Related papers: Remarks on counting negative eigenvalues of Schr\"…

Spectral properties of the Schroedinger operator $A_{\lambda} = -\Delta +\lambda V$ on regular metric trees are studied. It is shown that as $\lambda$ goes to zero the behavior of the negative eigenvalues of $A_{\lambda}$ depends on the…

We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0,$ we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity.…

Consider a regular $d$-dimensional metric tree $\Gamma$ with root $o$. Define the Schroedinger operator $-\Delta - V$, where $V$ is a non-negative, symmetric potential, on $\Gamma$, with Neumann boundary conditions at $o$. Provided that $V$…

For a two-dimensional Schr\"odinger operator $H_{\alpha V}=-\Delta-\alpha V$ with the radial potential $V(x)=F(|x|), F(r)\ge 0$, we study the behavior of the number $N_-(H_{\alpha V})$ of its negative eigenvalues, as the coupling parameter…

For a normalized root system $R$ in $\mathbb R^N$ and a multiplicity function $k\geq 0$ let $\mathbf N=N+\sum_{\alpha \in R} k(\alpha)$. We denote by $dw(\mathbf{x})=\Pi_{\alpha \in R}|\langle \mathbf{x},\alpha…

The Schr\"odinger operator on a metric tree is a family of ordinary differential operators on its edges complemented by certain matching conditions at the vertices. The regular trees are highly symmetric. This allows one to construct an…

We present upper estimates for the number of negative eigenvalues of two-dimensional Schroedinger operators with potentials generated by Ahlfors regular measures of arbitrary dimension $\alpha\in (0, 2]$.The estimates are given in terms of…

The paper presents estimates for the number of negative eigenvalues of a two-dimensional Schr\"odinger operator in terms of $L\log L$ type Orlicz norms of the potential and proves a conjecture by N.N. Khuri, A. Martin and T.T. Wu.

In this article, we consider the semiclassical Schr\"odinger operator $P = - h^{2} \Delta + V$ in $\mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $\lambda_{k} ( P )$ as $h \to…

We prove a universal bound for the number of negative eigenvalues of Schr\"odinger operators with Neumann boundary conditions on bounded H\"older domains, under suitable assumptions on the H\"older exponent and the external potential. Our…

We consider a nonlocal differential--difference Schr\"odinger operator on a segment with the Neumann conditions and two translations in the free term. The values of the translations are denoted by $\alpha$ and $\beta$ and are treated as…

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…

We consider Schr\^odinger operators $H_\alpha$ given by equation (1.1) below. We study the asymptotic behavior of the spectral density $E(H_\alpha, \lambda)$ when $\lambda$ goes to $0$ and the $L^1\to L^\infty$ dispersive estimates…

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…

Let L be a Schr\"odinger operator of the form L=-\Delta+V, where the nonnegative potential V satisfies a reverse H\"older inequality. Using the method of L-harmonic extensions we study regularity estimates at the scale of adapted H\"older…

We derive H\"older regularity estimates for operators associated with a time independent Schr\"odinger operator of the form $-\Delta+V$. The results are obtained by checking a certain condition on the function $T1$. Our general method…

In the present paper we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators $L=-(\nabla-i\mathbf{a})^TA(\nabla-i\mathbf{a})+V$. The latter class includes, in…

Let $H:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. Let $\|\nabla^\alpha e^{-tH}\|_{(L^{p,\sigma}\to L^{q,\theta})}$ be the operator norm…

We investigate the negative part of the spectrum of the operator $-\partial^2 - \mu$ on $L^2(\mathbb R)$, where a locally finite Radon measure $\mu \geq 0$ is serving as a potential. We obtain estimates for the eigenvalue counting function,…