Related papers: Singular Schr\"odinger operators with prescribed s…

We study the Schr\"odinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and…

We use B\'{e}zout's theorem and Bernstein-Khovanskii-Kushnirenko theorem to analyze the level sets of the extrema of the spectral band functions of discrete periodic Schr\"odinger operators on $\mathbb{Z}^2$. These approaches improve upon…

We study discrete spectral quantities associated to Schr\"odinger operators of the form $-\Delta_{\mathbb{R}^d}+V_N$, $d$ odd. The potential $V_N$ models a highly disordered crystal; it varies randomly at scale $N^{-1} \ll 1$. We use…

We study the manner in which spectral shift functions associated with self-adjoint one-dimensional Schr\"odinger operators on the finite interval $(0,R)$ converge in the infinite volume limit $R\to\infty$ to the half-line spectral shift…

We study two- and three-dimensional matrix Schr\"odinger operators with $m\in \mathbb N$ point interactions. Using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results obtained by…

We study singular Schrodinger operators with an attractive interaction supported by a closed smooth surface A in R^3 and analyze their behavior in the vicinity of the critical situation where such an operator has empty discrete spectrum and…

We consider the discrete eigenvalues of the operator $H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$ is localized on $\R^d,\ \ d\ge1$. For $\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate…

We proved that Schr\"odinger operators with unbounded potentials $(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n$ have purely singular continuous spectrum on the set $\{E:…

We continue the investigation of the existence of absolutely continuous (a.c.) spectrum for the discrete Schr\"odinger operator $\Delta+V$ on $\ell^2(\Z^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying the long range condition…

Our paper investigates one-dimensional Schr\"odinger operators defined as closed operators on $L^2(\mathbb{R})$ or $L^2(\mathbb{R}_+)$ that are exactly solvable in terms of confluent functions (or, equivalently, Whittaker functions). We…

We study the relation between spectral flow and index theory within the framework of (unbounded) KK-theory. In particular, we consider a generalised notion of 'Dirac-Schr\"odinger operators', consisting of a self-adjoint elliptic…

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 the Schr{\"o}dinger operator $-\Delta +V(x)$ in $L^2({\bf R}^3)$ with a real short-range (integrable) potential $V$. Using the associated Fredholm determinant, we present new trace formulas, in particular, the ones in terms of…

We show that a generic quasi-periodic Schr\"odinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling…

This paper is devoted to estimates of the exponential decay of eigenfunctions of difference operators on the lattice Z^n which are discrete analogs of the Schr\"{o}dinger, Dirac and square-root Klein-Gordon operators. Our investigation of…

In this paper, new self-adjoint realizations of the Dirac operator in dimension two and three are introduced. It is shown that they may be associated with the formal expression $\mathcal{D}_0+|F\delta_\Sigma\rangle\langle G\delta_\Sigma|$,…

This work focuses on the analysis of the spectral $\zeta$-function associated with a Schr\"{o}dinger operator endowed with a P\"oschl--Teller potential. We construct the spectral $\zeta$-function using a contour integral representation and,…

A derivation of the spectral determinant of the Schr\"odinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of Kostrykin and Schrader.…

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…

The problem of self-adjoint extensions of Dirac-type operators in manifolds with boundaries is analysed. The boundaries might be regular or non-regular. The latter situation includes point-like interactions, also called delta-like…