We prove new results on the stability of the absolutely continuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate $(1+x)^{-\frac{1}{3}-ε}$ or if it is continuously differentiable with derivative from the Hölder space $C_α(R),$ with any $α>0.$
In this paper, we consider the perturbed Stark operator \begin{equation*} Hu=H_0u+qu=-u^{\prime\prime}-xu+qu, \end{equation*} where $q$ is the power-decaying perturbation. The criteria for $q$ such that $H=H_0+q$ has at most one eigenvalue (finitely many, infinitely many eigenvalues) are obtained. All the results are quantitative and are generalized to the perturbed Stark type operator.
We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is $|…
We consider the 3-dimensional Stark operator perturbed by a complex-valued potential. We obtain an estimate for the number of eigenvalues of this operator as well as for the sum of imaginary parts of eigenvalues situated in the upper half-plane.
We consider the Stark operator perturbed by a compactly supported potential (of a certain class) on the real line. We prove the following results: (a) upper and lower bounds on the number of resonances in complex discs with large radii, (b) the trace formula in terms of resonances only, (c) all resonances determine the potential uniquely.
The structure of the spectrum of random operators is studied. It is shown that if the density of states measure of some subsets of the spectrum is zero, then these subsets are empty. In particular follows that absolute continuity of the IDS implies singular spectra of ergodic operators is either empty or of positive measure. Our results apply to Anderson and alloy type models, perturbed Landau Hamiltonians, almost periodic potentials and models which are not ergodic.
Let $H_0$ be a periodic operator on $\R^+$(or periodic Jacobi operator on $\N$). It is known that the absolutely continuous spectrum of $H_0$ is consisted of spectral bands $\cup[α_l,β_l]$. Under the assumption that $\limsup_{x\to \infty} x|V(x)|<\infty$ ($\limsup_{n\to \infty} n|V(n)|<\infty$), the asymptotical behaviour of embedded eigenvalues approaching to the spectral boundary is established.
For an arbitrary Hermitian period-$T$ Jacobi operator, we assume a perturbation by a Wigner-von Neumann type potential to devise subordinate solutions to the formal spectral equation for a (possibly infinite) real set, $S$, of the spectral parameter. We employ discrete Levinson type techniques to achieve this, which allow the analysis of the asymptotic behaviour of the solution. This enables us to construct infinitely many spectral singularities on the absolutely continuous spectrum of the periodic Jacobi operator, which are stable with respect to an $l^1$-perturbation. An analogue of the quan…
We are interested in diagonal perturbations of a periodic Jacobi operator that introduce embedded eigenvalues in its essential spectrum. Embedding multiple points in the essential spectrum has been known to be difficult, given that eigenvalues are destroyed easily by small perturbations. However, given a finite or countably infinite set of points within an absolutely continuous band of the original periodic operator (subject only to a very weak non-resonance condition) we are able to construct a diagonal perturbation that preserves the essential spectrum and places eigenvalues in all of those …
We consider the Schrödinger operator $Hy=-y"+(p+q)y$ with a periodic potential $p$ plus a compactly supported potential $q$ on the real line. The spectrum of $H$ consists of an absolutely continuous part plus a finite number of simple eigenvalues below the spectrum and in each spectral gap $\g_n\ne \es, n\ge1$. We prove the following results: 1) the distribution of resonances in the disk with large radius is determined, 2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, 3) if $H$ has infinitely many open gaps in the continuous spectrum, then for any sequ…
We apply weighted Mourre commutator theory to prove the limiting absorption principle for the discrete Schr{ö}dinger operator perturbed by the sum of a Wigner-von Neumann and long-range type potential. In particular, this implies a new result concerning the absolutely continuous spectrum for these operators even for the one-dimensional operator. We show that methods of classical Mourre theory based on differential inequalities and on the generator of dilation cannot apply to the mentionned Schr{ö}dinger operators.
In this paper, we consider the Schrödinger equation, \begin{equation*} Hu=-u^{\prime\prime}+(V(x)+V_0(x))u=Eu, \end{equation*} where $V_0(x)$ is 1-periodic and $V (x)$ is a decaying perturbation. By Floquet theory, the spectrum of $H_0=-\nabla^2+V_0$ is purely absolutely continuous and consists of a union of closed intervals (often referred to as spectral bands). Given any finite set of points $\{ E_j\}_{j=1}^N$ in any spectral band of $H_0$ obeying a mild non-resonance condition, we construct smooth functions $V(x)=\frac{O(1)}{1+|x|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$. Gi…
We study perturbations of the self-adjoint periodic Sturm--Liouville operator \[ A_0 = \frac{1}{r_0}\left(-\frac{\mathrm d}{\mathrm dx} p_0 \frac{\mathrm d}{\mathrm dx} + q_0\right) \] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment c…
We consider decaying oscillatory perturbations of periodic Schrödinger operators on the half line. More precisely, the perturbations we study satisfy a generalized bounded variation condition at infinity and an $L^p$ decay condition. We show that the absolutely continuous spectrum is preserved, and give bounds on the Hausdorff dimension of the singular part of the resulting perturbed measure. Under additional assumptions, we instead show that the singular part embedded in the essential spectrum is contained in an explicit countable set. Finally, we demonstrate that this explicit countable set …
We consider the first order periodic systems perturbed by a $2N\ts 2N$ matrix-valued periodic potential on the real line. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has branch points, which we call resonances. We prove the existence of real or complex resonances. We determine the asymptotics of the periodic, anti-p…
We consider the unperturbed operator $H_0 : = (-i \nabla - A)^2 + W$, self-adjoint in $L^2(\R^2)$. Here $A$ is a magnetic potential which generates a constant magnetic field $b>0$, and the edge potential $W$ is a non-decreasing non constant bounded function depending only on the first coordinate $x \in \R$ of $(x,y) \in \R^2$. Then the spectrum of $H_0$ has a band structure and is absolutely continuous; moreover, the assumption $\lim_{x \to \infty}(W(x) - W(-x)) < 2b$ implies the existence of infinitely many spectral gaps for $H_0$. We consider the perturbed operators $H_{\pm} = H_0 \pm V$ whe…
We study ergodic Schrödinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for Boshernitzan's criterion under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, …
