Related papers: Schr\"odinger operators with negative potentials a…

We prove resolvent estimates in the Euclidean setting for Schr\"{o}dinger operators with potentials in Lebesgue spaces: $-\Delta+V$. The $(L^{2}, L^{p})$ estimates were already obtained by Blair-Sire-Sogge, but we extend their result to…

We analyze properties of semigroups generated by Schr\"odinger operators $-\Delta+V$ or polyharmonic operators $-(-\Delta)^m$, on metric graphs both on $L^p$-spaces and spaces of continuous functions. In the case of spatially constant…

For Schr\"odinger operator $H=-\Delta+ V({\mathbf x})\cdot$, acting in the space $L_2(\mathbb R^d)\,(d\ge 3)$, necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum.are obtained without assumption that…

This note proves convexity resp. concavity of the ground state energy of one dimensional Schr\"odinger operators as a function of an endpoint of the interval for convex resp. concave potentials.

Motivated by the recent results in arXiv:1601.05679 about the quark-antiquark potential in $\mathcal N=4$ SYM, we reconsider the problem of computing the asymptotic weak-coupling expansion of the ground state energy of a certain class of 1d…

We give an abstract definition of a one-dimensional Schr\"odinger operator with $\delta'$-interaction on an arbitrary set~$\Gamma$ of Lebesgue measure zero. The number of negative eigenvalues of such an operator is at least as large as the…

In this paper, we investigate the existence of multiple positive solutions to the following multi-critical Schr\"{o}dinger equation \begin{equation} \label{p} \begin{cases} -\Delta u+\lambda V(x)u=\mu…

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 the existence and concentration behavior of the bound states for the following logarithmic Schr\"odinger equation \begin{equation*} \begin{cases} -\varepsilon^2\Delta v+V(x)v=v\log v^2 \ \ &\text {in}\ \ \mathbb R^N,\\ v(x)\to 0 \…

We prove existence of positive ground state solutions to the pseudo-relativistic Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-\Delta +m^2} u +Vu = \left( W * |u|^{\theta} \right)|u|^{\theta -2} u \quad\text{in…

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 consider the difference $f(-\Delta +V)-f(-\Delta)$ of functions of Schr\"odinger operators in $L^2(\mathbb R^d)$ and provide conditions under which this difference is trace class. We are particularly interested in non-smooth functions…

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schr\"odinger equation $$ -\Delta u+ u=u^{2^*-1}+\lambda u^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the…

For Schr\"odinger operators with potentials that are asymptotically homogeneous of degree $-2$, the size of the coupling determines whether it has finite or infinitely many negative eigenvalues. In the latter case the asymptotic…

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…

In this note we consider the self-adjoint Schr\"odinger operator $\mathsf{A}_\alpha$ in $L^2(\mathbb{R}^d)$, $d\geq 2$, with a $\delta$-potential supported on a Lipschitz hypersurface $\Sigma\subseteq\mathbb{R}^d$ of strength $\alpha\in…

We study localization properties of low-lying eigenfunctions $$(-\Delta +V) \phi = \lambda \phi \qquad \mbox{in}~\Omega$$ for rapidly varying potentials $V$ in bounded domains $\Omega \subset \mathbb{R}^d$. Filoche & Mayboroda introduced…

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem $L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$…

We show that whole-line Schr\"odinger operators with finitely many bound states have no embedded singular spectrum. In contradistinction, we show that embedded singular spectrum is possible even when the bound states approach the essential…

In this note we provide an explicit lower bound on the spectral gap of one-dimensional Schr\"odinger operators with non-negative bounded potentials and subject to Neumann boundary conditions.