相关论文: Convergence of an exact quantization scheme

Let $\varphi:\mathbb{D} \to \mathbb{D}$ be a holomorphic map with a fixed point $\alpha\in\mathbb{D}$ such that $0\leq |\varphi'(\alpha)|<1$. We show that the spectrum of the composition operator $C_\varphi$ on the Fr\'echet space $…

The two-dimensional relativistic configurational $\vec r$-space is proposed and the exactly solvable finite-difference model of the harmonic oscillator in this space is constructed. The wave functions of the stationary states and the…

The norm resolvent convergence of discrete Schr\"odinger operators to a continuum Schr\"odinger operator in the continuum limit is proved under relatively weak assumptions. This result implies, in particular, the convergence of the spectrum…

We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum…

The operator \[ A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2} \] is considered in $L_{2}({\mathbb{R}}^{2})$, where $g_{j}(x_{1},x_{2})$, $j=1,2,$ are periodic in $x_{1}$ with…

This paper studies the observability inequalities for the Schr\"{o}dinger equation associated with an anharmonic oscillator $H=-\frac{\d^2}{\d x^2}+|x|$. We build up the observability inequality over an arbitrarily short time interval…

It is shown that Schroedinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points…

We show that positive absolutely norm attaining operators can be characterized by a simple property of their spectra. This result clarifies and simplifies a result of Ramesh. As an application we characterize weighted shift operators which…

A new model for the finite one-dimensional harmonic oscillator is proposed based upon the algebra u(2)_{\alpha}. This algebra is a deformation of the Lie algebra u(2) extended by a parity operator, with deformation parameter {\alpha}. A…

We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding…

We introduce the notion of Schr\"odinger integral operators and prove sharp local and global regularity results for these (including propagators for the quantum mechanical harmonic oscillator). Furthermore we introduce general classes of…

We prove that any $n$-dimensional Hamiltonian operator with pure point spectrum is completely integrable via self-adjoint first integrals. Furthermore, we establish that given any closed set $\Sigma\subset\mathbb R$ there exists an…

In this paper, we present an analysis of the equation $\ddot{x} - (1/2x) \dot{x}^2 + 2 \omega^2 x - 1/8x = 0$, where $\omega > 0$ and $x = x(t)$ is a real-valued variable. We first discuss the appearance of this equation from a…

We investigate scattering, localization and dispersive time-decay properties for the one-dimensional Schr\"odinger equation with a rapidly oscillating and spatially localized potential, $q_\epsilon=q(x,x/\epsilon)$, where $q(x,y)$ is…

In the model suggested by Smilansky one studies an operator describing the interaction between a quantum graph and a system of K one-dimensional oscillators attached at different points of the graph. This paper is a continuation of our…

This paper is devoted to the homogenization of Shr\"odinger type equations with periodically oscillating coefficients of the diffusion term, and a rapidly oscillating periodic time-dependent potential. One convergence theorem is proved and…

We study Schr\"odinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the…

We establish spectral convergence results of approximations of unbounded non-selfadjoint linear operators with compact resolvents by operators that converge in generalized strong resolvent sense. The aim is to establish general assumptions…

We study the one-dimensional Schr\"odinger operators $$ S(q)u:=-u"+q(x)u,\quad u\in \mathrm{Dom}\left(S(q)\right), $$ with $1$-periodic real-valued singular potentials $q(x)\in H_{\operatorname{per}}^{-1}(\mathbb{R},\mathbb{R})$ on the…

We have applied a collocation approach to obtain the numerical solution to the stationary Schr\"odinger equation for systems of coupled oscillators. The dependence of the discretized Hamiltonian on scale and angle parameters is exploited to…