lvert t\right\rvert})$ through relations (2.11)–(2.12). Thus, our goal now is to derive monotonicity estimates for $\alpha_{t}(\mathcal{A}_{s}(t,\chi))$ . Indeed, through (RME), we can control the growth of $\mathcal{A}_{s}(t,\chi)$ as follows:

Theorem 2.3 (monotonicity estimate).

Suppose (RME) holds for $n\geq 1$ and (C1) holds for some $C_{V}>0$ . Then, for any $c>\kappa$ and $\chi\in\mathcal{X}$ , there exists $C>0,\,\xi\in\mathcal{X}$ depending only on $n,c,\chi$ such that for all $s>0$ , $t\in\mathbb{R}$ :

(ME)

\displaystyle\alpha_{t}\left(\mathcal{A}_{s}(t,\chi)\right)\leq{\mathcal{A}_{s}(0,\chi)}+{Cs^{-1}}{\mathcal{A}_{s}(0,\xi)}+Cs^{-1}(s^{-n}\left\lvert t\right\rvert+C_{V}).

This theorem is proved in Section 3.2.

Remark 8.

Estimate (ME) shows that the expectation of $\mathcal{A}_{s}(t,\chi)$ is bounded by a time-decaying envelope for $\left\lvert t\right\rvert\leq s$ ; see Figure 2 below. Indeed, assume for simplicity $C_{V}=0$ . We evaluate the expectation of both side of (ME) on a state $\psi\in\mathcal{D}$ and use the duality $\left\langle\psi_{0},\,{\alpha_{t}(A)}\psi_{0}\right\rangle=\left\langle\psi_{t},\,{A}\psi_{t}\right\rangle$ (see (2.1)) to find, for $\left\lvert t\right\rvert\leq s$ ,

(2.13)

\displaystyle\left\langle\psi_{t},\,{\mathcal{A}_{s}(t,\chi)}\psi_{t}\right\rangle\leq\left\langle\psi_{0},\,{\mathcal{A}_{s}(0,\chi)}\psi_{0}\right\rangle+C{\left\lvert t\right\rvert^{-1}}\left\langle\psi_{0},\,{\mathcal{A}_{s}(0,\xi)}\psi_{0}\right\rangle+C\left\lvert t\right\rvert^{-n}\left\lVert\psi_{0}\right\rVert^{2}.

Figure 2. Schematic diagram illustrating the monotone envelop on the r.h.s. of (2.13).

Here we sketch the proof of (ME) for $t\geq 0$ . Integrating (RME), dropping certain non-negative terms, and using the $L^{1}$ bound on $G(t)$ from (C1), we find

(2.14)		$\displaystyle\delta s^{-1}\int_{0}^{t}{\alpha_{t}\left(\mathcal{A}_{s}(t,\chi^{\prime})\right)}\leq$	$\displaystyle\mathcal{A}_{s}(0,\chi)+C{s^{-2}{\int_{0}^{t}\alpha_{t}\left(\mathcal{A}_{s}\left(t,\xi^{\prime}\right)\right)}+Cs^{-1}(s^{-n}t+C_{V})},$
(2.15)		$\displaystyle\alpha_{t}\left(\mathcal{A}_{s}(t,\chi)\right)\leq$	$\displaystyle\mathcal{A}_{s}(0,\chi)+C{s^{-2}\int_{0}^{t}{\alpha_{t}\left(\mathcal{A}_{s}\left(t,\xi^{\prime}\right)\right)}+Cs^{-1}(s^{-n}t+C_{V})}.$

Since the l.h.s. term in (2.14) is of the same form as the second term on the r.h.s., we can apply the same estimate to the latter, while introducing another cutoff function, say $\eta$ , in the same class $\mathcal{X}$ as $\chi$ and $\xi$ . Iterating this procedure $n$ times on (2.14) until no integral is present in the r.h.s., we obtain

(2.16)

\displaystyle\int_{0}^{t}{\alpha_{t}\left(\mathcal{A}_{s}(t,\chi^{\prime})\right)}\leq C\left(s\mathcal{A}_{s}(0,\chi)+\mathcal{A}_{s}(0,\xi)+\cdots+s^{-(n-2)}\mathcal{A}_{s}(0,\eta)+s^{-n}t+C_{V}\right),

for $n$ cutoff functions $\xi,\ldots,\eta\in\mathcal{X}$ . Lastly, we apply (2.16) to bound the integral in the r.h.s. of (2.15) (which carries a prefactor of $s^{-2}$ ). This, together with some additional algebraic properties of the ASTLOs, yields (ME). See Section 3.2 for detailed derivations.

2.3.4. Concluding estimate (2.4).

Theorem 2.3 establishes the approximate monotonicity for ASTLOs as alluded to in Section 2.3. It remains now to conclude the desired propagation estimate (2.4).

Using the geometric properties (2.11)–(2.12) and estimate (2.13) for $\mathcal{A}_{s}(t,\chi)$ , we obtain

Theorem 2.4.

Suppose (ME) holds for $n\geq 1$ . Then, for any $c>\kappa$ , there exists $C=C(n,c)>0$ such that (2.4) holds for all $t\neq 0$ .

This theorem is proved in Section 3.3. Note that the statement of Theorem 2.4 no longer involves the ASTLOs.

Proof of Theorem 2.1.

Using the relation

	$\displaystyle\partial_{t}\alpha_{t}(\mathcal{A}_{s}(t,\chi))=$	$\displaystyle\alpha_{t}(D_{H_{0}}\mathcal{A}_{s}(t,\chi))+\alpha_{t}(i[V(t),\mathcal{A}_{s}(t,\chi)])$
(2.17)		$\displaystyle\leq$	$\displaystyle\alpha_{t}(D_{H_{0}}\mathcal{A}_{s}(t,\chi))+\left\lVert[V(t),\mathcal{A}_{s}(t,\chi)]\right\rVert,$

together with Lemma 5.5, which shows $\left\lVert[V(t),\mathcal{A}_{s}(t,\chi)]\right\rVert\leq Cs^{-1}G(t)$ for some absolute constant $C>0$ , we verify that (H) holds. This, together with Thms. 2.2–2.4, yields inequality (2.4) under conditions (C1)–(C2) for $s\geq t$ . Thus the proof is complete. ∎

2.4. Discussion

We are interested in the spectral localization properties of abstract dispersive evolutions. We have proposed a modular paradigm that accomplishes this goal, as long as the underlying evolution satisfies (H) with a reference operator satisfying (C1)–(C2). Our method is geometric in nature and traces back to the line of works by Enss, Hunziker, Sigal, Skibsted, Soffer, and others, who have laid out the foundation of the geometric method for scattering theory of the Schrödinger operators (see [HunSig2, HunSig3] for reviews).

Indeed, the asymptotic localization theory of general dispersive evolutions (1.1), which we consider here, bears an intrinsic similarity to the scattering theory of the standard Schödinger operators. Both problems concern with the semiclassical behaviour of particles for large time. Notice however that the parameter $s>0$ in (ASTLO), essentially a semiclassical parameter, does not come with the model (1.1), but is determined by the problem directly. The precise choice of $s$ is given in (3.48).

A main technical advantage of our localization theory based on the analysis of ASTLOs lies in its flexibility. Whereas strictly monotone quantities along a given evolution equation are rare to find, the approximately monotone ASTLOs are rather easy to engineer. One reason is that the underlying evolution does not enter directly into our analysis, but only through assumptions (C1)–(C2) on the commutators between the reference operator $\phi$ and the system Hamiltonian.

For example, suppose (1.1) is posed on $\mathbb{R}^{d}$ and the reference operator $\phi$ is a multiplication operator by a function $\phi:\mathbb{R}^{d}\to\mathbb{R}$ with $\operatorname{supp}\phi\subset X^{\mathrm{c}}$ for some $X\subset\mathbb{R}^{d}$ . Then, for generic pseudo-differential operators $\bar{V}(t)$ satisfying relevant domain conditions and $V(t):={\bf 1}_{X}\bar{V}(t){\bf 1}_{X}$ , we have $V(t)\phi=\phi V(t)=0$ , and so (C1) is satisfied with $C_{V}=0$ . Consequently, the ASTLOs satisfy the differential identity (H) with equality. Since the evolution only enters our analysis through (RME) and the latter depends only on (C2) and (H), we conclude that (RME) and all subsequent modular theorems hold. See Section 4 for more details.

Moreover, since the system Hamiltonian does not enter directly into the main technical assumption (C2), but only through its commutators with $\phi$ in (ASTLO), we can derive conditional localization properties when the commutator assumption (C2) fails for the obvious choice of $\phi$ . Consider the case $H_{0}=-\Delta$ acting on $\mathbb{R}^{d}$ . Let $d_{X}$ be a smoothed distance function to a smooth bounded domain $X$ . The obvious choice $\bar{\phi}=d_{X}$ does not satisfies (C2), since $[-\Delta,d_{X}]=-\Delta d_{X}-2\nabla d_{X}\cdot\nabla$ is unbounded. However, with an energy cutoff $g=g_{E}(H_{0})$ , where $E\in\sigma(H_{0})$ and $g_{E}$ is a smooth cutoff function supported in $\mathbb{R}_{\leq E}$ , one can check that, with the microlocalized position operator $\phi=gd_{X}g$ , the (microlocal) group velocity $i[H_{0},\phi]$ (together with higher commutators) is bounded. Using this microlocalized version of $\phi$ in (ASTLO) and running the paradigm above, we obtain energy-dependent spacetime localization estimates for $H_{0}=-\Delta$ . See [MVBvNL] for concrete results of this nature, with applications to von-Neuman-Linblad equations in Markovian open quantum dynamics. Related propagation bounds involving microlocal cutoff are obtained in [MR4254070, MR4604685] for linear and nonlinear quantum dynamics involving standard Schrödinger operators.

Lastly, since our method is based on monotonicity estimate in the form of operator inequalities, we can reduce localization theory for quantum many-body problems to the corresponding $1$ -body problems. Consider an abstract second quantization map, $\operatorname{\mathrm{d}\Gamma}$ , mapping $1$ -body observables $A$ acting on $\mathfrak{h}$ to many-body observables $\hat{A}$ acting on a Fock space $\mathcal{F}$ over $\mathfrak{h}$ . We assume the map $\operatorname{\mathrm{d}\Gamma}$ is positive-preserving, i.e., for any self-adjoint $1$ -body operators $A$ , $B$ ,

(2.18)

\displaystyle A\leq B\implies\hat{A}\leq\hat{B},

and, with $\hat{\alpha}_{t}$ denoting the many-body evolution of observables on $\mathcal{F}$ ,

(2.19)

\displaystyle\operatorname{\mathrm{d}\Gamma}(\alpha_{t}(A))=\hat{\alpha}_{t}(\hat{A}).

Then, applying $\operatorname{\mathrm{d}\Gamma}$ on both sides of (ME) yields the many-body approximate monotonicity estimate

(2.20)

\displaystyle\hat{\alpha}_{t}\left(\hat{\mathcal{A}}_{s}(t,\chi)\right)\leq{\hat{\mathcal{A}}_{s}(0,\chi)}+C{s^{-1}}\left[{\hat{\mathcal{A}}_{s}(0,\xi)}+(\left\lvert t\right\rvert{s^{-n}+C_{V})N}\right],

where $N=\operatorname{\mathrm{d}\Gamma}(\mathbf{1})$ is the number operator. See [FLS1, FLS2, LRSZ, LRZ] for related results based on this technique for quantum many-body systems arising from condensed matter physics.

3. Proofs of Theorems 2.2–2.4

In this section, we proved the main results presented in Section 2.3.

We begin with the precise definition of (ASTLO). Fix $c>0$ , together with a densely defined self-adjoint operator $\phi$ . For each $s>0$ , we define a class of observables by functional calculus:

(3.1)

\begin{array}[]{ccrcl}{\mathcal{A}_{s}}&\colon&{\mathbb{R}\times L^{\infty}(\mathbb{R})}&\longrightarrow&{\mathcal{B}(\mathfrak{h})}\\ \mbox{}&\mbox{}&{(t,\chi)}&\longmapsto&{\chi\left(\frac{\phi-c\left\lvert t\right\rvert}{s}\right)}\end{array}.

For a parameter $0<\delta<1$ , we define a class $\mathcal{X}\equiv\mathcal{X}_{\delta}$ as follows:

(3.2)

\displaystyle\mathcal{X}:=

\displaystyle\left\{\chi\in C^{\infty}(\mathbb{R},\mathbb{R}_{\geq 0})\left|\begin{aligned} &\operatorname{supp}\chi\subset(0,\infty),\,\chi^{\prime}\geq 0,\\ &\sqrt{\chi^{\prime}}\in C_{c}^{\infty},\,\operatorname{supp}\chi^{\prime}\subset(0,\delta)\end{aligned}\right.\right\}.

Then, for any $s,t$ , the operator $\mathcal{A}_{s}(t,\chi),\,\chi\in\mathcal{X}$ is bounded on $\mathfrak{h}$ and non-negative definite, with $\left\lVert\mathcal{A}_{s}(t,\chi)\right\rVert\leq\left\lVert\chi\right\rVert_{L^{\infty}}$ . Typical examples of functions in $\mathcal{X}$ are suitably smoothed characteristic functions of $\mathbb{R}_{\geq 0}$ as in Figure 1.

In what follows, we will use two properties of the space $\mathcal{X}$ , which can be readily verified:

(X1)

If $\xi(x)=\int_{0}^{x}w^{2}(y)\,dy$ for some $w\in C_{c}^{\infty}$ with $\operatorname{supp}w\subset(0,\delta)$ , then $\xi\in\mathcal{X}$ .
(X2)

For any $\xi_{1},\,\xi_{2}\in\mathcal{X}$ and $c\geq 0$ , there exists $\xi\in\mathcal{X}$ with $\xi\geq\xi_{1}+c\xi_{2}$ and $\xi^{\prime}\geq\xi_{1}^{\prime}+c\xi_{2}^{\prime}$ .

In principle, the class $\mathcal{X}$ could be replaced by suitable classes of functions satisfying the abstract properties (X1)–(X2).

In view of relation (H), to prove Theorems 2.2–2.4, it suffices to derive an estimate for the Heisenberg derivative $D_{H_{0}}\mathcal{A}_{s}(t,\chi)$ associated with the free Hamiltonian $H_{0}$ . Thus, in Sections 3.1 and 3.2, we only work with the free evolution and write $H\equiv H_{0}$ and

(3.3)

\displaystyle DA(t)=\frac{\partial}{\partial t}A(t)+i[H,A(t)],

so that, with $\alpha_{t}$ denoting the unitary evolution generated by $H_{0}$ , the Heisenberg equation (2.9) reads

(3.4)

\displaystyle\partial_{t}\alpha_{t}(A(t))=\alpha_{t}(DA(t)).

3.1. Proof of Theorem 2.2

Let $\vec{\kappa}:=(\kappa_{1},\ldots,\kappa_{n+1})$ as in (C2) and set $\delta:=c-\kappa$ . Recall in this subsection $\alpha_{t}$ denotes the free evolution and $H\equiv H_{0}$ .

The main result of this section is the following differential operator inequality:

Theorem 3.1.

Suppose condition (C2) holds for some $n\geq 1$ . Then, for all $c>\kappa$ and $\chi\in\mathcal{\mathcal{X}}$ , there exists a constant $C>0$ and functions $\xi_{k}\in\mathcal{X},\,k=2,\ldots,n$ (dropped if $n=1$ ) depending only on $n,\,\vec{\kappa}$ , and $\chi$ , such that for all $t\in\mathbb{R},\,s>0$ , the following operator inequality holds on $\mathfrak{h}$ :

(3.5)

\displaystyle\partial_{t}\alpha_{t}\left(\mathcal{A}_{s}(t,\chi)\right)\leq-\delta s^{-1}\alpha_{t}\left(\mathcal{A}_{s}(t,\chi^{\prime})\right)+\sum_{k=2}^{n}s^{-k}\alpha_{t}\left(\mathcal{A}_{s}\left(t,\xi_{k}^{\prime}\right)\right)+C{s^{-(n+1)}}.

(The sum in the r.h.s. is dropped if $n=1$ .)

This theorem is proved at the end of this section. Estimate (3.5), together with property (X2) and relation (H), implies Theorem 2.2.

Remark 9.

Identity (3.4) plays a crucial role in our analysis, and it is precisely in (3.4) that the Hamiltonian structure of (4.1) is used. Indeed, for a heat-type equation $\partial_{t}u=-Hu$ with self-adjoint $H$ , we have formally, instead of (3.4),

\partial_{t}\alpha_{t}(A(t))=\alpha_{t}(\partial_{t}A(t)-\left\{H,A(t)\right\}),

where the brace denotes the anti-commutator. The change $[\cdot,\cdot]\to\left\{\cdot,\cdot\right\}$ renders key expansion formulae below unavailable, and thus new machinery is needed to handle heat type equations. We will not seek to pursue this problem presently.

We begin with the following lemma:

Lemma 3.2.

Suppose the assumption of Theorem 3.1 holds. Then there exist $\xi_{k}=\xi_{k}(n,\vec{\kappa},\chi)\in\mathcal{X},\,k=2,\ldots,n$ (dropped if $n=1$ ), together with a constant $C=C(n,\vec{\kappa},\chi)>0,$ such that the following operator inequality holds on $\mathfrak{h}$ :

(3.6)

\displaystyle i[H,\mathcal{A}_{s}(t,\chi)]\leq s^{-1}\kappa\mathcal{A}_{s}(t,\chi^{\prime})+\sum_{k=2}^{n}s^{-k}\mathcal{A}_{s}(t,\xi_{k}^{\prime})+C{s^{-(n+1)}}\quad(t\in\mathbb{R},s>0).

(The sum in the r.h.s. is dropped if $n=1$ .)

Proof.

Within this proof, we fix $t$ and write $\mathcal{A}_{s}(\chi)\equiv\mathcal{A}_{s}(t,\chi)$ . Also, we set $B_{k}\equiv\pm i\operatorname{ad}^{k}_{\phi}(H)$ for $k=1,...,n+1$ . (The sign is irrelevant for our argument.)

1. By condition (C2), there exists $C=C(n,\vec{\kappa})>0,$ such that

(3.7)

\displaystyle\left\lVert B_{k}\right\rVert\leq C,\quad k=1,\ldots,n+1.

This, together with the definition of $\mathcal{X}$ (see (3.2)), implies that the hypotheses of Lemma 5.4 are satisfied for $\chi\in\mathcal{X}$ , and so there hold the commutator expansion

(3.8)

\displaystyle[H,\mathcal{A}_{s}(\chi)]=\sum_{k=1}^{n}\frac{s^{-k}}{k!}\mathcal{A}_{s}(\chi^{(k)})B_{k}+s^{-(n+1)}R_{n+1},

with some $C=C(n,\vec{\kappa},\chi)>0$ such that (c.f. (5.20)–(5.21))

(3.9)

\displaystyle\left\lVert R_{n+1}\right\rVert\leq C.

Adding commutator expansion (3.8) to its adjoint and dividing the result by two, we obtain

(3.10)	$\displaystyle i[H,\mathcal{A}_{s}(\chi)]=$	$\displaystyle\mathrm{I}+\mathrm{II}+\mathrm{III},$
(3.11)	$\displaystyle\mathrm{I}=$	$\displaystyle\frac{1}{2}s^{-1}\left(\mathcal{A}_{s}(\chi^{\prime})B_{1}+B_{1}^{*}\mathcal{A}_{s}(\chi^{\prime})\right),$
(3.12)	$\displaystyle\mathrm{II}=$	$\displaystyle\frac{1}{2}\sum_{k=2}^{n}\frac{s^{-k}}{k!}\left(\mathcal{A}_{s}(\chi^{(k)})B_{k}+B_{k}^{*}\mathcal{A}_{s}(\chi^{(k)})\right),$
(3.13)	$\displaystyle\mathrm{III}=$	$\displaystyle\frac{1}{2}s^{-(n+1)}\left(R_{n+1}+R_{n+1}^{*}\right),$

where the term $\mathrm{II}$ is dropped for $n=1$ .

2. We first bound the term $\mathrm{I}$ in line (3.11). Let $u:=\sqrt{\chi^{\prime}}$ , which is well defined and lies in $C_{c}^{\infty}(\mathbb{R})$ by (3.2). Then, by (3.7) and Lemma 5.4, expansion (3.8) also holds for $u$ . This expansion, together with the fact that $\operatorname{ad}_{\phi}^{l}(B_{k})=B_{k+l}$ , implies

	$\displaystyle\quad\mathcal{A}_{s}(\chi^{\prime})B_{1}+B_{1}^{*}\mathcal{A}_{s}(\chi^{\prime})$
	$\displaystyle=\mathcal{A}_{s}(u)^{2}B_{1}+B_{1}\mathcal{A}_{s}(u)^{2}$
	$\displaystyle=2\mathcal{A}_{s}(u)B_{1}\mathcal{A}_{s}(u)+\mathcal{A}_{s}(u)[\mathcal{A}_{s}(u),B_{1}]+[B_{1},\mathcal{A}_{s}(u)]\mathcal{A}_{s}(u)$
	$\displaystyle=2\mathcal{A}_{s}(u)B_{1}\mathcal{A}_{s}(u)$
(3.14)		$\displaystyle\quad+\sum_{l=1}^{n-1}\frac{s^{-l}}{l!}\left({\mathcal{A}_{s}(u)B_{1+l}\mathcal{A}_{s}(u^{(l)})+\mathcal{A}_{s}(u^{(l)})B_{1+l}^{*}\mathcal{A}_{s}(u)}\right)$
(3.15)		$\displaystyle\quad+s^{-n}(\mathcal{A}_{s}(u)\mathrm{Rem}_{1}+\mathrm{Rem}_{1}^{*}\mathcal{A}_{s}(u)),$

where line (3.14) is dropped for $n=1$ and, for some $C=C(n,\vec{\kappa},\chi)>0$ ,

(3.16)

\displaystyle\left\lVert\mathrm{Rem}_{1}\right\rVert\leq C.

We will bound the terms in (3.14)–(3.15) using the operator estimate

(3.17)

\displaystyle\pm(P^{*}Q+Q^{*}P)

\displaystyle\leq P^{*}P+Q^{*}Q.

For the terms in line (3.14), we use (3.17) with

(3.18)

\displaystyle P=\mathcal{A}_{s}(u),\quad Q:=B_{1+l}\mathcal{A}_{s}(u^{(l)}),\quad l=1,\ldots,n-1,

yielding

		$\displaystyle s^{-l}({\mathcal{A}_{s}(u)B_{1+l}\mathcal{A}_{s}(u^{(l)})+\mathcal{A}_{s}(u^{(l)})B_{1+l}^{*}\mathcal{A}_{s}(u)})$
(3.19)		$\displaystyle\leq$	$\displaystyle s^{-l}\left(\mathcal{A}_{s}(u)^{2}+\\|B_{1+l}\\|^{2}(\mathcal{A}_{s}(u^{(l)}))^{2}\right).$

For the remainder terms in (3.15), we apply (3.17) with

(3.20)

\displaystyle P=\mathcal{A}_{s}(u),\quad Q=\mathrm{Rem}_{1},

to obtain

(3.21)

\displaystyle s^{-n}(\mathcal{A}_{s}(u)\mathrm{Rem}_{1}+\mathrm{Rem}_{1}^{*}\mathcal{A}_{s}(u))

\displaystyle\leq s^{-n}\left(\mathcal{A}_{s}(u)^{2}+\|\mathrm{Rem}_{1}\|^{2}\right).

Combining (3.1) and (3.21) in (3.11) yields

(3.22)			$\displaystyle\mathrm{I}\leq s^{-1}\mathcal{A}_{s}(u)B_{1}\mathcal{A}_{s}(u)$
		$\displaystyle\quad+\frac{1}{2}\sum_{l=1}^{n-1}\frac{s^{-(l+1)}}{l!}\left(\mathcal{A}_{s}(u)^{2}+\\|B_{1+l}\\|^{2}(\mathcal{A}_{s}(u^{(l)}))^{2}\right)+\frac{1}{2}s^{-(n+1)}\\|\mathrm{Rem}_{1}\\|^{2}.$

This bound the term $\mathrm{I}$ (3.11).

3. For $n\geq 2$ , the term $\mathrm{II}$ in line (3.12) is bounded similarly as in Step 2. For $k=2,...,n$ , we take $\theta^{k}\in C_{c}^{\infty}(\mathbb{R})$ with

(3.23)

\displaystyle\operatorname{supp}\theta^{k}\subset(0,\delta),\quad\theta^{k}\equiv 1\text{ on }\operatorname{supp}\chi^{(k)}.

We claim that for some bounded operator $\mathrm{Rem}_{k}=O(1)$ ,

		$\displaystyle s^{-k}\left(\mathcal{A}_{s}(\chi^{(k)})B_{k}+B_{k}^{*}\mathcal{A}_{s}(\chi^{(k)})\right)$
(3.24)		$\displaystyle=$	$\displaystyle s^{-k}\left(\mathcal{A}_{s}(\chi^{(k)})B_{k}\mathcal{A}_{s}(\theta^{k})+\mathcal{A}_{s}(\theta^{k})B_{k}^{*}\mathcal{A}_{s}(\chi^{(k)})\right)+s^{-(n+1)}\mathrm{Rem}_{k}.$

For this, it suffices to show that

(3.25)

\displaystyle\mathcal{A}_{s}(\chi^{(k)})B_{k}=\mathcal{A}_{s}(\chi^{(k)})B_{k}\mathcal{A}_{s}(\theta^{k})+s^{-(n+1-k)}\mathrm{Rem}_{k}.

Using relation (3.23), commutator expansion (3.8), and the fact that $\operatorname{ad}_{\phi}^{l}(B_{k})=B_{k+l}$ , we have

	$\displaystyle\quad\mathcal{A}_{s}(\chi^{(k)})B_{k}$
	$\displaystyle=\mathcal{A}_{s}(\chi^{(k)})\mathcal{A}_{s}(\theta^{k})B_{k}$
	$\displaystyle=\mathcal{A}_{s}(\chi^{(k)})B_{k}\mathcal{A}_{s}(\theta^{k})+\mathcal{A}_{s}(\chi^{(k)})[\mathcal{A}_{s}(\theta^{k}),B_{k}]$
	$\displaystyle=\mathcal{A}_{s}(\chi^{(k)})B_{k}\mathcal{A}_{s}(\theta^{k})$
(3.26)		$\displaystyle\quad+\sum_{l=1}^{n-k}\frac{s^{-l}}{l!}\mathcal{A}_{s}(\chi^{(k)})\mathcal{A}_{s}((\theta^{k})^{(l)})B_{k+l}+s^{-(n+1-k)}\mathcal{A}_{s}(\chi^{(k)})\mathrm{Rem}_{k},$

where the $l$ -sum is dropped for $k=n$ and

(3.27)

\displaystyle\mathrm{Rem}_{k}\leq C,\quad k=2,\ldots,n,

for some $C=C(n,\vec{\kappa},\chi)>0$ .

Since $\theta^{k}\equiv 1$ on $\operatorname{supp}(\chi^{(k)})$ , we have $\operatorname{supp}((\theta^{k})^{(l)})\cap\operatorname{supp}(\chi^{(k)})=\emptyset$ for all $l\geq 1$ and so in line (3.26),

\displaystyle\mathcal{A}_{s}(\chi^{(k)})\mathcal{A}_{s}((\theta^{k})^{(l)})B_{k+l}=0,\quad l=1,\ldots n-k.

Estimate (3.25) follows from here. Thus we conclude claim (3.1).

Now, we apply estimate (3.17) on the first term on the r.h.s. of (3.1) with

(3.28)

\displaystyle P=\mathcal{A}_{s}(\chi^{(k)}),\quad Q=B_{k}\mathcal{A}_{s}(\theta^{k}),

and then sum over $k$ to obtain

(3.29)

\displaystyle\mathrm{II}\leq

\displaystyle\frac{1}{2}\sum_{k=1}^{n-1}\frac{s^{-k}}{k!}\left((\mathcal{A}_{s}(\chi^{(k)}))^{2}+\|B_{k}\|^{2}(\mathcal{A}_{s}(\theta^{k}))^{2}\right)+\frac{1}{2}s^{-(n+1)}\left\lVert\mathrm{Rem}_{k}\right\rVert^{2}.

This bounds the term $\mathrm{II}$ in line (3.12).

4. Plugging (3.22), (3.29) back to (3.10) and using bounds (3.7), (3.9), (3.16), and (3.27), we find that for some $C=C(n,\vec{\kappa},\chi)>0$ ,

(3.30)		$\displaystyle i[H,\mathcal{A}_{s}(\chi)]\leq s^{-1}\mathcal{A}_{s}(u)B_{1}\mathcal{A}_{s}(u)$
	$\displaystyle+C\sum_{k=2}^{n}{s^{-k}}\left(\mathcal{A}_{s}(u)^{2}+(\mathcal{A}_{s}(u^{(k-1)}))^{2}+\mathcal{A}_{s}(\chi^{(k)}))^{2}+(\mathcal{A}_{s}(\theta^{k}))^{2}\right)+Cs^{-(n+1)}.$

Now, for $k=2,\ldots,n$ , we choose, with $C$ , $u$ , $\theta^{k}$ from (3.30),

		$\displaystyle w_{k}\in C_{c}^{\infty},\quad\operatorname{supp}w_{k}\subset(0,\delta),$
(3.31)		$\displaystyle w_{k}^{2}\geq$	$\displaystyle C\left(u^{2}+(u^{(k-1)})^{2}+(\chi^{(k)})^{2}+(\theta^{k})^{2}\right),$

which is possible since the r.h.s. of (3.31) is supported in $(0,\delta)$ by construction. Then the function

(3.32)

\displaystyle\xi_{k}(x):=\int_{0}^{x}w_{k}^{2}(y)\,dy

lies in $\mathcal{X}$ by identity (X1). Thus, by (3.30), the desired estimate (3.6) holds with the choice of $\xi_{k}$ from (3.32). This completes the proof of Lemma 3.2. ∎

Proof of Theorem 3.1.

To prove estimate (3.5), we first apply the differential identity (3.4) with $A(t)=\mathcal{A}_{s}(t,\chi)$ for each $s,\,\chi$ . This yields

(3.33)

\partial_{t}\alpha_{t}(\mathcal{A}_{s}(t,\chi))=\alpha_{t}(\partial_{t}\mathcal{A}_{s}(t,\chi))+\alpha_{t}(i[H,\mathcal{A}_{s}(t,\chi)]).

By definition (3.1), we find

(3.34)

\displaystyle\partial_{t}\mathcal{A}_{s}(t,\chi)=-s^{-1}c\,\mathcal{A}_{s}(t,\chi^{\prime}).

By estimate (3.6), we find

(3.35)

\displaystyle i[H,\mathcal{A}_{s}(t,\chi)]

\displaystyle\leq s^{-1}\kappa\mathcal{A}_{s}(t,\chi^{\prime})+\sum_{k=2}^{n}s^{-k}\mathcal{A}_{s}(t,\xi_{k}^{\prime})+C{s^{-(n+1)}},

where $C=C(n,\vec{\kappa},\chi)>0$ and the second term in the r.h.s. is dropped for $n=1$ . Plugging (3.34) and (3.35) back to (3.33) and using the positive-preserving property of evolution $\alpha_{t}$ yields (3.5). ∎

3.2. Proof of Theorem 2.3

Recall in this subsection $\alpha_{t}$ denotes the free evolution and $H\equiv H_{0}$ . Our main result is the following:

Theorem 3.3.

Suppose (3.5) holds. Then there exist $C>0$ and a function $\xi\in\mathcal{X}$ (dropped if $n=1$ ) depending only on $n,\,\vec{\kappa}$ , $\chi$ , and $\delta$ such that for all $t\in\mathbb{R},\,s>0$ , the following operator inequality holds on $\mathfrak{h}$ :

(3.36)

\displaystyle\alpha_{t}\left(\mathcal{A}_{s}(t,\chi)\right)\leq{\mathcal{A}_{s}(0,\chi)}+{s^{-1}}{\mathcal{A}_{s}(0,\xi)}+C\left\lvert t\right\rvert s^{-(n+1)}.

(The second term on the r.h.s. is dropped for $n=1$ .)

Using Estimate (3.36), together with relation (H) and the $L^{1}$ bound on $G(t)$ from assumption (C1), we arrive at Theorem 2.3.

Proof of Theorem 3.3.

Within this proof, we fix $s>0$ and all constants $C>0$ depend only on $n$ , $\chi$ , $\vec{\kappa}$ , and $\delta=c-\kappa$ . For simplicity, below we consider the case for $t\geq 0$ . For negative times the argument is similar.

1. For ease of notation, for any function $f\in L^{\infty}$ , we write

(3.37)

\displaystyle f[t]:=\alpha_{t}(\mathcal{A}_{s}(t,f)).

Note in particular that $f[0]\equiv\mathcal{A}_{s}(0,f)$ .

To begin with, we claim the following holds: There exist $\tilde{\xi}_{k}\in\mathcal{X}$ , $2\leq k\leq n$ (dropped for $n=1$ ), depending only on $n,\vec{\kappa},\chi$ , and $\delta$ , such that for all $t\geq 0\,,s>0$ ,

(3.38)

\displaystyle\int_{0}^{t}\chi^{\prime}[t]dr\leq C\left(s\chi[0]+\sum_{k=2}^{n}s^{-k+2}\tilde{\xi}_{k}[0]+ts^{-n}\right),

where the sum is dropped if $n=1$ .

To prove (3.38), we bootstrap the recursive monotonicity estimate (3.5). For each fixed $s$ , integrating formula (3.4) with $A(t)\equiv\mathcal{A}_{s}(t,\chi)$ in $t$ gives

(3.39)

\displaystyle\chi[t]-\int_{0}^{t}\partial_{r}\chi[r]\,dr=\chi[0].

We apply inequality (3.5) to the second term on the l.h.s. of (3.39) to obtain, after transposing the $O(s^{-1})$ -term,

(3.40)

\displaystyle\chi[t]+s^{-1}\delta\int_{0}^{t}\chi^{\prime}[r]\,dr\leq\chi[0]+\sum_{k=2}^{n}s^{-k}\int_{0}^{t}\xi_{k}^{\prime}[r]\,dr+C{ts^{-(n+1)}},

where $\delta=c-\kappa$ , $\xi_{k}=\xi_{k}(n,\vec{\kappa},\chi)\in\mathcal{X}$ , and the second term in the r.h.s. is dropped for $n=1$ .

Since $s\,,\delta>0$ , estimate (3.40) implies, after dropping $\chi[t]$ on the l.h.s., which is non-negative-definite due to the positive-preserving property of the evolution $\alpha_{t}$ , and multiplying both sides by $s\delta^{-1}>0$ , that

(3.41)

\displaystyle\int_{0}^{t}\chi^{\prime}[r]\,dr\leq\frac{1}{\delta}\left(s\chi[0]+\sum_{k=2}^{n}s^{-k+1}\int_{0}^{t}\xi^{\prime}_{k}[r]\,dr+Cts^{-n}\right),

where the second term in the r.h.s. is dropped for $n=1$ .

If $n=1$ , then (3.41) gives (3.38). If $n\geq 2$ , we proceed to apply (3.41) to the term $\int_{0}^{t}\xi^{\prime}_{2}[r]\,dr$ up to $(n-1)$ -th order to get

(3.42)

\int_{0}^{t}\xi_{2}^{\prime}[r]\,dr\leq\frac{1}{\delta}\left(s\xi_{2}[0]+\sum_{k=2}^{n-1}s^{-k+1}\int_{0}^{t}\eta_{k}^{\prime}[r]\,dr+Cts^{-(n-1)}\right),

where the sum is dropped for $n=2$ and, for $n\geq 3$ ,

\eta_{k}=\eta_{k}(n,\vec{\kappa},\xi_{2})\in\mathcal{X},\quad k=2,\ldots,n-1.

Plugging (3.42) back to (3.41), we find

(3.43)

\displaystyle\int_{0}^{t}\chi^{\prime}[r]\,dr\leq

\displaystyle\frac{1}{\delta}\left(s\chi[0]+\frac{1}{\delta}\xi_{2}[0]+\sum_{k=3}^{n}s^{-k+1}\int_{0}^{t}\rho_{k}^{\prime}[r]\,dr+\left(1+\frac{1}{\delta}\right)Cts^{-n}\right),

where the third term in the r.h.s. is dropped for $n=2$ and the functions $\rho_{k}\in\mathcal{X}$ , $\rho_{k}^{\prime}\geq\xi_{k}^{\prime}+\tfrac{1}{\delta}\eta_{k}^{\prime}$ for $k=3,\ldots,n$ (see (X2)). Bootstrapping this procedure, we arrive at (3.38) for $n\geq 2$ .

2. Now we use (3.38) to derive the desire estimate (ME).

Dropping the second term in the l.h.s. of (3.40), which is non-negative since $\delta>0$ and $\chi^{\prime}[r]\geq 0$ for all $r$ , we obtain

(3.44)

\displaystyle\chi[t]\leq\chi[0]+\sum_{k=2}^{n}s^{-k}\int_{0}^{t}\xi_{k}^{\prime}[r]\,dr+C{ts^{-(n+1)}},

where the second term is dropped for $n=1$ (in which case we are done). If $n\geq 2$ , then for each $k=2,\ldots,n$ , we apply estimate (3.38) to the $k$ -th summand in the second term in the r.h.s. of (3.44), with remainder expanded to $(n-k+1)$ -th order. This way we obtain

(3.45)

\displaystyle\chi[t]\leq\chi[0]+C\left[\sum_{k=2}^{n}\sum_{l=2}^{n-k}s^{-(k-1)}\left(\tilde{\xi}_{k}[0]+s^{-(l+k-2)}\tilde{\xi}_{k,l}[0]\right)\right]+Cts^{-(n+1)}.

where the $k$ -sum is dropped for $n=1$ , the $l$ -sum is dropped if $n-k\leq 1$ , and $C$ , $\tilde{\xi}_{k,l}$ are chosen according to (3.38).

Using property (X2), we can choose $\xi\in\mathcal{X}$ such that for $C,\,\tilde{\xi}_{k},\,\tilde{\xi}_{k,l}$ as in (3.45),

(3.46)

\displaystyle\xi\geq C\left[\sum_{k=2}^{n}\sum_{l=2}^{n-k}\left(\tilde{\xi}_{k}[0]+\tilde{\xi}_{k,l}[0]\right)\right].

With this choice of $\xi$ , we conclude the desired estimate, (3.36), from (3.45). This completes the proof of Theorem 3.3. ∎

3.3. Proof of Theorem 2.4

Recall that $\phi$ is a densely defined self-adjoint operator on $\mathfrak{h}$ and $P_{{a}}$ denotes the spectral cutoff operator defined in (2.3). Our goal now is to choose a function $s=s(t)$ s.th. the geometric inequalities (2.11)–(2.12) hold, whereby eliminating the adiabatic parameter $s$ and the ASTLOs from (ME) so as to conclude the desired estimate (2.4).

Our main result is the following proposition:

Proposition 3.4.

Let $\delta,\,c^{\prime}>0$ . For functions $f(t)>c^{\prime}\left\lvert t\right\rvert$ and $\eta\in C^{1}\cap L^{\infty}(\mathbb{R},\mathbb{R}_{\geq 0})$ with

(3.47)

\eta\not\equiv 0,\quad\operatorname{supp}\eta\subset(0,\infty),\quad\operatorname{supp}\eta^{\prime}\subset(0,\delta),

let

(3.48)

\displaystyle s:=\delta^{-1}(f(t)-c^{\prime}\left\lvert t\right\rvert),\quad\mathcal{A}(t,\eta):=\eta(s^{-1}(\phi-c^{\prime}\left\lvert t\right\rvert)).

Then the following estimates hold:

(3.49)			$\displaystyle\left\lVert\eta\right\rVert_{L^{\infty}}^{-1}\mathcal{A}(0,\eta)\leq P_{{0}},$
(3.50)			$\displaystyle P_{{f(t)}}\leq\left\lVert\eta\right\rVert_{L^{\infty}}^{-1}\mathcal{A}(t,\eta)\quad(t\neq 0).$

Proof.

First, by (3.47), we have $\operatorname{supp}\eta\big{(}\frac{\cdot}{s}\big{)}\subset(0,\infty)$ for $s>0$ . This implies

(3.51)

\displaystyle\left\lVert\eta\right\rVert_{L^{\infty}}^{-1}\mathcal{A}(0,\eta)\equiv\left\lVert\eta\right\rVert_{L^{\infty}}^{-1}\eta\left(\phi/s\right)\leq\theta(\phi)\equiv P_{{0}},

where $\theta:\mathbb{R}\to\mathbb{R}$ is the characteristic function of the half-line $(0,\infty)$ (see Figure 3). Thus (3.49) follows.

Figure 3. Schematic diagram illustrating (3.51)

Next, again by (3.47), we have $\left\lVert\eta\right\rVert_{L^{\infty}}^{-1}\eta(\mu)\equiv 1$ for $\mu>\delta$ and so, by definition (3.48),

(3.52)

\displaystyle\left\lVert\eta\right\rVert_{L^{\infty}}^{-1}\mathcal{A}(t,\eta)\equiv\left\lVert\eta\right\rVert_{L^{\infty}}^{-1}\eta\left(\delta\frac{\phi-c^{\prime}\left\lvert t\right\rvert}{f(t)-c^{\prime}\left\lvert t\right\rvert}\right)\equiv\mathbf{1},

on the subspace $\operatorname{Ran}P_{{f(t)}}$ . Since $P_{{f(t)}}\equiv\theta(\phi-f(t))$ , estimate (3.52) implies

(3.53)

\displaystyle\left\lVert\eta\right\rVert_{L^{\infty}}^{-1}\mathcal{A}(t,\eta)\geq\theta(\phi-f(t)),

see Figure 4. Thus (3.50) follows.

Figure 4. Schematic diagram illustrating (3.53).

This completes the proof of Proposition 3.4. ∎

We now use Proposition 3.4 and Theorem 2.3 to prove Theorem 2.4.

First, for $c>\kappa$ as in the statement of Theorem 2.4, we set

(3.54)

\displaystyle\delta:=\frac{1}{3}(c-\kappa)>0,\quad c^{\prime}:=\kappa+\delta.

Fix any $\chi\in\mathcal{X}$ . We apply Theorem 2.3 with $c^{\prime}>\kappa$ to get a constant $C>0$ and a function $\xi\in\mathcal{X}$ such that

(3.55)

\displaystyle\alpha_{t}\left(\mathcal{A}_{s}(t,\chi)\right)\leq{\mathcal{A}_{s}(0,\chi)}+Cs^{-1}{\mathcal{A}_{s}(0,\xi)}+Cs^{-1}\left(s^{-n}\left\lvert t\right\rvert+C_{V}\right).

Next, we apply Proposition 3.4 with

(3.56)

\displaystyle f(t):=c\left\lvert t\right\rvert>c^{\prime}\left\lvert t\right\rvert,\quad s:=\delta^{-1}(c-c^{\prime})\left\lvert t\right\rvert>\left\lvert t\right\rvert,

where the inequalities are ensured by the choice (3.54). The function $\chi$ clearly satisfies condition (3.47). If the function $\xi\not\equiv 0$ in (3.55), then $\xi$ also satisfy (3.47). (If $\xi\equiv 0$ then we drop the second term in the r.h.s. of (3.55)). Hence, applying (3.49)–(3.50) with $\eta=\chi,\xi$ and $\mathcal{A}\equiv\mathcal{A}_{s}$ as in (3.48), we conclude the desired estimate, (2.4), from estimate (3.55).

This completes the proof of Theorem 2.4.∎

4. Applications to nonlocal dispersive equations

In this section, we apply the general localization theory laid out in Section 2 to study a large class of nonlocal dispersive evolution models.

We consider the following nonlocal non-autonomous Schrödinger equation:

(4.1)

i\partial_{t}\psi=H(t)\psi.

Here $\psi=\psi(\cdot,t),\,t\in\mathbb{R}$ is a differentiable path of vectors in the Hilbert space $\mathfrak{h}:=L^{2}(\mathbb{R}^{d},\mathbb{C}),\,d\geq 1$ . The Hamiltonian $H(t)=H_{0}+V(t)$ consists of a nonlocal part

(4.2)

H_{0}[\psi](x)=p.v.\int_{y\in\mathbb{R}^{d}}(\psi(x)-\psi(y))K(x,y),

for some symmetric (and possibly singular) integral kernel $K$ with $K(y,x)=\overline{K(x,y)}$ , together with a time-dependent potential $V(t).$

As a standing assumption, we assume that $H_{0}$ is self-adjoint on a dense domain $\mathcal{D}\equiv\mathcal{D}(H_{0})\subset\mathfrak{h}$ and $V(t)$ is uniformly bounded for all $t$ . Consequently, $H(t)$ is self-adjoint on $\mathcal{D}$ and so, by standard perturbation theory, admits bounded propagator $U(t,s)$ with $t,s\in\mathbb{R}$ (see e.g. [GS, Theorem 25.32]).

Our main technical assumption is the following: For some integer $n\geq 1$ and function $\phi\in\dot{W}^{1,\infty}$ (i.e., weakly differentiable with $\nabla\phi\in L^{\infty}),$ the operators

(4.3)

B_{p}[f](x):=\int_{y\in\mathbb{R}^{d}}{K(x,y)}(\phi(x)-\phi(y))^{p}f(x)

satisfy, for $p=1,\ldots,n+1$ and some $\kappa_{p}>0$ ,

(4.4)

\displaystyle\left\lVert B_{p}\right\rVert_{L^{2}\to L^{2}}\leq\kappa_{p}.

We show in Appendix A that condition (4.4) amounts to the main technical condition (C2) in the general theory (see Section 2.1) and that a sufficient condition for (4.4) is

(4.5)

\max_{1\leq p\leq n+1}\sup_{x\in\mathbb{R}^{d}}\int_{y\in\mathbb{R}^{d}}\left\lvert K(x,y)\right\rvert\left\lvert x-y\right\rvert^{p}<\infty.

Typical examples of the form (4.2) satisfying (4.5) include the nonlocal diffusion operators

(4.6)

H_{0}=1-J*,

where $J$ is a radial function with profile satisfying

(4.7)

\sup_{1\leq p\leq n+1}\int_{0}^{\infty}r^{p+d-1}\left\lvert J(r)\right\rvert\,dr<\infty,

e.g., $J(x)=(1+\left\lvert x\right\rvert^{2})^{-a/2}$ with $a>d+n+1$ . By interpolation, mild singularity is allowed at $0$ , e.g., $J(x)=O(|x|^{-b})$ with $b<d+1$ .

Condition (4.4) are also verified by certain fractional differential operators. In his seminal work [MR0177312], Calderón proved that (4.4) holds for $p=1$ and $H_{0}=(-\Delta)^{1/2}$ , or equivalently,

(4.8)

\displaystyle K(x,y)=\frac{1}{\left\lvert x-y\right\rvert^{d+1}}.

The boundedness of commutators of more general singular integral operators and fractional elliptic operators are subsequently established in [MR0412721, MR0358205, MR0763911, MR3286493, MR3547014, MR4443495, MR3555319], among many others, under various conditions on $H_{0}$ , $K$ and for various classes of functions $\phi$ (typically belonging to $\dot{W}^{1,\infty}$ or BMO). As the scheme below indicates, boundedness of singular integral operators of the form (4.3) would lead to similar propagation estimates in the corresponding dynamical models.

Evolution equations involving nonlocal operators of the form (4.2) have received much research attention in recent years. For recent results concerning evolution equations involving (4.6) subject to similar conditions as (4.7), see e.g. [MR2257732, MR2542582, MR3285829, MR3289358, MR3440113, MR4182983] and, for applications to natural sciences, [MR3469920, MR4409816], as well as the references therein. For regularity theory of nonlocal evolution equations, see [MR3626038, MR3771838, MR3959442]. For an excellent recent review on nonlocal diffusion operators with integrable kernels, see [MR4187861].

Note however that all of the cited works above are concerned with, instead of Hamiltonian evolution equation as in (4.1), gradient flows of the form $\partial_{t}\psi=-H\psi$ with $H$ of the form (4.2). This distinction should be made clear since the Hamiltonian structure of (4.1) is used crucially in proving the recursive monotonicity estimate (RME) for $\mathcal{A}_{s}(t,\chi)$ (wherefore in all us results in Section 2 as well). See Remark 9 below for a discussion.

Eq. (4.1) arises, among others, from the study of nonlinear nonlocal Schrödinger (NLS) equations of the form

(4.9)

i\partial_{t}\psi=H_{0}\psi+W\psi+f(\left\lvert\psi\right\rvert^{2})\psi,\quad f\in C(\mathbb{R}_{\geq 0},\mathbb{R}),

where $W$ is a bounded external potential (possibly time-dependent). Eq. (4.9) has a Hamiltonian structure inherited from the nonlocal generalization of the Ginzburg-Landau free-energy functional in the presence of external potential:

E(\psi)=\frac{1}{4}\iint K(x,y)\left\lvert\psi(x)-\psi(y)\right\rvert^{2}+\int W(x)\left\lvert\psi(x)\right\rvert^{2}+F(\left\lvert\psi(x)\right\rvert^{2}),\quad F^{\prime}=f.

Indeed, if $\psi^{(0)}_{t}\in L^{\infty}\cap L^{2}$ solves (4.9), then $\psi^{(0)}_{t}$ satisfies (4.1) with $V(t):=W+f(\lvert{\psi^{(0)}_{t}}\rvert)$ bounded for all $t$ . This convolution-type model for phase transitions was proposed in [MR1463804] and the associated $L^{2}$ -gradient flow (the nonlocal Allen-Cahn equation) has been studied in [MR1463804, MR1712445, MR1933014, MR2257732, MR2542582]. See [MR1712445, Sect. 1] for a discussion on the connection between $E(\psi)$ above and the classical Ginzburg-Landau energy functional.

4.1. Results

Under the standing assumption, the evolution of a state $\psi_{s}\in\mathcal{D}$ from time $s$ according to (4.1) is given by

(4.10)

\psi_{t}=U(t,s)\psi_{s},

where $U(t,s),\,s,t\in\mathbb{R}$ is the propagator for $H(t)=H_{0}+V(t)$ in (4.1). The evolution of an observable $A$ , dual to the evolution of states $\psi_{s}\mapsto\psi_{t}=U(t,s)\psi_{s}$ w.r.t. the coupling $(A,\psi_{s})\mapsto\left\langle\psi_{s},\,A\psi_{s}\right\rangle$ , is given by

(4.11)

\alpha_{t,s}(A):=U(t,s)^{*}AU(t,s),

where $U(t,s)^{*}$ is the backward propagator.

For $\psi_{0}\in\mathcal{D}$ and $A\in\mathcal{B}(\mathfrak{h})$ , we denote by $\psi_{t}=U(t,0)\psi_{0}$ and $\alpha_{t}(A)=\alpha_{t,0}(A)$ the evolution of states and observables, respectively. Let $\kappa=\kappa_{1}$ and $\vec{\kappa}=(\kappa_{1},\ldots,\kappa_{n+1})$ be as in (4.4). Our main result in this section is the following:

Theorem 4.1.

Suppose (4.4) holds for $n\geq 1$ and $\phi\in\dot{W}^{1,\infty}$ . Then, for every $c>\kappa$ , there exists $C=C(n,c,\left\lVert\nabla\phi\right\rVert_{L^{\infty}},\vec{\kappa})>0$ such that for any function $f(t)>c\left\lvert t\right\rvert$ and $t\neq 0$ ,

	$\displaystyle\left\lVert{\bf 1}_{\left\{x\mid\phi(x)>f(t)\right\}}\psi_{t}\right\rVert^{2}\leq$	$\displaystyle(1+C(f(t)-c\left\lvert t\right\rvert)^{-1})\left\lVert{\bf 1}_{\left\{x\mid\phi(x)>0\right\}}\psi_{0}\right\rVert^{2}$
(4.12)			$\displaystyle+C\left\lvert t\right\rvert(f(t)-c\left\lvert t\right\rvert)^{-(n+1)}\left\lVert\psi_{0}\right\rVert^{2}.$

Proof.

We derive estimate (4.14) as a consequence of Thms. 2.2–2.3 and Proposition 3.4. Fix $t\neq 0$ and $\chi\in\mathcal{X}$ (see (3.2)) with $\chi(\mu)\equiv 1$ for $\mu\geq 1$ . Below, all estimates are independent of these parameters.

First, we verify the assumptions of Theorem 2.2. Since $H=H_{0}+V$ in (4.1) with $[V,\phi]=0$ , the evolution condition (H) is satisfied with $H_{0}$ given by (4.2). By Lemma A.1, the Hamiltonian $H_{0}$ from (4.2) and $\phi$ verify the commutator condition (C2), with $\kappa_{p}$ depending on $\operatorname{\mathrm{Lip}}(\phi)$ . We have shown that the assumptions of Theorem 2.2 hold. Thus, by Thms. 2.2–2.3, estimate (ME) holds.

Next, define $s=s(t):=f(t)-c\left\lvert t\right\rvert>0$ and denote by $\mathcal{A}(t,\chi)\equiv\mathcal{A}_{s}(t,\chi)$ the ASTLOs from (3.1) with this choice of $s$ . Then, by estimate (ME), there exists a constant $C>0$ and a function $\xi\in\mathcal{X}$ such that

	$\displaystyle\left\langle\psi_{t},\,{\mathcal{A}(t,\chi)}\psi_{t}\right\rangle\leq$	$\displaystyle\left\langle\psi_{0},\,{\mathcal{A}(0,\chi)}\psi_{0}\right\rangle+(f(t)-c\left\lvert t\right\rvert)^{-1}\left\langle\psi_{0},\,{\mathcal{A}_{s}(0,\xi)}\psi_{0}\right\rangle$
(4.13)			$\displaystyle+C\left\lvert t\right\rvert(f(t)-c\left\lvert t\right\rvert)^{-(n+1)}\left\lVert\psi_{0}\right\rVert^{2}.$

Lastly, we use Proposition 3.4. The function $\chi$ clearly satisfies condition (3.47). If the function $\xi\not\equiv 0$ in (4.1), then $\xi$ also satisfy (3.47). (If $\xi\equiv 0$ then we drop the second term in the r.h.s. of (4.1)). Hence, applying (3.49)–(3.50) with $\eta=\chi,\xi$ in (4.1) and using that $P_{{a}}\equiv{\bf 1}_{\left\{x\mid\phi(x)>a\right\}}$ for all $a>0$ (see (2.3)), we conclude the desired estimate, (4.14), from estimate (4.1). This completes the proof of Theorem 4.1. ∎

Theorem 4.1 grants control over the localization of states $\psi_{t}$ w.r.t. to a fixed reference geometry, $\phi(x)$ , and a height function $f(t)$ . The growth rate of $f(t)$ in turn determines the decay estimate of the probability leakage as in (4.1).

Specifically, let $X\subset\mathbb{R}^{d}$ and $d_{X}(x)=\inf_{y\in X}\left\lvert x-y\right\rvert$ . Taking $\phi=d_{X}$ and using the facts that $\left\lVert\nabla d_{X}\right\rVert_{L^{\infty}}\leq 1$ , ${\bf 1}_{\left\{d_{X}>c\left\lvert t\right\rvert\right\}}\equiv{\bf 1}_{X_{c\left\lvert t\right\rvert}^{\mathrm{c}}}$ , we conclude from Theorem 4.1 that

Corollary 4.1 (Localization of scattering states).

Suppose (4.4) holds for $n\geq 1$ and $\phi=d_{X}$ . Then, for every $c>\kappa$ , there exists $C=C(n,c,\vec{\kappa})>0$ such that for all subset $X\subset\mathbb{R}^{d}$ , functions $f(t)>c\left\lvert t\right\rvert$ , and $t\neq 0$ ,

(4.14)

\left\lVert\mathbf{1}_{X_{f(t)}^{\mathrm{c}}}\psi_{t}\right\rVert^{2}\leq(1+C(f(t)-c\left\lvert t\right\rvert)^{-1})\left\lVert\mathbf{1}_{X^{\mathrm{c}}}\psi_{0}\right\rVert^{2}+C\left\lvert t\right\rvert(f(t)-c\left\lvert t\right\rvert)^{-(n+1)}\left\lVert\psi_{0}\right\rVert^{2}.

To see that (4.14) controls the localization of evolving states according to (4.1), fix $\epsilon>0$ and define $f(t)=(c+\epsilon)\left\lvert t\right\rvert$ . Assuming the initial condition $\psi$ at $t=0$ is localized in $X$ in the sense that $\left\lVert\mathbf{1}_{X^{\mathrm{c}}}\psi_{0}\right\rVert\leq\epsilon$ , we conclude from (4.14) that $\left\lVert\mathbf{1}_{X_{ct}^{\mathrm{c}}}\psi_{t}\right\rVert_{L^{2}}^{2}\lesssim\epsilon+\left\lvert t\right\rvert^{-1}+\epsilon^{-(n+1)}\left\lvert t\right\rvert^{-n}$ for all $t\neq 0$ .

As a consequence of the localization estimate (4.14), we have the following a priori estimate on the propagation speed of traveling wave solutions to the nonlinear nonlocal Schrödinger equation (4.9):

Corollary 4.2.

Suppose (4.5) holds for $n\geq 1$ . Suppose $\psi_{t}\in L^{2}\cap L^{\infty},\,t\geq 0$ solves the NLS equation (4.9) and $\psi_{t}=U(\cdot-\beta t)$ for some fixed velocity $\beta\in\mathbb{R}^{d}$ and profile $U$ with the following property: There exists a bounded subset $X\subset\mathbb{R}^{d}$ such that $\left\lVert\mathbf{1}_{X^{c}}U\right\rVert^{2}<\left\lVert U\right\rVert^{2}/2$ . Then $\left\lvert\beta\right\rvert\leq\kappa$ .

Proof.

Since $\psi_{t}$ solves (4.1), by freezing coefficients, $\psi_{t}$ satisfies (4.14) and therefore we have

(4.15)

\left\lVert\mathbf{1}_{X_{ct}^{\mathrm{c}}}U(x-\beta t)\right\rVert^{2}\leq\left\lVert U\right\rVert^{2}/2+Ct^{-n},

for all $c>\kappa$ . Suppose now $\left\lvert\beta\right\rvert>\kappa$ . Then, on the one hand, we can choose $c\in(\kappa,\left\lvert\beta\right\rvert)$ such that (4.15) holds. On the other hand, since $c<\left\lvert\beta\right\rvert$ , there is a large $T\gg 1$ depending only on $\left\lvert\beta\right\rvert-c$ and $\operatorname{diam}(X)$ such that

(4.16)

\left\lVert\mathbf{1}_{X_{ct}^{\mathrm{c}}}U(\cdot-\beta t)\right\rVert^{2}\geq\left\lVert\mathbf{1}_{X}U\right\rVert^{2}>\left\lVert U\right\rVert^{2}/2

for all $t\geq T$ (see Figure 5). This is a contradiction to (4.15).

Figure 5. Schematic diagram illustrating relation (4.16).

The proof of Corollary 4.2 is complete. ∎

5. Technical lemmas

5.1. Remainder estimates

In this section and the next one, we present some estimates and commutator expansions, first derived in [SigSof] and then improved in [HunSig3, Skib] etc. Below, we adapt some of the arguments from [HunSig3] and results from [MVBvNL].

Throughout this section we fix an integer $\nu\geq 0$ . For integers $p\geq 0$ and smooth functions $f\in C^{\nu+2}(\mathbb{R})$ , we define a weighted norm

(5.1)

\displaystyle\mathcal{N}(f,p):=\sum_{m=0}^{\nu+2}\int_{\mathbb{R}}\left\langle x\right\rangle^{m-p-1}\left\lvert f^{(m)}(x)\right\rvert\,dx.

Note that

(5.2)

\displaystyle p\leq p^{\prime}\implies\mathcal{N}(f,p^{\prime})\leq\mathcal{N}(f,p),

and we have the following property:

Lemma 5.1.

Let $p\geq 0$ be an integer. Suppose $f\in C^{\nu+2}(\mathbb{R})$ and there exist $C_{0},\,\rho>0$ such that for $m=0,\ldots,\nu+2$ ,

(5.3)

\displaystyle\left\lVert\left\langle x\right\rangle^{m-p+\rho}f^{(m)}(x)\right\rVert_{L^{\infty}}\leq C_{0}.

Then there exists $C>0$ depending only on $\rho,\,C_{0},\,\nu$ such that

(5.4)

\displaystyle\mathcal{N}(f,p)\leq C.

Proof.

We have

	$\displaystyle\mathcal{N}(f,p)\leq$	$\displaystyle\sum_{m=0}^{\nu+2}\left\lVert\left\langle x\right\rangle^{m-p+\rho}f^{(m)}(x)\right\rVert\int_{\mathbb{R}}\left\langle x\right\rangle^{-1-\rho}dx$
	$\displaystyle\leq$	$\displaystyle(\nu+3)C_{0}\int_{\mathbb{R}}\left\langle x\right\rangle^{-1-\rho}dx,$

and the integral converges for $\rho>0$ . ∎

Write $z=x+iy\in\mathbb{C}$ and $\partial_{\bar{z}}=\partial_{x}+i\partial_{y}$ . In what follows, as in [HunSig3, eq.(B.5)], for $f\in C^{\nu+2}(\mathbb{R})$ , we take $\tilde{f}(z)$ to be an almost analytic extension of $f$ defined by

(5.5)

\widetilde{f}(z):=\eta\left(\frac{y}{\left\langle x\right\rangle}\right)\sum_{k=0}^{\nu+1}f^{(k)}(x)\frac{(iy)^{k}}{k!},

where $\eta\in C_{c}^{\infty}(\mathbb{R})$ is a cutoff function with $\eta(\mu)\equiv 1$ for $\left\lvert\mu\right\rvert\leq 1$ , $\eta(\mu)\equiv 0$ for $\left\lvert\mu\right\rvert\geq 2$ , and $\left\lvert\eta^{\prime}(\mu)\right\rvert\leq 1$ for all $\mu$ . This $\widetilde{f}(z)$ induces a measure on $\mathbb{C}$ as

(5.6)

\displaystyle d\widetilde{f}(z):=-\frac{1}{2\pi}\partial_{\bar{z}}\widetilde{f}(z)dx\,dy.

In the remainder of this section, we derive integral estimate for various functions against the measure (5.6).

The next result is obtained by adapting the argument in [HunSig3, Lem. B.1]:

Lemma 5.2 (Remainder estimate).

Let $0\leq p\leq\nu$ . Let $f\in C^{\nu+2}(\mathbb{R})$ satisfy (5.4). Then the extension $\widetilde{f}$ from (5.5) satisfies the following estimate for some $C=C(f,\nu,p)>0:$

(5.7)

\displaystyle\int\left\lvert d\widetilde{f}(z)\right\rvert\left\lvert\mathrm{Im}(z)\right\rvert^{-(p+1)}\leq C.

Proof.

Differentiating formula (5.5), we obtain the estimate

(5.8)

\displaystyle\left\lvert\partial_{\bar{z}}\widetilde{f}(z)\right\rvert\leq\eta\left(\frac{y}{\left\langle x\right\rangle}\right)\frac{\left\lvert y\right\rvert^{\nu+1}}{(\nu+1)!}\left\lvert f^{(\nu+2)}(x)\right\rvert+\sum_{k=0}^{\nu+1}\rho\left(\frac{y}{\left\langle x\right\rangle}\right)\frac{\left\lvert y\right\rvert^{k}}{k!}\left\lvert\frac{1}{\left\langle x\right\rangle}f^{(k)}(x)\right\rvert,

where

(5.9)

\displaystyle\rho(\mu):=\left\lvert\eta^{\prime}(\mu)\right\rvert\left\langle\mu\right\rangle

is supported on $1<\left\lvert\mu\right\rvert<2$ .

For each fixed $x$ , we define

(5.10)

\displaystyle G(x):=p.v.\int\left\lvert\partial_{\bar{z}}f(z)\right\rvert\left\lvert y\right\rvert^{-(p+1)}\,dy

by integrating (5.8) against $\left\lvert y\right\rvert^{-(p+1)}$ . Using that $\eta(y/\left\langle x\right\rangle)\equiv 0$ for $\left\lvert y\right\rvert>\left\langle x\right\rangle$ and $\rho(y/\left\langle x\right\rangle)\equiv 0$ for $\left\lvert y\right\rvert\leq\left\langle x\right\rangle$ or $\left\lvert y\right\rvert\geq 2\left\langle x\right\rangle$ , we find

(5.11)		$\displaystyle G(x)\leq$	$\displaystyle\int_{\left\lvert y\right\rvert\leq\left\langle x\right\rangle}\frac{\left\lvert y\right\rvert^{\nu-p}}{(\nu+1)!}\eta\left(\frac{y}{\left\langle x\right\rangle}\right)\,dy\left\lvert f^{(\nu+2)}(x)\right\rvert$
(5.12)			$\displaystyle+\sum_{k=0}^{\nu+1}\int_{\left\langle x\right\rangle<\left\lvert y\right\rvert<2\left\langle x\right\rangle}\rho\left(\frac{y}{\left\langle x\right\rangle}\right)\frac{\left\lvert y\right\rvert^{k-p-1}}{k!}\,dy\left\lvert\frac{1}{\left\langle x\right\rangle}f^{(k)}(x)\right\rvert.$

Since $0\leq\eta(\mu)\leq 1$ and $\nu\geq p$ , the integral in line (5.11) converges and can be bounded as

(5.13)

\displaystyle\int_{\left\lvert y\right\rvert\leq\left\langle x\right\rangle}\frac{\left\lvert y\right\rvert^{\nu-p}}{(p+1)!}\eta\left(\frac{y}{\left\langle x\right\rangle}\right)\,dy\left\lvert f^{(p+2)}(x)\right\rvert\leq\frac{2\left\langle x\right\rangle^{\nu-p+1}}{(p+1)!}\left\lvert f^{(p+2)}(x)\right\rvert.

To bound line (5.12), we use that $\rho(y/\left\langle x\right\rangle)<\sqrt{5}$ and $\left\lvert y\right\rvert^{k-p-1}\leq\left\langle x\right\rangle^{k-p-1}$ for $\left\langle x\right\rangle<\left\lvert y\right\rvert<2\left\langle x\right\rangle$ , $0\leq k\leq p+1$ (see (5.9)). Thus each integral in line (5.12) can be bounded as

		$\displaystyle\sum_{k=0}^{\nu+1}\int_{\left\langle x\right\rangle<\left\lvert y\right\rvert<2\left\langle x\right\rangle}\rho\left(\frac{y}{\left\langle x\right\rangle}\right)\frac{\left\lvert y\right\rvert^{k-p-1}}{k!}\,dy\left\lvert\frac{1}{\left\langle x\right\rangle}f^{(k)}(x)\right\rvert$
(5.14)		$\displaystyle\leq$	$\displaystyle{\sum_{k=0}^{p+1}\frac{4\sqrt{5}\left\langle x\right\rangle^{k-p-1}}{k!}\left\lvert f^{(k)}(x)\right\rvert+\sum_{k=p+1}^{\nu+1}\frac{\sqrt{5}\cdot 2^{k-p+1}\left\langle x\right\rangle^{k-p-1}}{k!}\left\lvert f^{(k)}(x)\right\rvert}.$

Combining (5.13)–(5.14) in (5.12), we conclude that

(5.15)

\displaystyle\left\lvert G(x)\right\rvert\leq CF(x),\quad F(x):=\sum_{m=0}^{\nu+2}\left\langle x\right\rangle^{m-p-1}\left\lvert f^{(m)}(x)\right\rvert.

Let $G_{\lambda}(x):=\mathbf{1}_{[-\lambda,\lambda]}G(x)$ with $\lambda>0$ . Then $G_{\lambda}\in L^{1}$ and $\left\lvert G_{\lambda}(x)\right\rvert\leq CF(x)$ for any $\lambda$ . By assumption (5.4) and definition(5.1), we have $\left\lVert F\right\rVert_{L^{1}}=\mathcal{N}(f,p)<\infty$ and so $F\in L^{1}$ . Therefore, sending $\lambda\to\infty$ and using the dominated convergence theorem yields $G\in L^{1}$ with

(5.16)

\displaystyle\left\lVert G\right\rVert_{L^{1}}\leq C\left\lVert F\right\rVert_{L^{1}}.

Recalling definition (5.10), we find $(2\pi)^{-1}\left\lVert G\right\rVert_{L^{1}}=$ l.h.s. of (5.7). Thus we conclude (5.7) from (5.16). ∎

5.2. Commutator expansions

In this section, we take $\widetilde{f}(z)$ , $d\widetilde{f}(z)$ to be as in (5.5)–(5.6).

We frequently use the following result, taken from [HunSig3, Lems. B.2]:

Lemma 5.3.

Let $f\in C^{\nu+2}(\mathbb{R})$ satisfy (5.4) for some $p\geq 0$ . Then for any self-adjoint operator $A$ on $\mathfrak{h}$ ,

(5.17)

\displaystyle\frac{1}{p!}f^{(p)}(A)=\int_{\mathbb{C}}d\widetilde{f}(z)(z-A)^{-(p+1)},

where the integral converges absolutely in operator norm and is uniformly bounded in $A$ .

Remark 10.

Condition (5.4) ensures that $f^{(p)}$ is bounded independent of $A$ and the remainder estimate in Lemma 5.2 ensures the norm convergence of the r.h.s. of (5.17).

We call equation (5.17) the Helffer-Sjöstrand (HS) representation. The HS representation (5.17), together with the remainder estimate (5.7), implies the following commutator expansion:

Lemma 5.4.

Let $n\geq 1$ . Let $f\in C^{n+3}(\mathbb{R})$ satisfy (5.4) with $p=1$ . Let $A$ be an operator on $\mathfrak{h}$ . Let $\phi$ be a densely defined self-adjoint operator on $\mathfrak{h}$ . Let $\mathcal{A}_{s}(f):=f(s^{-1}(\phi-\alpha))$ for some fixed $\alpha$ and all $s>0$ .

Suppose

(5.18)

B_{k}:=\operatorname{ad}^{k}_{\phi}(A)\in\mathcal{B}(\mathfrak{h})\quad(1\leq k\leq n+1).

Then $[A,\mathcal{A}_{s}(f)]\in\mathcal{B}(\mathfrak{h})$ , and we have the expansion

(5.19)		$\displaystyle[A,\mathcal{A}_{s}(f)]=$	$\displaystyle\sum_{k=1}^{n}(-1)^{k}\frac{s^{-k}}{k!}B_{k}\mathcal{A}_{s}(f^{(k)})+(-1)^{n+1}s^{-(n+1)}\mathrm{Rem}_{\rm left}(s)$
(5.20)		$\displaystyle=$	$\displaystyle\sum_{k=1}^{n}\frac{s^{-k}}{k!}\mathcal{A}_{s}(f^{(k)})B_{k}+s^{-(n+1)}\mathrm{Rem}_{\rm right}(s),$

where the remainders are defined by these relations and given explicitly by (5.28)–(5.29).

Moreover, there exists $c>0$ depending only on $n$ and $\mathcal{N}(f,n+1)$ , such that

(5.21)

\displaystyle\left\lVert\mathrm{Rem}_{\rm left}(s)\right\rVert_{\rm op}+\left\lVert\mathrm{Rem}_{\rm right}(s)\right\rVert_{\rm op}\leq

\displaystyle c\left\lVert B_{n+1}\right\rVert,

Remark 11.

Note that $f$ needs not to be bounded. By (5.3), it suffices for $f$ to have strictly sublinear growth.

Proof of Lemma 5.4.

Within this proof we write $R=(z-\phi_{s,\alpha})^{-1}$ with $\phi_{s,\alpha}=s^{-1}(\phi-\alpha)$ .

Since $R$ is bounded, it follows that

(5.22)

\big{[}A,R\big{]}=s^{-1}R\operatorname{ad}_{\phi}(A)R

holds in the sense of quadratic forms on $\mathcal{D}(A)$ . Since $\operatorname{ad}_{\phi}(A)$ is bounded by assumption, the r.h.s. of (5.22) is bounded and so $[A,R]$ extends to an bounded operator on $\mathfrak{h}$ . Using (5.22), we proceed by commuting successively the commutators $B_{k}:=\operatorname{ad}^{k}_{\phi}(A)$ to left and right, respectively. Iteratively, we obtain

	$\displaystyle[A,R]$
(5.23)	$\displaystyle=$	$\displaystyle\sum_{k=1}^{n}(-1)^{k}s^{-k}B_{k}R^{k+1}+(-1)^{n+1}s^{-(n+1)}RB_{n+1}R^{n+1}$
(5.24)	$\displaystyle=$	$\displaystyle\sum_{k=1}^{n}s^{-k}R^{k+1}B_{k}+s^{-(n+1)}R^{n+1}B_{n+1}R,$

which hold on all of $\mathfrak{h}$ since $B_{k}$ ’s are bounded operators by assumption (5.18).

Let $\eta^{\lambda}\in C_{c}^{\infty}(\mathbb{R})$ , $\lambda>0$ be cutoff functions with $\eta^{\lambda}(x)\equiv 1$ for $\left\lvert x\right\rvert\leq\lambda$ , $\eta(x)\equiv 0$ for $\left\lvert\mu\right\rvert\geq\lambda+1$ , and $\left\lVert\eta^{\lambda}\right\rVert_{C^{n+3}}\leq C$ for all $\lambda$ . Set $f^{\lambda}:=\eta^{\lambda}f$ . Since $f^{\lambda}\in C_{c}^{n+3}$ , it satisfies (5.4) for all $p\geq 0$ . (Note that $f$ itself, a priori, does not satisfy (5.4) with $p=0$ .) Thus the HS representation 5.17 holds with $p=0$ and so

(5.25)

\displaystyle[A,\mathcal{A}_{s}(f^{\lambda})]=\int d\widetilde{f^{\lambda}}(z)\big{[}A,R\big{]},

which holds a priori on $\mathcal{D}(A)$ . Plugging expansions (5.23)–(5.24) into (5.25) yields

	$\displaystyle\quad[A,\mathcal{A}_{s}(f)^{\lambda}]$
(5.26)		$\displaystyle=\sum_{k=1}^{n}(-1)^{k}\frac{s^{-k}}{k!}B_{k}\int d\widetilde{f^{\lambda}}(z)R^{k+1}+(-1)^{n+1}s^{-(n+1)}\mathrm{Rem}_{\rm left}^{\lambda}(s),$
(5.27)		$\displaystyle=\sum_{k=1}^{n}\frac{s^{-k}}{k!}\int d\widetilde{f^{\lambda}}(z)R^{k+1}B_{k}+s^{-(n+1)}\mathrm{Rem}_{\rm right}^{\lambda}(s),$

where

(5.28)		$\displaystyle\mathrm{Rem}_{\rm left}^{\lambda}(s)$	$\displaystyle=\int d\widetilde{f^{\lambda}}(z)RB_{n+1}R^{({n+1})},$
(5.29)		$\displaystyle\mathrm{Rem}_{\rm right}^{\lambda}(s)$	$\displaystyle=\int d\widetilde{f^{\lambda}}(z)R^{({n+1})}B_{n+1}R.$

Since the operator $B_{n+1}$ is bounded independent of $\lambda,\,z$ , and $\left\lVert R\right\rVert\leq\left\lvert\mathrm{Im}(z)\right\rvert^{-1}$ , we have

		$\displaystyle\left\lVert\mathrm{Rem}_{\rm left}^{\lambda}(s)\right\rVert_{\rm op}+\left\lVert\mathrm{Rem}_{\rm right}^{\lambda}(s)\right\rVert_{\rm op}$
	$\displaystyle\leq$	$\displaystyle 2\\|B_{n+1}\\|\int\|d\widetilde{f^{\lambda}}(z)\|\left\lVert R\right\rVert_{\mathrm{op}}^{n+2}$
(5.30)		$\displaystyle\leq$	$\displaystyle 2\\|B_{n+1}\\|\int\|d\widetilde{f^{\lambda}}(z)\|\|\mathrm{Im}(z)\|^{-(n+2)}.$

Similarly we could bound the sums in (5.26)–(5.27). Thus we see $[A,\mathcal{A}_{s}(f^{\lambda})]$ extends to a bounded operator on $\mathfrak{h}$ for each $\lambda$ .

By (5.2) and the assumption $\mathcal{N}(f,1)\leq C$ , $f$ satisfies condition (5.4) with $p=1,\ldots,n+1$ . Hence, sending $\lambda\to\infty$ in (5.26)–(5.29) and using (5.17) for $p=1,\ldots,n$ and remainder estimate (5.7) for $p=n+1$ , we conclude that $[A,\mathcal{A}_{s}(f)]\in\mathcal{B}(\mathfrak{h})$ and expansions (5.20) and estimate (5.21) hold.∎

The following lemma is a direct consequence of the estimates proved in Lemma 5.4:

Lemma 5.5.

Let $G(t)$ be defined as in (C1). Then for any $s>0$ , $\chi\in\mathcal{E}$ , the commutator $[\mathcal{A}_{s}(\chi),V(t)]$ extends to bounded operators for all times and satisfies, for some absolute constant $C>0$ ,

(5.31)

\displaystyle\left\lVert[V(t),\mathcal{A}_{s}(\chi)]\right\rVert\leq Cs^{-1}G(t)\qquad(t\in\mathbb{R}).

Proof.

By relations (5.22) and (5.25), we have for all $t\in\mathbb{R}$ that

(5.32)

\displaystyle[V(t),\mathcal{A}_{s}(\chi)]=s^{-1}\int d\widetilde{\chi^{\lambda}}(z)R\big{[}V(t),\phi\big{]}R.

This, together with the definition $G(t)\equiv\left\lVert[\phi,V(t)]\right\rVert$ , yields

(5.33)

\displaystyle\left\lVert[V(t),\mathcal{A}_{s}(\chi)]\right\rVert\leq s^{-1}G(t)\int|d\widetilde{\chi^{\lambda}}(z)||\mathrm{Im}(z)|^{-2}.

Since the integral in (5.33) is uniformly bounded by some absolute constant for all $\lambda$ , sending $\lambda\to\infty$ yields the desired estimate (5.31). ∎

Acknowledgments

The Author is supported by National Key R & D Program of China Grant 2022YFA100740, China Postdoctoral Science Foundation Grant 2024T170453, and the Shuimu Scholar program of Tsinghua University. He was also supported by DNRF Grant CPH-GEOTOP-DNRF151, DAHES Fellowship Grant 2076-00006B, DFF Grant 7027-00110B, and the Carlsberg Foundation Grant CF21-0680 during the completion of this paper. He thanks G. Grubb for introduction to the subject matter in Section 4, R. Frank for helpful discussion and pointing out reference [MR0177312], and C. Jiang for useful comments on earlier draft of the paper. He thanks S. Breteaux, J. Faupin, M. Lemm, C. Rubiliani, M. Sigal, and D. Ouyang for fruitful collaborations, and especially M. Lemm for pointing out (1.5) as well as other helpful comments.

Parts of this work were done while the Author was visiting MIT. Earlier version of parts of this work has appeared as a chapter in the Author’s PhD thesis at the University of Copenhagen.

Declarations

•

Conflict of interest: The Author has no conflicts of interest to declare that are relevant to the content of this article.
•

Data availability: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Appendix A Commutator estimates

In this appendix, we prove that condition (4.5) implies uniform estimates on multiple commutators with (multiplication operator by) Lipschitz functions. In particular, this implies that (4.4) holds with $H_{0}$ from (4.2) and $\phi=d_{X}$ .

Lemma A.1.

Let $n\geq 1$ . Suppose $A$ is an operator acting on $L^{2}(\mathbb{R}^{d})$ as

(A.1)

A[u](x)=\int_{\mathbb{R}^{d}}(V(x)u(x)-u(y))K(x,y)\,dy

for $V\in L^{\infty}(\mathbb{R}^{d})$ and integral kernel $K(x,y)$ satisfying

(A.2)

\displaystyle M:=

\displaystyle\sup_{1\leq p\leq n+1}\left(\sup_{x\in\mathbb{R}^{d}}\int_{y\in\mathbb{R}^{d}}\left\lvert K(x,y)\right\rvert\left\lvert x-y\right\rvert^{p}\right)\left(\sup_{y\in\mathbb{R}^{d}}\int_{x\in\mathbb{R}^{d}}\left\lvert K(x,y)\right\rvert\left\lvert x-y\right\rvert^{p}\right)<\infty.

Then for every Lipschtiz function $f$ on $\mathbb{R}^{d}$ such that for some $L>0$ ,

(A.3)

\left\lvert f(x)-f(x)\right\rvert\leq L\left\lvert x-y\right\rvert\quad(x,y\in\mathbb{R}^{d}),

there holds

(A.4)

\left\lVert\operatorname{ad}^{k}_{f}(A)\right\rVert\leq L^{k}M\quad(1\leq k\leq n+1).

Proof.

We first prove that for each fixed $f:\mathbb{R}^{d}\to\mathbb{C}$ and all $1\leq k\leq n+1$ , we have

(A.5)

\operatorname{ad}^{k}_{f}(A)[u]=-\int(f(y)-f(x))^{k}K(x,y)u(y)\,dy.

We prove this by a simple induction. Clearly, the $V$ term in (A.1) does not contribute to the commutators $\operatorname{ad}^{k}_{f}(A)$ , since $[V,f]\equiv 0$ . Hence below we take $V\equiv 0$ in (A.1).

For the base case $k=1$ , we compute, for fixed $f$ and every $u$ ,

	$\displaystyle A[fu](x)=$	$\displaystyle-\int K(x,y)f(y)u(y)\,dy,$
	$\displaystyle f(x)A[u](x)=$	$\displaystyle-\int f(x)K(x,y)u(y)\,dy.$

Taking the difference yields (A.5) with $k=1$ . Now assume (A.5) holds for $k$ . Then we have

	$\displaystyle\operatorname{ad}^{k}_{f}(A)[fu](x)=-$	$\displaystyle\int(f(y)-f(x))^{k}K(x,y)f(y)u(y)\,dy,$
	$\displaystyle f(x)\operatorname{ad}^{k}_{f}(A)[u]=-$	$\displaystyle\int f(x)(f(y)-f(x))^{k}K(x,y)u(y)\,dy.$

Since $\operatorname{ad}^{k+1}_{f}(A)=[\operatorname{ad}^{k}_{f}(A),f]$ , taking the difference of the last two expressions yields (A.5) for $k+1$ . This completes the induction.

Formula (A.5), together with the Schur test for integral operators, implies

(A.6)

\displaystyle\left\lVert\operatorname{ad}^{k}_{f}(A)\right\rVert^{2}\leq

\displaystyle\left(\sup_{x\in\mathbb{R}^{d}}\int_{y\in\mathbb{R}^{d}}\left\lvert K(x,y)\right\rvert\left\lvert f(x)-f(y)\right\rvert^{k}\right)\left(\sup_{y\in\mathbb{R}^{d}}\int_{x\in\mathbb{R}^{d}}\left\lvert K(x,y)\right\rvert\left\lvert f(x)-f(y)\right\rvert^{k}\right).

Now we compute, using assumptions (A.2) and (A.3), that

\displaystyle\sup_{x\in\mathbb{R}^{d}}\int_{y\in\mathbb{R}^{d}}\left\lvert K(x,y)\right\rvert\left\lvert f(x)-f(y)\right\rvert^{k}\leq L^{k}\sup_{x\in\mathbb{R}^{d}}\int_{y\in\mathbb{R}^{d}}\left\lvert K(x,y)\right\rvert\left\lvert x-y\right\rvert^{k}\leq L^{k}M.

This bounds the first term in the r.h.s. of (A.6). Similarly we can derive the same bound for the second term in the r.h.s. of (A.6). Plugging the results back to (A.6) yields estimate (A.4). ∎

Spectral localization estimates for abstract linear Schrödinger equations

Spectral localization estimates for abstract linear Schrödinger equations

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

1.1. Organization of the paper

Notation

2. Setup and main result

2.1. Main result

Theorem 2.1 (Spectral localization).

Remark 1.

Remark 2.

Remark 3.

Remark 4.

2.2. Literature review and discussion

2.3. Key steps for proving Theorem 2.1

2.3.1. Construction of ASTLOs.

Remark 5.

2.3.2. Differential inequality for ASTLOs.

Theorem 2.2 (Recursive monotonicity estimate).

Remark 6.

Remark 7.

2.3.3. Monotonicity estimate for ASTLOs.

Theorem 2.3 (monotonicity estimate).

Remark 8.

2.3.4. Concluding estimate (2.4).

Theorem 2.4.

Proof of Theorem 2.1.

2.4. Discussion

3. Proofs of Theorems 2.2–2.4

3.1. Proof of Theorem 2.2

Theorem 3.1.

Remark 9.

Lemma 3.2.

Proof.

Proof of Theorem 3.1.

3.2. Proof of Theorem 2.3

Theorem 3.3.

Proof of Theorem 3.3.

3.3. Proof of Theorem 2.4

Proposition 3.4.

Proof.

4. Applications to nonlocal dispersive equations

4.1. Results

Theorem 4.1.

Proof.

Corollary 4.1 (Localization of scattering states).

Corollary 4.2.

Proof.

5. Technical lemmas

5.1. Remainder estimates

Lemma 5.1.

Proof.

Lemma 5.2 (Remainder estimate).

Proof.

5.2. Commutator expansions

Lemma 5.3.

Remark 10.

Lemma 5.4.

Remark 11.

Proof of Lemma 5.4.

Lemma 5.5.

Proof.

Acknowledgments

Declarations

Appendix A Commutator estimates

Lemma A.1.

Proof.

References