Searcharxiv⌕ Search

arXiv subjects

P. D. Hislop

Publications and source records attributed to P. D. Hislop.

13 recordsLinked to original sources
arXiv2026

A semiclassical approach to spectral estimates for random Landau Schrodinger operators

We prove spectral properties for random Landau Schrödinger operators on $L^2(\mathbb{R}^2)$ with bounded, random potentials supported in a square $Λ_L \subset \mathbb{R}^2$ of side length $L>0$, using semiclassical pseudodifferential calculus. The semiclassical parameter $h$ is the inverse of the magnetic field strength $B > 0$. By means of the Grushin method, we are led to the analysis of an effective Hamiltonian on $L^2 (\mathbb{R})$, the principal term of which is a sum of certain compact, self-adjoint pseudodifferential operators. By analyzing these operators, we prove semiclassical Wegner and Minami estimates for the random Landau Schrodinger operator in energy intervals in the spectral bands around each Landau level.

math-ph↗
arXiv2020

Some remarks on spectral averaging and the local density of states for random Schrödinger operators on $L^2 ( R^d )$

We prove some local estimates on the trace of spectral projectors for random Schrödinger operators restricted to cubes $Λ\subset R^d$. We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schrödinger operators on cubes $Λ\subset R^d$, for $d \geq 1$. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit.

math-ph↗
arXiv2019

Dependence of the density of states on the probability distribution -- part II: Schrödinger operators on $\mathbb{R}^d$ and non-compactly supported probability measures

We extend our results in \cite{hislop_marx_1} on the quantitative continuity properties, with respect to the single-site probability measure, of the density of states measure and the integrated density of states for random Schrödinger operators. For lattice models on $\mathbb{Z}^d$, with $d \geq 1$, we treat the case of non-compactly supported probability measures with finite first moments. For random Schrödinger operators on $\mathbb{R}^d$, with $d \geq 1$, we prove results analogous to those in \cite{hislop_marx_1} for compactly supported probability measures. The method of proof makes use of the Combes-Thomas estimate and the Helffer-Sjöstrand formula.

math-ph↗
arXiv2018

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 $E'$ in the localization region and 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. The method of proof contains new ingredients that simplify the proof of the rank one case \cite{klopp,shirley,trinh}, extends to models for which the eigenvalues are degenerate, and applies to models for which the potential is not sign definite \cite{tautenhahn-veselic1} in dimensions $d \geq 1$.

math-ph↗
arXiv2015

Some trace monotonicity properties and applications

We present some results on the monotonicity of some traces involving functions of self-adjoint operators with respect to the natural ordering of their associated quadratic forms. We also apply these results to complete a proof of the Wegner estimate for continuum models of random Schrödinger operators as given in \cite{co-hi1}.

math.FA↗
arXiv2013

Some remarks on resonances in even-dimensional Euclidean scattering

The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions $d \geq 2$. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto $Λ$, the Riemann surface of the logarithm function. We relate the multiplicities of the poles of the continued scattering matrix to the multiplicities of the poles of the resolvent. Moreover, we show that the poles of the scattering matrix on the $m$th sheet of $Λ$ are related to the zeros of a scalar function defined on the physical sheet. This paper contains a number of results about "pure imaginary" resonances. As an example, in contrast with the odd-dimensional case, we show that in even dimensions there are no "purely imaginary" resonances on any sheet of $Λ$ for Schrödinger operators with potentials $0 \leq V \in L_0^\infty (\R^d)$.

math-ph↗
arXiv2010

Conductivity and the current-current correlation measure

We review various formulations of conductivity for one-particle Hamiltonians and relate them to the current-current correlation measure. We prove that the current-current correlation measure for random Schrödinger operators has a density at coincident energies provided the energy lies in a localization regime. The density vanishes at such energies and an upper bound on the rate of vanishing is computed. We also relate the current-current correlation measure to the localization length.

math-ph↗
arXiv2010

Resonances for manifolds hyperbolic at infinity: optimal lower bounds on order of growth

Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of growth $(n+1)$ generically for such manifolds.

math.SP↗
arXiv2009

Resonances for Schrodinger operators with compactly supported potentials

We describe the generic behavior of the resonance counting function for a Schrödinger operator with a bounded, compactly-supported real or complex valued potential in $d \geq 1$ dimensions. This note contains a sketch of the proof of our main results \cite{ch-hi1,ch-hi2} that generically the order of growth of the resonance counting function is the maximal value $d$ in the odd dimensional case, and that it is the maximal value $d$ on each nonphysical sheet of the logarithmic Riemann surface in the even dimensional case. We include a review of previous results concerning the resonance counting functions for Schrödinger operators with compactly-supported potentials.

math-ph↗
arXiv2008

Maximal order of growth for the resonance counting functions for generic potentials in even dimensions

We prove that the resonance counting functions for Schrödinger operators $H_V = - Δ+ V$ on $L^2 (\R^d)$, for $d \geq 2$ {\it even}, with generic, compactly-supported, real- or complex-valued potentials $V$, have the maximal order of growth $d$ on each sheet $Λ_m$, $m \in \Z \backslash \{0 \}$, of the logarithmic Riemann surface. We obtain this result by constructing, for each $m \in \Z \backslash \{0 \}$, a plurisubharmonic function from a scattering determinant whose zeros on the physical sheet $Λ_0$ determine the poles on $Λ_m$. We prove that the order of growth of the counting function is related to a suitable estimate on this function that we establish for generic potentials. We also show that for a potential that is the characteristic function of a ball, the resonance counting function is bounded below by $C_m r^d$ on each sheet $Λ_m$, $m \in \Z \backslash \{0\}$.

math-ph↗
arXiv2007

Localization for Schrodinger operators with random vector potentials

We prove Anderson localization at the internal band-edges for periodic magnetic Schr{ö}dinger operators perturbed by random vector potentials of Anderson-type. This is achieved by combining new results on the Lifshitz tails behavior of the integrated density of states for random magnetic Schr{ö}dinger operators, thereby providing the initial length-scale estimate, and a Wegner estimate, for such models.

math-ph↗
arXiv1994

Landau Hamiltonians with Random Potentials: Localization and the Density of States

We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finite-area Green's functions. The proof of the decay estimates for the Green's functions uses fundamental results from two-dimensional bond percolation theory.

funct-an↗