Absolutely Continuous Spectrum for Random Schrödinger Operators on Tree-Strips of Finite Cone Type

A tree-strip of finite cone type is the product of a tree of finite cone type with a finite set. We consider random Schrödinger operators on these tree-strips, similar to the Anderson model. We prove that for small disorder, the spectrum is almost surely, purely, absolutely continuous in a certain set.     

On Spectral Problems of Discrete Schrödinger Operators

Applications of Mathematics - A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the...     

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.     

Stability of the Absolutely Continuous Spectrum of Random Schrödinger Operators on Tree Graphs

According to the Smolukowski-Kramers approximation, we show that the solution of the semi-linear stochastic damped wave equations μ u _tt (t,x)=Δu(t,x)?u _t (t,x)+b(x,u(t,x))+Q (t),u(0)=u ₀, u _t (0)=v ₀, endowed with Dirichlet boundary conditions, converges as μ goes to zero to the solution of the semi-linear stochastic heat equation u _t (t,x)=Δ u(t,x)+b(x,u(t,x))+Q (t),u(0)=u ₀, endowed with Dirichlet boundary conditions. Moreover we consider relations between asymptotics for the heat and for the wave equation. More precisely we show that in the gradient case the invariant measure of the heat equation coincides with the stationary distributions of the wave equation, for any μ>0.     

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.     

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.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

On Spectral Properties of Translationally Invariant Magnetic Schrödinger Operators

We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr?dinger operator with such a potential. In particular, we show that the spectrum of is absolutely continuous and we find its location. Then we study the long-time behaviour of solutions exp of the time dependent Schr?dinger equation. It turns out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to an evolution of a one-dimensional free particle but “exits” to +∞ and −∞ in the direction of the potential might be essentially different. Submitted: June 7, 2007. Accepted: August 20, 2007.     

A Weak Gordon Type Condition for Absence of Eigenvalues of One-dimensional Schrödinger Operators

We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. The criterion leads to sharp quantitative bounds on the eigenvalues. We apply our result to quasiperiodic measures as potentials.     

Gauge Optimization and Spectral Properties of Magnetic Schrödinger Operators

We establish new necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with positive scalar potentials. We also derive two-sided estimates for the bottoms of the spectrum and essential spectrum. The main idea is to optimize the gauges of the magnetic field on cubes, thus reducing the quadratic form on the cubes to ones without magnetic field (but with appropriately adjusted scalar potentials).     

Lifshits Tails for Random Schrödinger Operators with Nonsign Definite Potentials

This paper is devoted to the study of Lifshits tails for random Schr?dinger operator acting on of the form , where H ₀ is a -periodic Schr?dinger operator, λ is a positive coupling constant, are i.i.d and bounded random variables and V is the single site potential with changing sign. We prove that, in the weak disorder regime, at an open band edge, a true Lifshits tail for the random Schr?dinger operator occurs under a certain set of conditions on H ₀ and on V. Submitted: April 17, 2007. Accepted: December 13, 2007.     

A Characterization of Hardy Spaces Associated with Certain Schrödinger Operators

Let {K _t } _t>0 be the semigroup of linear operators generated by a Schrödinger operator ?L = Δ ? V (x) on ? ^d , d ≥ 3, where V (x) ≥ 0 satisfies Δ ^?1 V ∈ L ^∞. We say that an L ¹-function f belongs to the Hardy space \({H^{1}_{L}}\) if the maximal function ? _L f (x) = sup _t>0 |K _t f (x)| belongs to L ¹ (? ^d ). We prove that the operator (?Δ)^1/2 L ^?1/2 is an isomorphism of the space \({H^{1}_{L}}\) with the classical Hardy space H ¹(? ^d ) whose inverse is L ^1/2(?Δ)^?1/2. As a corollary we obtain that the space \({H^{1}_{L}}\) is characterized by the Riesz transforms \(R_{j}=\frac {\partial }{\partial x_{j}}L^{-1\slash 2}\) .     

A Spectral Property of Discrete Schrödinger Operators with Non-Negative Potentials

In the context of an infinite weighted graph of bounded degree, we give a sufficient condition for the discrete Schrödinger operator with a non-negative potential to have a strictly positive bottom of the spectrum. The main result is a discrete analogue of a theorem of Shen in the setting of complete Riemannian manifolds.     

Spectral Theory of Semibounded Schrödinger Operators with δ′-Interactions

We study spectral properties of Hamiltonians H _X,β,q with δ′-point interactions on a discrete set ${X = \{x_k\}_{k=1}^\infty \subset (0, +\infty)}$ . Using the form approach, we establish analogs of some classical results on operators H _q =  ?d²/dx ² + q with locally integrable potentials ${q \in L^1_{\rm loc}[0, +\infty)}$ . In particular, we establish the analogues of the Glazman–Povzner–Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with δ-interactions, spectral properties of operators H _X,β,q are closely connected with those of ${{\rm H}_{X,q}^N = \oplus_{k}{\rm H}_{q,k}^N}$ , where ${{\rm H}_{q,k}^N}$ is the Neumann realization of ?d²/dx ² + q in L ²(x _k-1,x _k ).     

Weighted Spectral Gap for Magnetic Schrödinger Operators with a Potential Term

We prove that singular Schrödinger equations with external magnetic field admit a representation with a positive Lagrangian density whenever their “nonmagnetic” counterpart is nonnegative. In this case the operator has a weighted spectral gap as long as the strength of the magnetic field is not identically zero. We provide estimates of the weight in the spectral gap, including the versions with L ^p -norm and with a magnetic gradient term, and applications to an increase of the best Hardy constant due to the presence of a magnetic field. The paper also shows existence of the ground state for the nonlinear magnetic Schrödinger equation with the periodic magnetic field.     

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 \(\mathcal {L}f = 0.\)

     

A-Priori Decay for Eigenfunction of Perturbed Periodic Schrödinger Operators

. Recently Neumayr and Metzner [1] have shown that the connected N-point density-correlation functions of the two-dimensional and the one-dimensional Fermi gas at one-loop order generically (i.e.for nonexceptional energy-momentum configurations) vanish/are regular in the small momentum/small energy-momentum limits. Their result is based on an explicit analysis in the sequel of the results of Feldman et al. [2]. In this note we use Ward identities to give a proof of the same fact - in a considerably shortened and simplified way - for any dimension of space.     

Spectral deformations of one-dimensional Schrödinger operators

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operatorsH=?d ²/dx ²+V in $H = - d^2 /dx^2 + V in \mathcal{L}^2 (\mathbb{R})$ . Our technique is connected to Dirichlet data, that is, the spectrum of the operatorH ^D onL ²((?∞,x ₀)) ⊕L ²((x ₀, ∞)) with a Dirichlet boundary condition atx ₀. The transformation moves a single eigenvalue ofH ^D and perhaps flips which side ofx ₀ the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such asV(x)→∞ as |x|→∞, whereV is uniquely determined by the spectrum ofH and the Dirichlet data, our result implies that the specific Dirichlet data allowed are determined only by the asymptotics asE→∞.