Related papers: Gordon type Theorem for measure perturbation

It is known that the spectrum of Schr\"odinger operators with sparse potentials consists of singular continuous spectrum. We give a sufficient condition so that the edge of the singular continuous spectrum is not an eigenvalue and construct…

Schr\"odinger perturbations of transition densities by singular potentials may fail to be comparable with the original transition density. For instance this is so for the transition density of a subordinator perturbed by any…

We consider the nonlinear Schr\"odinger equation on the $d$-dimensional torus $\mathbb T^d$, with the nonlinearity of polynomial type $|u|^{2\sigma}u$. For any $\sigma \in \mathbb N$ and $s>\frac d2$ we prove that adding to this equation a…

We utilize the theory of de Branges spaces to show when certain Schr\"odinger operators with strongly singular potentials are uniquely determined by their associated spectral measure. The results are applied to obtain an inverse uniqueness…

We prove a generalization of the well-known theorems by Borg and Hochstadt for periodic self-adjoint Schr\"odinger operators without a spectral gap, respectively, one gap in their spectrum, in the matrix-valued context. Our extension of the…

We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original…

We prove an analogue of Beurling's theorem on the H-type groups of certain dimensions after establishing the Gutzmer's formula for the H-type groups. We also obtain some other versions of the theorem using the modified Radon transform.

In this work we take into consideration a generalization of Gauge Theories based on the analysis of the structural characteristics of Maxwell theory, which can be considered as the prototype of such kind of theories (Maxwell-like). Such…

A general form for ladder operators is used to construct a method to solve bound-state Schr\"odinger equations. The characteristics of supersymmetry and shape invariance of the system are the start point of the approach. To show the…

Recently developed simple approach for the exact/approximate solution of Schrodinger equations with constant/position-dependent mass, in which the potential is considered as in the perturbation theory, is shown to be equivalent to the one…

Recently the KAM theory has been extended to multidimensional PDEs. Nevertheless all these recent results concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the…

We prove unique continuation properties for solutions of evolution Schr\"odinger equation with time dependent potentials. In the case of the free solution these correspond to uncertainly principles referred to as being of Morgan type. As an…

We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the…

We discuss resonances for Schr\"odinger operators with compactly supported potentials on the line and the half-line. We estimate the sum of the negative power of all resonances and eigenvalues in terms of the norm of the potential and the…

Using the recently defined concept of Taylor measures, we propose a generalization of Taylor's theorem to measurable, non-analytic functions, that do not require differentiation. We study consequences of the generalization, including the…

Let L be a Schroedinger operator with potential W in L^{(n+1)/2}. We prove that there is no embedded eigenvalue. The main tool is an Lp Carleman type estimate, which builds on delicate dispersive estimates established in a previous paper.…

This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl-Totik theory of regular measures, especially the case of OPRL and OPUC. Links…

We study sufficient conditions for the absence of positive eigenvalues of magnetic Schr\"odinger operators in $\mathbb{R}^d,\, d\geq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends…

In our previous work, we introduced a new class of bounded potentials of the one-dimensional Schr\"odinger operator on the real axis, and a corresponding family of solutions of the KdV hierarchy. These potentials, which we call primitive,…

We prove an analog of Perron-Frobenius theorem for multilinear forms with nonnegative coefficients, and more generally, for polynomial maps with nonnegative coefficients. We determine the geometric convergence rate of the power algorithm to…