MAGNETIC SCHR ODINGER OPERATORS AND LANDSCAPE FUNCTIONS JEREMY G. HOSKINS HADRIAN QUAN AND STEFAN STEINERBERGER

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MAGNETIC SCHR ¨

ODINGER OPERATORS

AND LANDSCAPE FUNCTIONS

JEREMY G. HOSKINS, HADRIAN QUAN, AND STEFAN STEINERBERGER

Abstract. We study localization properties of low-lying eigenfunctions of

magnetic Schr¨odinger operators

2(−i∇ − A(x))2φ+V(x)φ=λφ,

where V: Ω →R≥0is a given potential and A: Ω →Rdinduces a mag-

netic ﬁeld. We extend the Filoche-Mayboroda inequality and prove a reﬁned

inequality in the magnetic setting which can predict the points where low-

energy eigenfunctions are localized. This result is new even in the case of

vanishing magnetic ﬁeld. Numerical examples illustrate the results.

1. Introduction

1.1. Introduction. A fundamental problem in mathematics and physics is to un-

derstand the behavior of low-energy eigenstates of Schr¨odinger operators. Given

an open and bounded domain Ω ⊂Rd,and a potential V: Ω →R≥0, this amounts

to characterizing solutions of the equation

−∆φ(x) + V(x)φ(x) = λφ(x) in Ω

φ= 0 on ∂Ω

for small values of λ. Our primary focus in this paper is on the regime where V

oscillates rapidly at small scales. It is known that solutions of this equation may

become strongly localized [3]. Moreover, it is also understood that in this setting

the boundary conditions are not tremendously important. We mention that in the

case V= 0 it is possible to have fascinating localization phenomena that stem from

Neumann boundary conditions (see [13, 19, 30]) but these will not be discussed

here. Our main question is whether it is possible to predict eﬃciently, using only

V, where such highly-localized eigenfunctions will be concentrated.

1.2. Landscape. This question has received renewed attention in recent years after

the introduction of the landscape function by Filoche & Mayboroda [14]. They

deﬁne the landscape function as u:Ω:R→Rsatisfying

(−∆ + V)u= 1 in Ω

u= 0 on ∂Ω

and prove that uexerts pointwise control on all eigenfunctions (−∆ + V)φ=λφ

|φ(x)| ≤ λu(x)kφkL∞(Ω).

2010 Mathematics Subject Classiﬁcation. 35J10, 65N25 (primary), 82B44 (secondary).

Key words and phrases. Localization, Eigenfunction, Schr¨odinger Operator, Regularization.

S.S. is supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation.

arXiv:2210.02646v1 [math.AP] 6 Oct 2022

The landscape function turns out to have remarkable predictive power: the ﬁrst

few eigenfunctions tend to localize close to the maxima of u. Indeed, one can think

of 1/u as a suitable regularization of the potential Vwith the property that eigen-

functions localize in its local minima. This has inspired considerable subsequent

work, we refer to [4, 5, 6, 1, 2, 8, 9, 14, 15, 16, 18, 20, 21, 22, 24, 27].

1.3. Local Landscape. Another approach was proposed by the third author in

[32, 33]: there it was shown that, on suﬃciently small scales, there exists a canonical

smoothing of the potential Vgiven by the Wiener integral

Vt(x) = E1

tZt

V(ωx(s))ds,

where ωx(s) is a Brownian motion started at xat time s. As is shown in [33],

this molliﬁer arises naturally when looking for local approximations of the identity

with good error estimates. Moreover, there is a probability-free way of computing

this object. Since Brownian motion is, in free space, distributed according to a

Gaussian and since expectation is linear, we can express Vt∼V∗kt, where

kt(x) = 1

tZt

exp −kxk2/(4s)

(4πs)d/2ds

with an exponentially small error provided that tdist(x, Ωc)2. Note that con-

volution with ktcan be performed rapidly using the Fast Fourier Transform. The

local landscape is, in a suitable sense, the canonical molliﬁer for small t(see [33])

and was empirically shown to have predictive power roughly comparable to that of

the landscape function [25] for larger values of t.

2. Magnetic Schr¨

odinger Operators

Our goal is to extend and unify these two approaches for magnetic Schr¨odinger

operators. Let Ω ⊂Rdbe a bounded domain with smooth boundary (mainly for

ease of exposition, this could be relaxed). We consider the eigenvalue problem

2(−i∇ − A(x))2φ(x) + V(x)φ(x) = λφ(x)

where V: Ω →R≥0,A: Ω →Rdand φis subject to Dirichlet boundary condi-

tions. The main technical ingredient in our approach is a Feynman-Kac formula

for magnetic Schr¨odinger operators due to Broderix, Hundertmark & Leschke [7].

In particular, we will work with the regularity assumptions A∈C1(Rd,Rd) and

V∈C1(Rd,R≥0) which could be somewhat weakened (see [7] for a discussion).

In our setting of interest where Vis wildly oscillatory, one does not expect the

boundary conditions to have a signiﬁcant impact on the behavior of eigenfunctions

inside the domain. The only prior work in this direction is due to Poggi [28] who

proposes a diﬀerent extension of the landscape function leading to diﬀerent results.

2.1. A Magnetic Filoche-Mayboroda inequality. We start by extending the

Filoche-Mayboroda Landscape function to the magnetic setting. We consider the

solution of the equation, henceforth referred to as the landscape function,

−1

2∆ + Vu= 1 in Ω

u= 0 on ∂Ω.

The factor 1/2 in front of the Laplacian is equivalent to the usual Laplacian up

to scaling Vand λby a factor of 2: it is the probabilistic normalization of the

Laplacian chosen to facilitate comparison with the result in [7].

Theorem (Filoche-Mayboroda [14]).For any eigenfunction (−(1/2)∆+V)φ=λφ

subject to Dirichlet boundary conditions and all x∈Ω

|φ(x)|

kφkL∞≤λ·u(x).

Note that the quantity on the left-hand side always assumes the value 1 at the

global extrema of any eigenfunction. Therefore, eigenfunctions corresponding to

eigenvalue λcan only localize at points x∈Ω where u(x)·λ≥1. Our ﬁrst main

result shows that this inequality extends to magnetic Schr¨odinger operators.

Theorem 1. Let Ω⊂Rdbe a bounded domain with smooth boundary, let A∈

C1(Ω,Rd), let V∈C1(Ω,R≥0)and let φbe a solution of

2(−i∇ − A(x))2φ(x) + V(x)φ(x) = λφ(x) (1)

subject to Dirichlet boundary conditions. Then, for all x∈Ω,

|φ(x)|

kφkL∞≤λ·u(x).

An interesting aspect of the inequality is that the magnetic potential Adoes not

explicitly appear in the inequality (it appears implicitly in λ). Indeed, if Ais much

smoother than the potential V(in the sense of being virtually constant on the

scale of the eigenfunction), this is not particularly surprising since, intuitively, it

contributes only a phase modulation to the corresponding eigenfunction (see §3).

2.2. A Reﬁned Localization Inequality. We also prove a reﬁned localization

inequality which is stronger than Theorem 1 (and will imply Theorem 1).

Theorem 2. Under the same assumptions as Theorem 1, for all x∈Ωand t > 0,

|ψ(x0)|

kψkL∞≤EωχΩ(ωx0, t) exp λ·t−Zt

V(ωx0(s))ds

where expectation is taken with respect to all Brownian motions ωx(s)started at

ωx(0) = xrunning for tunits of time and

χΩ(ω, t) = (1if ∀0≤s≤t:ω(s)∈Ω

0otherwise.

Note that when tis suﬃciently small compared to the distance d(x0, ∂Ω) between

x0and the boundary of the domain (in particular, when td(x0, ∂Ω)2), then

virtually all Brownian motions will stay inside the domain for tunits of time: the

diﬀerence between having the term χΩ(ω, t) and omitting it is exponentially small.

One important fact is that Theorem 2 contains all the information contained in

Theorem 1 and more: this is made precise in the following Corollary.

Corollary. Under the same assumption as Theorem 1, we have

inf

t>0

EωχΩ(ωx0, t) exp λ·t−Zt

V(ωx0(s))ds≤λ·u(x).

We refer to §3 where this is illustrated on concrete examples.

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MAGNETICSCHRODINGEROPERATORSANDLANDSCAPEFUNCTIONSJEREMYG.HOSKINS,HADRIANQUAN,ANDSTEFANSTEINERBERGERAbstract.Westudylocalizationpropertiesoflow-lyingeigenfunctionsofmagneticSchrodingeroperators12(irA(x))2+V(x)=;whereV:!R0isagivenpotentialandA:!Rdinducesamag-neticeld.WeextendtheFiloche-Mayboro...

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MAGNETIC SCHR ODINGER OPERATORS AND LANDSCAPE FUNCTIONS JEREMY G. HOSKINS HADRIAN QUAN AND STEFAN STEINERBERGER

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