Related papers: On Spectral Theory for Schr\"odinger Operators wit…

Given a complex, separable Hilbert space $\mathcal{H}$, we consider self-adjoint $L^2$-realizations of differential expressions $\tau = - (d^2/dx^2) I_{\mathcal{H}} + V(x)$, on half-lines and on the real line (assuming the limit-point…

We develop Weyl-Titchmarsh theory for self-adjoint Schr\"odinger operators $H_{\alpha}$ in $L^2((a,b);dx;\cH)$ associated with the operator-valued differential expression $\tau =-(d^2/dx^2)+V(\cdot)$, with $V:(a,b)\to\cB(\cH)$, and $\cH$ a…

We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schr\"odinger operators on $[a,\infty)$, $a\in\mathbb{R}$, with a regular finite end point $a$ and the case of Schr\"odinger…

In the paper we consider a functional-difference operator $H=U+U^{-1}+V$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{\pi i\tau}$ and $\tau>0$. The operator $H$ has applications in the conformal…

We study the spectral theory of operators, generated as direct sums of self-adjoint extensions of quasi-differential minimal operators on a multi-interval set (self-adjoint vector-operators), acting in a Hilbert space. Spectral theorems for…

We study the 1-D Schr\"odinger operators in Hilbert space $L^{2}(\mathbb{R})$ with real-valued Radon measure $q'(x)$, $q\in \mathrm{BV}_{loc}(\mathbb{R})$ as potentials. New sufficient conditions for minimal operators to be bounded below…

We study spectral properties of Hamiltonians $\rH_{X,\gB,q}$ with $\delta'$-point interactions on a discrete set $X={x_k}_{k=1}^\infty\subset\R_+$. %at the centers $x_n$ on the positive half line in terms of energy forms. Using the form…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity.…

We are concerned with the non-normal Schr\"odinger operator $$ H=-\Delta+V $$ on $ L^2(\mathbb R^n)$, where $V\in W^{1,\infty}_{\text{loc}}(\mathbb{R}^n)$ and $\operatorname{Re} (V(x))\ge c|x|^2-d$ for some $c,d>0$. The spectrum of this…

We give an exposition on the $L^2$ theory of the perturbed Fourier transform associated with a Schr\"odinger operator $H=-d^2/dx^2 +V$ on the real line, where $V$ is a real-valued \mbox{finite} measure. In the case $V\in L^1\cap L^2$, we…

The goal of this paper is the spectral analysis of the Schr\"{o}dinger type operator $H=L+V$, the perturbation of the Taibleson-Vladimirov multiplier $L=\mathfrak{D}^{\alpha}$ by a potential $V$. Assuming that $V$ belongs to a certain class…

Let $H$ be the discrete Schr\"odinger operator $Hu(n):=u(n-1)+u(n+1)+v(n)u(n)$, $u(0)=0$ acting on $l^2({\bf Z}^+)$ where the potential $v$ is real-valued and $v(n)\to 0$ as $n\to \infty$. Let $P$ be the orthogonal projection onto a closed…

Let $(-A,B,C)$ be a linear system in continuous time $t>0$ with input and output space ${\bf C}$ and state space $H$. The function $\phi_{(x)}(t)=Ce^{-(t+2x)A}B$ determines a Hankel integral operator $\Gamma_{\phi_{(x)}}$ on $L^2((0, \infty…

We investigate the spectral properties of the Schr\"odinger operators in $L^2(\mathbb{R}^n)$ with a singular interaction supported by an infinite family of concentric spheres $$…

The paper deals with the distribution of eigenvalues of the compact fractal pseudodifferential operator $T^\mu_\tau$, \[ \big( T^\mu_\tau f\big)(x) = \int_{\mathbb{R}^n} e^{-ix\xi} \, \tau(x,\xi) \, \big( f\mu \big)^\vee (\xi) \, \mathrm{d}…

A spectral theory of linear operators on a rigged Hilbert space is applied to Schr\"odinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach…

Based on the recent work \cite{KKK} for compact potentials, we develop the spectral theory for the one-dimensional discrete Schr\"odinger operator $$ H \phi = (-\De + V)\phi=-(\phi_{n+1} + \phi_{n-1} - 2 \phi_n) + V_n \phi_n. $$ We show…

We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space.) The models allow for a scale of complexity, indexed by the natural numbers…

We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[\tau f = \frac{1}{r} \left(- \big(p[f' + s…

Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schr\"odinger operators with matrix-valued potentials, with special emphasis on distributional…