Intrinsic Ultracontractivity for Schrödinger Operators with Mixed Boundary Conditions

Let be a domain in , . Let be a divergence form uniformly elliptic operator with Dirichlet boundary conditions on and Neumann boundary conditions on , where is a closed subset of . We prove intrinsic ultracontractivity for the semigroup associated to the Schrödinger operator , where is a potential in the Kato class, provided that is locally Lipschitz and is given by the boundary of either a Hölder domain of order or a uniformly Hölder domain of order , . Our results extend to the mixed boundary case the results of Bañuelos, Bass and Burdzy, Bass and Hsu, and Davies and Simon.     

Schrödinger Equations with Fractional Laplacians

It is shown that the unique solution of } can be represented as { } where X=(X _t , t≥ 0) is a stable process whose generator is (-Δ) ^α/2 with X ₀ =0 . Accepted 24 July 2000. Online publication 13 November 2000.     

Schrödinger Operators on Zigzag Nanotubes

We consider the Schr?dinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We describe all finite gap potentials. We show that the mapping: potential all eigenvalues is a real analytic isomorphism for some class of potentials. Submitted: October 5, 2006. Accepted: December 15, 2006.     

Eigenfunction Expansions for Schrödinger Operators on Metric Graphs

We construct an expansion in generalized eigenfunctions for Schr?dinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.      

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)

     

Ballistic Transport for One-Dimensional Quasiperiodic Schrödinger Operators

In this paper, we show that one-dimensional discrete multifrequency quasiperiodic Schrödinger operators with smooth potentials demonstrate ballistic motion on the set of energies on which the corresponding Schrödinger cocycles are smoothly reducible to constant rotations. The proof is performed by establishing a local version of strong ballistic transport on an exhausting sequence of subsets on which reducibility can be achieved by a conjugation uniformly bounded in the C^ℓ-norm. We also establish global strong ballistic transport under an additional integral condition on the norms of conjugation matrices. The latter condition is quite mild and is satisfied in many known examples. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

Fractional time-dependent Schrödinger equation on the Heisenberg group

Let Δ be the Kohn sublaplacian on the Heisenberg group , . In this paper we estimate the L ²-norm of the local maximal function of the unitary group of operators generated by L, by the Sobolev W _γ,ε -norm for some γ > 0 and for all ε > 0. Research supported in part by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389. The first author was also supported by the MNiSW research grant N201 012 31/1020.     

Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential

Potential Analysis - We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential...     

Inverse Scattering Theory for Discrete Schrödinger Operators on the Hexagonal Lattice

We consider the spectral theory and inverse scattering problem for discrete Schrödinger operators on the hexagonal lattice. We give a procedure for reconstructing finitely supported potentials from the scattering matrices for all energies. The same procedure is applicable for the inverse scattering problem on the triangle lattice.     

Integrated Density of States for Ergodic Random Schrödinger Operators on Manifolds

We consider the Riemannian universal covering of a compact manifold M = X/ and assume that is amenable. We show the existence of a (nonrandom) integrated density of states for an ergodic random family of Schrödinger operators on X.     

Spectral Properties of Schrödinger Operators on Perturbed Lattices

We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for the resolvent, construct a spectral representation, and define the S-matrix. Our theory covers the square, triangular, diamond, Kagome lattices, as well as the ladder, the graphite and the subdivision of square lattice.     

Anderson Localization for Time Periodic Random Schrödinger Operators

Abstract

We prove that at large disorder, Anderson localization in Z ^d is stable under localized time-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schrödinger equations.     

Approximation of the Fractional Schrödinger Propagator on Compact Manifolds

Let L be a second order positive, elliptic differential operator that is self-adjoint with respect to some C~∞ density dx on a compact connected manifold M. We proved that if 0 α 1,α/2 s α and f ∈ H~s(M) then the fractional Schr?dinger propagator e~(it Lα/2) on M satisfies e~(it Lα/2) f(x)-f(x) = o(t~(s/α-ε)) almost everywhere as t → 0~+, for any ε 0.     

Homogeneous Schr?dinger Operators on Half-Line

The differential expression L_m=-?_x²+(m²-1/4)x^-2{L_m=-partial_x^2+(m^2-1/4)x^{-2}} defines a self-adjoint operator H _m on L ²(0, ∞) in a natural way when m ² ≥ 1. We study the dependence of H _m on the parameter m show that it has a unique holomorphic extension to the half-plane Re m > −1, and analyze spectral and scattering properties of this family of operators.     

Unique Continuation for Fractional Schrödinger Equations with Rough Potentials

This article deals with the weak and strong unique continuation principle for fractional Schrödinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods extend to “variable coefficient” versions of fractional Schrödinger equations and operators on non-flat domains.     

Inverse Problems,Trace Formulae for Discrete Schrödinger Operators

We study discrete Schrödinger operators with compactly supported potentials on Z ^d . Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely reconstructed from the S-matrix of all energies. We also study the spectral shift function \({\xi(\lambda)}\) for the trace class potentials, and estimate the discrete spectrum in terms of the moments of \({\xi(\lambda)}\) and the potential.