Related papers: 2D Schr\"{o}dinger operators with singular potenti…

We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on $L^2(\mathbb{R}^2)$: $H^\lambda_{\rm edge}=-\Delta+\lambda^2…

In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator $\mathcal{A}_h = -h^2\Delta+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $\Omega\subset\mathbb{R}^n\,$, as $h\rightarrow0\,$. $V$ is…

We consider Schr\"odinger operators with a strongly attractive singular interaction supported by a finite curve $\Gamma$ of lenghth $L$ in $\R^3$. We show that if $\Gamma$ is $C^4$-smooth and has regular endpoints, the $j$-th eigenvalue of…

We establish observability inequalities for various problems involving fractional Schr\"odinger operators $(-\Delta)^{\alpha/2}+V$, $\alpha>0$, on a compact Riemannian manifold. Observability from an open set for the corresponding…

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…

Given $n\geq 2$, we put $r=\min\{i\in\mathbb{N}; i>n/2 \}$. Let $\Sigma$ be acompact, $C^{r}$-smooth surface in $\mathbb{R}^{n}$ which contains the origin. Let further $\{S_{\epsilon}\}_{0\le\epsilon<\eta}$ be a family of measurable subsets…

This paper investigates uniqueness results for perturbed periodic Schr\"odinger operators on $\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -\Delta + V + v$, where $\Delta$ is the discrete Laplacian, $V: \mathbb{Z}^d…

We consider operators of the form H+V where H is the one-dimensional harmonic oscillator and V is a zero-order pseudo-differential operator which is quasi-periodic in an appropriate sense (one can take V to be multiplication by a periodic…

We consider a quantum particle moving in the one dimensional lattice Z and interacting with a indefinite sign external field v. We prove that the associated discrete Schroedinger operator H can have one or two eigenvalues, situated as below…

We consider two-dimensional Schroedinger operators with an attractive potential in the form of a channel of a fixed profile built along an unbounded curve composed of a circular arc and two straight semi-lines. Using a test-function…

Let ${\mathsf D}$ and ${\mathsf H}$ be the self-adjoint, one-dimensional Dirac and Schr\"odinger operators in $L^{2}(\mathbb{R};\mathbb{C}^{2})$ and $L^{2}(\mathbb{R};\mathbb{C})$ respectively. It is well known that, in absence of an…

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent)…

The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the…

The discrete spectra of certain two-dimensional Schrodinger operators are numerically calculated. These operators have interesting spectral properties, i.e. their kernels are multi-dimensional and the deformations of potentials via the…

We consider a generalized Schr\"odinger operator in $L^2(\mathbb R^2)$ describing an attractive $\delta'$ interaction in a strong coupling limit. $\delta'$ interaction is characterized by a coupling parameter $\beta$ and it is supported by…

We consider discrete one-dimensional Schr\"odinger operators whose potentials are generated by H\"older continuous sampling along the orbits of a uniformly hyperbolic transformation. For any ergodic measure satisfying a suitable bounded…

I present an example of a discrete Schr"odinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than…

We study various direct and inverse spectral problems for the one-dimensional Schr\"{o}dinger equation with distributional potential and boundary conditions containing the eigenvalue parameter.

The periodic Schrodinger operator $ H $ on a discrete periodic graph is considered. We estimate the discrete spectrum of the perturbed operator $ H _ {-} (t) = H-tV $, $ t> 0 $, where the potential $ V \ ge 0 $ is decreasing and $t>0$ is…

In this paper we consider Schr\"{o}dinger operators on $M \times \mathbb{Z}^{d_2}$, with $M=\{M_{1}, \ldots, M_{2}\}^{d_1}$ (`quantum wave guides') with a `$\Gamma$-trimmed' random potential, namely a potential which vanishes outside a…