arXiv subjects

Peter D. Hislop

Publications and source records attributed to Peter D. Hislop.

At least 19 recordsLinked to original sources

arXiv2026

Ergodic Schrodinger operators on the Bethe lattice and a modified Thouless formula

The main result of this paper is a modified Thouless formula relating the density of states for ergodic Schrodinger operators on the Bethe lattice to the Lyapunov exponent. The modified Thouless formula consists of a Thouless-like term, involving the density of states, and a remainder term. The remainder term vanishes when the connectivity $κ$ equals one, yielding the usual Thouless formula for ergodic Schrodinger operators on $\mathbb{Z}$. We prove the remainder term is nontrivial for $κ\geq 2$. We also discuss the automorphism group of the Bethe lattice and its relation to ergodic Schrodinger operators. In particular, we clarify the use of the multiparameter noncommutative ergodic theorem in evaluating the limit of Green's functions along certain paths.

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arXiv2025

Dynamical localization and delocalization for random Schrodinger operators with $δ$-interactions in $\mathbb{R}^3$

We prove that the random Schrodinger operators on $\mathbb{R}^3$ with independent, identically distributed random variables and single-site potentials given by $δ$-functions on $\mathbb{Z}^3$, exhibit both dynamical localization and dynamical delocalization with probability one. That is, there are regions in the deterministic spectrum that exhibit dynamical localization, the nonspreading of wave packets, and regions in the deterministic spectrum where the models also exhibit nontrivial quantum transport, almost surely. These models are the first examples of ergodic, random Schrodinger operators exhibiting both dynamical localization and delocalization in dimension three or higher. The nontrivial transport is due to the presence of delocalized generalized eigenfunctions at positive energies $E > π^2$. The general idea of the proof follows [Hislop, Kirsch, Krishna (2024)] in which lower bounds on moments of the position operator are constructed using these generalized eigenfunctions. A new result of independent interest is a proof of the Combes-Thomas estimate on exponential decay of the Green's function for Schrodinger operators with infinitely-many $δ$-potentials.

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arXiv2025

Eigenvalue statistics for random polymer models: Localization and delocalization

We study the local eigenvalue statistics (LES) associated with one-dimensional lattice models of random polymers. We consider models constructed from two polymers. Each polymer is a finite interval of lattice points with a finite potential. These polymers are distributed along $\mathbb{Z}$ according to a Bernoulli distribution. The deterministic spectrum for these models is dense pure point, and is known to contain finitely-many critical energies. In this paper, we prove that the LES centered at these critical energies is described by a uniform clock process, and that the LES for the unfolded eigenvalues, centered at any other energy in the deterministic spectrum, is a Poisson point process. These results add to our understanding of these models that exhibit dynamical localization in any energy interval avoiding the critical energies [Damanik, Sims, Stolz] [De Bievre, Germinet], and nontrivial transport for wave packets with initial states supported at an integer point [Jitomirskaya, Schultz-Baldes, Stolz]. We show that the projection of these initial states onto spectral subspaces associated with any energy interval that contains all of the critical energies exhibit nontrivial transport, refining the connection between nontrivial transport and the critical energies. Finally, we also prove that the transition in the unfolded LES is sharp at the critical energies.

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arXiv2024

Asymptotic analysis of time-fractional quantum diffusion

We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrodinger equation in $\mathbb{R}^d$. We define the time-fractional derivative by the Caputo derivative and we consider the initial-value problem for the free evolution of wave packets in $\mathbb{R}^d$ governed by the time-fractional Schrodinger equation $ i^β\partial_t^αu = - Δu, ~~~~u(t=0) = u_0$, parameterized by two indices $α, β\in (0,1]$. We show distinctly different long-time evolution of the mean square displacement according to the relation between $α$ and $β$. In particular, asymptotically ballistic motion occurs only for $α=β$.

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arXiv2024

Edge currents for the time-fractional, half-plane, Schrodinger equation with constant magnetic field

We study the large-time asymptotics of the edge current for a family of time-fractional Schrodinger equations with a constant, transverse magnetic field on a half-plane $(x,y) \in \mathbb{R}_x^+ \times \mathbb{R}_y$. The TFSE is parameterized by two constants $(α, β)$ in $(0,1]$, where $α$ is the fractional order of the time derivative, and $β$ is the power of $i$ in the Schrodinger equation. We prove that for fixed $α$, there is a transition in the transport properties as $β$ varies in $(0,1]$: For $0 < β< α$, the edge current grows exponentially in time, for $α= β$, the edge current is asymptotically constant, and for $β> α$, the edge current decays in time. We prove that the mean square displacement in the $y\in \mathbb{R}$-direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin \cite{laskin2000_1} that the latter two cases, $α= β$ and $α< β$, are the physically relevant ones.

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arXiv2022

Local eigenvalue statistics for higher-rank Anderson models after Dietlein-Elgart

We use the method of eigenvalue level spacing developed by Dietlein and Elgart (arXiv:1712.03925) to prove that the local eigenvalue statistics (LES) for the Anderson model on $Z^d$, with uniform higher-rank $m \geq 2$, single-site perturbations, is given by a Poisson point process with intensity measure $n(E_0)~ds$, where $n(E_0)$ is the density of states at energy $E_0$ in the region of localization near the spectral band edges. This improves the result of Hislop and Krishna (arXiv:1809.01236), who proved that the LES is a compound Poisson process with Lévy measure supported on the set $\{1, 2, \ldots, m \}$. Our proofs are an application of the ideas of Dieltein and Elgart to these higher-rank lattice models with two spectral band edges, and illustrate, in a simpler setting, the key steps of the proof of Dieltein and Elgart.

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arXiv2021

On the local eigenvalue statistics for random band matrices in the localization regime

We study the local eigenvalue statistics $ξ_{ω,E}^N$ associated with the eigenvalues of one-dimensional, $(2N+1) \times (2N+1)$ random band matrices with independent, identically distributed, real random variables and band width growing as $N^α$, for $0 < α< \frac{1}{2}$. We consider the limit points associated with the random variables $ξ_{ω,E}^N [I]$, for $I \subset \mathbb{R}$, and $E \in (-2,2)$. For Gaussian distributed random variables with $0 \leq α< \frac{1}{7}$, we prove that this family of random variables has nontrivial limit points for almost every $E \in (-2,2)$, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables $ξ_{ω,E}^N [I]$ and associated quantities related to the intensities, as $N$ tends towards infinity, and employs known localization bounds of \cite{schenker, peled, et. al.}, and the strong Wegner and Minami estimates \cite{peled, et. al.}. Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for $0 < α< \frac{1}{2}$, we prove that any nontrivial limit points of the random variables $ξ_{ω,E}^N [I]$ are distributed according to Poisson distributions.

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arXiv2020

Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random models

We continue our study of the dependence of the density of states measure and related spectral functions of Schrödinger operators on the potential. Whereas our earlier work focused on random Schrödinger operators, we extend these results to Schrödinger operators on infinite graphs with deterministic potentials and ergodic potentials, and improve our results for random potentials. In particular, we prove the Lipschitz continuity of the DOSm for random Schrödinger operators on the lattice, recovering results of \cite{kachkovskiy, shamis}. For our treatment of deterministic potentials, we first study the density of states outer measure (DOSoM), defined for all Schrödinger operators, and prove a deterministic result of the modulus of continuity of the DOSoM with respect to the potential. We apply these results to Schrödinger operators on the lattice $Z^d$ and the Bethe lattice. In the former case, we prove the Lipschitz continuity of the DOSoM, and in the latter case, we prove that the DOSoM is $\frac{1}{2}$-log-Hölder continuous. Our technique combines the abstract Lipschitz property of one-parameter families of self-adjoint operators with a new finite-range reduction that allows us to study the dependency of the DOSoM and related functions on only finitely-many variables and captures the geometry of the graph at infinity.

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arXiv2020

The density of states and local eigenvalue statistics for random band matrices of fixed width

We prove that the local eigenvalue statistics for $d=1$ random band matrices with fixed bandwidth and, for example, Gaussian entries, is given by a Poisson point process and we identify the intensity of the process. The proof relies on an extension of the localization bounds of Schenker \cite{schenker} and the Wegner and Minami estimates. These two estimates are proved using averaging over the diagonal disorder. The new component is a proof of the uniform convergence and the smoothness of the density of states function. The limit function, known to be the semicircle law with a band-width dependent error \cite{bmp,dps,dl,mpk}, is identified as the intensity of the limiting Poisson point process. The proof of these results for the density of states relies on a new result that simplifies and extends some of the ideas used by Dolai, Krishna, and Mallick \cite{dkm}. These authors proved regularity properties of the density of states for random Schrödinger operators (lattice and continuum) in the localization regime. The proof presented here applies to the random Schrödinger operators on a class of infinite graphs treated by in \cite{dkm} and extends the results of \cite{dkm} to probability measures with unbounded support. The method also applies to fixed bandwidth RBM for $d=2,3$ provided certain localization bounds are known.

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arXiv2019

Eigenvalue statistics for Schrödinger operators with random point interactions on $\mathbb{R}^d$, $d=1,2,3$

We prove that the local eigenvalue statistics at energy $E$ in the localization regime for Schrödinger operators with random point interactions on $\mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity measure given by the density of states at $E$ times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrödinger operators in the continuum. The special structure of resolvent of Schrödinger operators with point interactions facilitates the proof of the Minami estimate for these models.

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arXiv2018

Transport of a quantum particle in a time-dependent white-noise potential

We show that a quantum particle in $\mathbb{R}^d$, for $d \geq 1$, subject to a white-noise potential, moves super-ballistically in the sense that the mean square displacement $\int \|x\|^2 \langle ρ(x,x,t) \rangle ~dx$ grows like $t^{3}$ in any dimension. The white noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. This is a known result in one dimension (see refs. Fischer, Leschke, Müller and Javannar, Kumar}. The energy of the system is also shown to increase linearly in time. We also prove that for the same white-noise potential model on the lattice $\mathbb{Z}^d$, for $d \geq 1$, the mean square displacement is diffusive growing like $t^{1}$. This behavior on the lattice is consistent with the diffusive behavior observed for similar models in the lattice $\mathbb{Z}^d$ with a time-dependent Markovian potential (see ref. Kang, Schenker).

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arXiv2018

Dependence of the density of states on the probability distribution for discrete random Schrödinger operators

We prove that the the density of states measure (DOSm) for random Schrödinger operators on $\mathbb{Z}^d$ is weak-$^*$ Hölder-continuous in the probability measure. The framework we develop is general enough to extend to a wide range of discrete, random operators, including the Anderson model on the Bethe lattice, as well as random Schrödinger operators on the strip. An immediate application of our main result provides quantitive continuity estimates for the disorder dependence of the DOSm and the integrated density of states (IDS) in the weak disorder regime. These results hold for a general compactly supported single-site probability measure, without any further assumptions. The few previously available results for the disorder dependence of the IDS valid for dimensions $d \geq 2$ assumed absolute continuity of the single-site measure and thus excluded the Bernoulli-Anderson model. As a further application of our main result, we establish quantitative continuity results for the Lyapunov exponent of random Schrödinger operators for $d=1$ in the probability measure with respect to the weak-$^*$ topology.

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arXiv2018

Compactness of iso-resonant potentials for Schrödinger operators in dimensions one and three

We prove compactness of a restricted set of real-valued, compactly supported potentials $V$ for which the corresponding Schrödinger operators $H_V$ have the same resonances, including multiplicities. More specifically, let $B_R(0)$ be the ball of radius $R > 0$ about the origin in $R^d$, for $d=1,3$. Let $\mathcal{I}_R (V_0)$ be the set of real-valued potentials in $C_0^\infty( \overline{B}_R(0); R)$ so that the corresponding Schrödinger operators have the same resonances, including multiplicities, as $H_{V_0}$. We prove that the set $\mathcal{I}_R (V_0)$ is a compact subset of $C_0^\infty (\overline{B}_R(0))$ in the $C^\infty$-topology. An extension to Sobolev spaces of less regular potentials is discussed.

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arXiv2017

Semiclassical Szegö limit of eigenvalue clusters for the hydrogen atom Zeeman Hamiltonian

We prove a limiting eigenvalue distribution theorem (LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian $H_V(h,B) = (1/2)( - i h {\mathbf \nabla} - {\mathbf A}(h))^2 - |x|^{-1}$, defined on $L^2 (R^3)$, in a constant magnetic field ${\mathbf B}(h) = {\mathbf \nabla} \times {\mathbf A}(h)=(0,0,ε(h)B)$ in the weak field limit $ε(h) \rightarrow 0$ as $h\rightarrow{0}$. We consider the Planck's parameter $h$ taking values along the sequence $h=1/(N+1)$, with $N=0,1,2,\ldots$, and $N\rightarrow\infty$. We prove a semiclassical $N \rightarrow \infty$ LEDT of the Szegö-type for the scaled eigenvalue shifts and obtain both ({\bf i}) an expression involving the regularized classical Kepler orbits with energy $E=-1/2$ and ({\bf ii}) a weak limit measure that involves the component $\ell_3$ of the angular momentum vector in the direction of the magnetic field. This LEDT extends results of Szegö-type for eigenvalue clusters for bounded perturbations of the hydrogen atom to the Zeeman effect. The new aspect of this work is that the perturbation involves the unbounded, first-order, partial differential operator $w(h, B) = \frac{(ε(h)B)^2}{8} (x_1^2 + x_2^2) - \frac{ ε(h)B}{2} hL_3 ,$ where the operator $hL_3$ is the third component of the usual angular momentum operator and is the quantization of $\ell_3$. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states $Ψ_{α,N}$, and their derivatives $L_3(h)Ψ_{α,N}$, in the large quantum number regime $N\rightarrow\infty$.

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arXiv2016

A bound on the averaged spectral shift function and a lower bound on the density of states for random Schrödinger operators on $\mathbb{R}^d$

We obtain a bound on the expectation of the spectral shift function for alloy-type random Schrödinger operators on $\mathbb{R}^d$ in the region of localisation, corresponding to a change from Dirichlet to Neumann boundary conditions along the boundary of a finite volume. The bound scales with the area of the surface where the boundary conditions are changed. As an application of our bound on the spectral shift function, we prove a reverse Wegner inequality for finite-volume Schrödinger operators in the region of localisation with a constant locally uniform in the energy. The application requires that the single-site distribution of the independent and identically distributed random variables has a Lebesgue density that is also bounded away from zero. The reverse Wegner inequality implies a strictly positive, locally uniform lower bound on the density of states for these continuum random Schrödinger operators.

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arXiv2015

Decorrelation estimates for random Schrödinger operators with non rank one perturbations

We prove decorrelation estimates for generalized lattice Anderson models on $Z^d$ constructed with finite-rank perturbations in the spirit of Klopp \cite{klopp}. These are applied to prove that the local eigenvalue statistics $ξ^ω_{E}$ and $ξ^ω_{E^\prime}$, associated with two energies $E$ and $E'$ satisfying $|E - E'| > 4d$, are independent. That is, if $I,J$ are two bounded intervals, the random variables $ξ^ω_{E}(I)$ and $ξ^ω_{E'}(J)$, are independent and distributed according to a compound Poisson distribution whose Lévy measure has finite support. We also prove that the extended Minami estimate implies that the eigenvalues in the localization region have multiplicity at most the rank of the perturbation.

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arXiv2015

The multiplicity of eigenvalues of the Hodge Laplacian on 5-dimensional compact manifolds

We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is a skew-adjoint operator, this implies the main result for the Hodge Laplacian.

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arXiv2015

Band functions in the presence of magnetic steps

We complete the analysis of the band functions for two-dimensional magnetic Schrödinger operators with piecewise constant magnetic fields. The discontinuity of the magnetic field can create edge currents that ow along the discontinuity that have been described by physicists. Properties of these edge currents are directly related to the behavior of the band functions. The effective potential of the fiber operator is an asymmetric double well (eventually degenerated) and the analysis of the splitting of the bands incorporates the asymmetry. If the magnetic field vanishes, the reduced operator has essential spectrum and we provide an explicit description of the band functions located below the essential spectrum. For non degenerate magnetic steps, we provide an asymptotic expansion of the band functions at infinity. We prove that when the ratio of the two magnetic fields is rational, a splitting of the band functions occurs and has a natural order, predicted by numerical computations.

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