Related papers: Schr\"odinger operators with $\delta$-potentials s…
We show that a Schr\"odinger operator $A_{\delta, \alpha}$ with a $\delta$-interaction of strength $\alpha$ supported on a bounded or unbounded $C^2$-hypersurface $\Sigma \subset \mathbb{R}^d$, $d\ge 2$, can be approximated in the norm…
Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity…
This paper deals with the approximation of a magnetic Schr\"odinger operator with a singular $\delta$-potential that is formally given by $(i \nabla + A)^2 + Q + \alpha \delta_\Sigma$ by Schr\"odinger operators with regular potentials in…
We study the Schr\"odinger operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\R^3)$ with a $\delta$ interaction supported by an infinite non-planar surface $\Gamma$ which is smooth, admits a global normal parameterization with a…
In this paper we consider the three-dimensional Schr\"{o}dinger operator with a $\delta$-interaction of strength $\alpha > 0$ supported on an unbounded surface parametrized by the mapping $\mathbb{R}^2\ni x\mapsto (x,\beta f(x))$, where…
We analyze a family of singular Schr\"odinger operators with local singular interactions supported by a hypersurface $\Sigma \subset \mathbb{R}^n, n \ge 2$, being the boundary of a Lipschitz domain, bounded or unbounded, not necessarily…
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 $$…
We investigate the spectral properties of self-adjoint Schr\"odinger operators with attractive $\delta$-interactions of constant strength $\alpha > 0$ supported on conical surfaces in ${\mathbb R}^3$. It is shown that the essential spectrum…
In this paper we study the spectrum of self-adjoint Schr\"odinger operators in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a smooth closed curve $\Sigma\subseteq \mathbb{R}^2$. Although these $\textit{oblique}$…
We consider self-adjoint Schr\"odinger operators in $L^2 (\mathbb{R}^d)$ with a $\delta$-interaction of strength $\alpha$ and a $\delta'$-interaction of strength $\beta$, respectively, supported on a hypersurface, where $\alpha$ and…
We prove and apply two theorems: First, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schr\"odinger operator on a bounded or unbounded domain, second, a perturbation and lifting estimate…
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)…
We investigate the operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\mathbb{R}^3)$, where $\Gamma$ is a smooth surface which is either compact or periodic and satisfies suitable regularity requirements. We find an asymptotic expansion…
We investigate Schr\"odinger operators with \delta- and \delta'-interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial…
In this paper we study spectral properties of a three-dimensional Schr\"odinger operator $-\Delta+V$ with a potential $V$ given, modulo rapidly decaying terms, by a function of the distance of $x \in \mathbb{R}^3$ to an infinite conical…
We shall consider the Schr\"odinger operators on $\mathbf{R}^2$ with random $\delta$ magnetic fields. Under some mild conditions on the positions and the fluxes of the $\delta$-fields, we prove the spectrum coincides with $[0,\infty)$ and…
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…
We consider the Schr\"odinger operator $\mathcal L_{\alpha}$ on the half-line with a periodic background potential and a perturbation which consists of two parts: a summable potential and the slowly decaying Wigner--von Neumann potential…
We study Stark Hamiltonians with a $\delta$-interaction supported on a compact hypersurface in $\mathbb R^d$. Let $\Sigma$ be a compact Lipschitz hypersurface and let $\alpha\in L^\infty(\Sigma;\mathbb R)$. We define the operator…
The spectral properties of two-dimensional Schr\"odinger operators with $\delta'$-potentials supported on star graphs are discussed. We describe the essential spectrum and give a complete description of situations in which the discrete…