Related papers: Schr\"odinger operators with $\delta$-interactions…

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 consider normalized Laplacians and their perturbations by periodic potentials (Schr\"odinger operators) on periodic discrete graphs. The spectrum of the operators consists of an absolutely continuous part (a union of a finite number of…

We consider a family of random Schr\"odinger operators on the discrete strip with decaying random $\ell^2$ matrix potential. We prove that the spectrum is almost surely pure absolutely continuous, apart from random, possibly embedded…

We study Schr\"odinger operators $H=-\Delta+V$ in $L^2(\Omega)$ where $\Omega$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential…

We study the two--dimensional magnetic Schr\"odinger operator with a penetrable circular wall modeled by a $\delta$--interaction. Using the boundary triple approach we classify all self--adjoint extensions and obtain Krein's resolvent…

Schr\"{o}dinger operators of the form $\Delta - W$ on $L^2_{\text{rad}}(\mathbb{R}^3)$, the space of radially symmetric square integrable functions are relevant in a variety of physical contexts. The potential $W$ is taken to be radially…

We consider discrete Schr\"odinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the…

Let $\Gamma\subset \mathbb{R}^2$ be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve $|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues of the Schr\"odinger operator…

This article is devoted to the spectral analysis of the electro-magnetic Schr\"odinger operator on the Euclidean plane. In the semiclassical limit, we derive a pseudo-differential effective operator that allows us to describe the spectrum…

In this paper we consider the two-dimensional Schr\"odinger operator with an attractive potential which is a multiple of the characteristic function of an unbounded strip-shaped region, whose thickness is varying and is determined by the…

We consider a discrete Schroedinger operator whose potential is the sum of a Wigner-von Neumann term and a summable term. The essential spectrum of this operator equals to the interval [-2,2]. Inside this interval, there are two critical…

We study Schr\"odinger operators with Floquet boundary conditions on flat tori obtaining a spectral result giving an asymptotic expansion of all the eigenvalues. The expansion is in $\lambda^{-\delta}$ with $\delta\in(0,1)$ for most of the…

We provide a purely variational proof of the existence of eigenvalues below the bottom of the essential spectrum for the Schr\"odinger operator with an attractive $\delta$-potential supported by a star graph, i.e. by a finite union of rays…

We consider non-self-adjoint Schr\"odinger operators $\Delta+V$ where $\Delta$ is the Laplace-Beltrami operator on a Zoll manifold $X$ and $V\in C^\infty(X,\mathbb C)$. We obtain asymptotic results on the pseudo-spectrum and numerical range…

We show that the spectrum of a discrete two-dimensional periodic Schr\"odinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least one dimension. In general, we show that…

In this paper we consider the Schr\"odinger operator in ${\mathbb R}^3$ with a long-range magnetic potential associated to a magnetic field supported inside a torus ${\mathbb{T}}$. Using the scheme of smooth perturbations we construct…

We study a family of discrete one-dimensional Schr\"odinger operators with power-like decaying potentials with rapid oscillations. In particular, for the potential $V(n)=\lambda n^{-\alpha}\cos(\pi \omega n^\beta)$, with $1<\beta<2\alpha$,…

For the discrete Schr\"odinger operator we obtain sharp estimates for the number of negative eigenvalues.

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…

We study the discrete Schr\"odinger operator $H$ in $\ZZ^d$ with the surface potential of the form $V(x)=g \delta(x_1) \tan \pi(\alpha \cdot x_2+ \omega)$, where for $x \in \ZZ^d$ we write $x=(x_1,x_2), \quad x_1 \in \ZZ^{d_1}, x_2 \in…