Related papers: Negative eigenvalues of non-local Schr\"{o}dinger …

We consider discrete one-dimensional random Schroedinger operators with decaying matrix-valued, independent potentials. We show that if the l^2-norm of this potential has finite expectation value with respect to the product measure then…

We study the discrete eigenvalues emerging from the threshold of the essential spectrum of one or two-dimensional Schr\"odinger operators with complex-valued $ L^p $-potentials in a weak coupling regime. We derive necessary and sufficient…

We show that the measure of the spectrum of Schr\"odinger operator with potential defined by non-constant function over any minimal aperiodic finite subshift tends to zero, as the coupling constant tends to infinity. We also obtained a…

We strengthen and generalise a result of Kirsch and Simon on the behaviour of the function $N_L(E)$, the number of bound states of the operator $L = \Delta+V$ in $\R^d$ below $-E$. Here $V$ is a bounded potential behaving asymptotically…

We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of armchair single-wall nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite…

We prove conditions on potentials which imply that the sum of the negative eigenvalues of the Schroeodinger operator is finite. We use a method for bounding eigenvalues based on estimates of the Hilbert-Schmidt norm of semigroup differences…

The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic…

We review recent advances in the spectral theory of Schr\"odinger operators with decaying potentials. The area has seen spectacular progress in the past few years, stimulated by several conjectures stated by Barry Simon starting at the 1994…

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 prove an asymptotic expansion for the eigenvalues and eigenfunctions of Schr\"{o}dinger-type operator with a confining potential and with principle part a periodic elliptic operator in divergence form. We compare the spectrum to the…

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…

In this paper we investigate the one-dimensional Schrodinger operator L(q) with complex-valued periodic potential q when q\inL_{1}[0,1] and q_{n}=0 for n=0,-1,-2,..., where q_{n} are the Fourier coefficients of q with respect to the system…

We give a short proof of the Cwikel-Lieb-Rozenblum (CLR) bound on the number of negative eigenvalues of Schr\"odinger operators. The argument, which is based on work of Rumin, leads to remarkably good constants and applies to the case of…

We study in dimension $d\geq2$ low-energy spectral and scattering asymptotics for two-body $d$-dimensional Schr\"odinger operators with a radially symmetric potential falling off like $-\gamma r^{-2},\;\gamma>0$. We consider angular…

We discuss Schr\"odinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous…

Consider the discrete 1D Schr\"odinger operator on $\Z$ with an odd $2k$ periodic potential $q$. For small potentials we show that the mapping: $q\to $ heights of vertical slits on the quasi-momentum domain (similar to the…

The singular real second order 1D Schrodinger operators are considered here with such potentials that all local solutions near singularities to the eigenvalue problem are meromorphic for all values of the spectral parameter. All…

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…

We study the low-energy asymptotics of the spectral shift function for Schr\"odinger operators with potentials decaying like $O(\frac{1}{|x|^2})$. We prove a generalized Levinson's for this class of potentials in presence of zero eigenvalue…

Several aspects of complex-valued potentials generating a real and positive spectrum are discussed. In particular, we construct complex-valued potentials whose corresponding Schr\"odinger eigenvalue problem can be solved analytically.