Related papers: On the domain of a magnetic Schr\"odinger operator…

We consider vector-valued magnetic Schr\"odinger operators $-\bm \Delta_{\bm a}+V$ with magnetic potential $\bm a \in L^2_{\mathrm{loc}}(\mathbb{R}^d;\mathbb{R}^d)$ and electric potential $V$ given by a matrix-valued function whose entries…

In this article we prove the property of unique continuation (also known for C^\infty functions as quasianalyticity) for solutions of the differential inequality |\Delta u| \leq |Vu| for V from a wide class of potentials (including…

The purpose of this paper is to study spectral properties of non-self-adjoint Schr\"odinger operators $-\Delta-\frac{(n-2)^2}{4|x|^{2}}+V$ on $\mathbb{R}^n$ with complex-valued potentials $V\in L^{p,\infty}$, $p>n/2$. We prove Keller type…

In this paper we consider the vector-valued Schr\"{o}dinger operator $-\Delta + V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L^1_{\rm loc}(\mathbb{R}^d)$ and, for every $x\in\mathbb{R}^d$, $V(x)$ is…

Let $\Omega$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-\Delta +V(x)$ be a Schr\"odinger operator on $\Omega$ with potential V locally bounded. We seek the weakest conditions we can find on the…

Laptev and Safronov conjectured that any non-positive eigenvalue of a Schr\"odinger operator $-\Delta+V$ in $L^2(\mathbb R^\nu)$ with complex potential has absolute value at most a constant times…

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…

We consider the Schr\"odinger operator $-\Delta+V$ for negative potentials $V$, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of $-\Delta+V$ is positive, provided that $V$ is greater than…

One-dimensional Schr\"odinger operators with singular perturbed magnetic and electric potentials are considered. We study the strong resolvent convergence of two families of the operators with potentials shrinking to a point. Localized…

We consider optimization problems for cost functionals which depend on the negative spectrum of Schr\"odinger operators of the form $-\Delta+V(x)$, where $V$ is a potential, with prescribed compact support, which has to be determined. Under…

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 the present paper we consider spectral optimization problems involving the Schr\"odinger operator $-\Delta +\mu$ on $\R^d$, the prototype being the minimization of the $k$ the eigenvalue $\lambda_k(\mu)$. Here $\mu$ may be a capacitary…

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}…

We prove three results giving sufficient and/or necessary conditions for discreteness of the spectrum of Schr\"odinger operators with non-negative matrix-valued potentials, i.e., operators acting on $\psi\in L^2(\mathbb{R}^n,\mathbb{C}^d)$…

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…

We propose a rigorous method for computing two-sided eigenvalue bounds of the Schr\"odinger operator $H=-\Delta+V$ with a confining potential on $\mathbb{R}^2$. The method combines domain truncation to a finite disk $D(R)$ on which the…

We prove eigenvalue bounds for Schr\"odinger operator $-\Delta_g+V$ on compact manifolds with complex potentials $V$. The bounds depend only on an $L^q$-norm of the potential, and they are shown to be optimal, in a certain sense, on the…

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 prove optimal spectral inequalities for Landau operators in full space and in arbitrary dimension. Spectral inequalities are lower bounds on the L 2 -mass of functions in spectral subspaces of finite energy when integrated over a…

Given a potential $V$ and the associated Schr\"odinger operator $-\Delta+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ enjoys suitable…