Related papers: Strong coupling asymptotics for a singular Schroed…

We consider discrete Schr\"odinger operators $H_{\mu Q}=\Delta+\mu Q$ with real periodic potentials $Q$ on periodic graphs, where $\Delta$ is the adjacency operator and $\mu\in\mathbb R$ is a coupling constant. The spectra of the operators…

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 deal with the nonlinear Schr\"odinger system \[ -\Delta u_i =\mu_i u_i^3 + \beta u_i \sum_{j\neq i} u_j^2 + \lambda_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(\Omega) \] in dimension 4, a problem with critical Sobolev exponent.…

This paper studies the sharp $L^p$-$L^q$ boundedness of the Bochner-Riesz operator $S^{\delta}_{\lambda}(\mathcal{L}_{\mathbf{A}})$ associated with a scaling-critical magnetic Schr\"odinger operator $\mathcal{L}_{\mathbf{A}}$ on…

We consider discrete one-dimensional Schroedinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of…

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…

I consider two-dimensional Schr\"odinger operator with degenerating magnetic field and in the generic situation I derive spectral asymptotics as $h\to +0$ and $\mu\to +\infty$ where $h$ and $\mu$ are Planck and coupling parameters…

Let P be the operator $-\Delta+V$ on R^d, where $V$ is a real potential with several inverse square singularities. The usual non-trapping type high-frequency inequality on the truncated resolvent of $P$ is shown, using semi-classical…

In this paper we introduce and study a family of self-adjoint realizations of the Laplacian in $L^2(\mathbb{R}^2)$ with a new type of transmission conditions along a closed bi-Lipschitz curve $\Sigma$. These conditions incorporate jumps in…

The paper studies the spectral properties of the Schr\"odinger operator $A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the free Laplacian…

We consider the two-dimensional Dirac operator with Lorentz-scalar $\delta$-shell interactions on each edge of a star-graph. An orthogonal decomposition is performed which shows such an operator is unitarily equivalent to an orthogonal sum…

We consider the operator $ L = - (d/dx)^2 + x^2 y + w(x) y , y \in L^2(\mathbb{R}) $, where $ w(x) = s [ \delta(x - b) - \delta(x + b)], b \neq 0,$ real, $s \in \mathbb{C}$. This operator has a discrete spectrum: eventually the eigenvalues…

We discuss the eigenvalues $E_j$ of Schr\"odinger operators $-\Delta+V$ in $L^2(\mathbb R^d)$ with complex potentials $V\in L^p$, $p<\infty$. We show that (A) $\mathrm{Re} E_j\to\infty$ implies $\mathrm{Im} E_j\to 0$, and (B) $\mathrm{Re}…

Spectral properties of Schr\"odinger operators on compact metric graphs are studied and special emphasis is put on differences in the spectral behavior between different classes of vertex conditions. We survey recent results especially for…

We consider $C=A+B$ where $A$ is selfadjoint with a gap $(a,b)$ in its spectrum and $B$ is (relatively) compact. We prove a general result allowing $B$ of indefinite sign and apply it to obtain a $(\delta V)^{d/2}$ bound for perturbations…

Let $\alpha,\beta$ be orientation-preserving homeomorphisms of $[0,\infty]$ onto itself, which have only two fixed points at $0$ and $\infty$, and whose restrictions to $\mathbb{R}_+=(0,\infty)$ are diffeomorphisms, and let…

We consider the Hardy-Schr\"odinger operator $ -\Delta_{\mathbb{B}^n}-\gamma{V_2}$ on the Poincar\'e ball model of the Hyperbolic space ${\mathbb{B}^n}$ ($n \geq 3$). Here $V_2$ is a well chosen radially symmetric potential, which behaves…

We consider a $2\times2$ block operator matrix ${\mathcal A}_\mu$ $($$\mu>0$ is a coupling constant$)$ acting in the direct sum of one- and two-particle subspaces of a bosonic Fock space. The location of the essential spectrum of ${\mathcal…

We characterize the domain of the Schr\"odinger operators $S=-\Delta+c|x|^{-\alpha}$ in $L^p(\mathbb{R}^N)$, with $0<\alpha<2$ and $c\in\mathbb{R}$. When $\alpha p< N$, the domain characterization is essentially known and can be proved…

We study the asymptotic behaviour of the spectral gap of Schr\"odinger operators in two and higher dimensions and in a limit where the volume of the domain tends to infinity. Depending on properties of the underlying potential, we will find…