Related papers: Estimates for the number of eigenvalues of two dim…

We prove dynamical upper bounds for discrete one-dimensional Schroedinger operators in terms of various spacing properties of the eigenvalues of finite volume approximations. We demonstrate the applicability of our approach by a study of…

We estimate sums of functions of negative eigenvalues of Schr\"odinger-type operators whose kinetic energy vanishes on a codimension one submanifold. Our main technical tool is the Stein-Tomas theorem and some of its generalizations.

We consider self-adjoint Schr\"odinger operators in $L^2 (\mathbb{R}^d)$ with a $\delta$-interaction of strength $\alpha$ and a $\delta'$-interaction of strength $\beta$, respectively, supported on a hypersurface, where $\alpha$ and…

We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of order gamma >= 3/4 for fourth order Schr\"odinger operators in arbitrary dimensions. We also consider some extensions to polyharmonic operators, and to systems of…

In 1976 Lieb and Thirring established upper bounds on sums of powers of the negative eigenvalues of a Schr\"odinger operator in terms of semiclassical phase-space integrals. Over the last 45 years the optimal constants in these…

For a large family of real-valued Radon measures m on R^d, including the Kato class, the operators -\Delta + C^2 \Delta^2 + m tend to the Schrodinger operator -\Delta +m in the norm resolvent sense as C tends to zero. If the measure is…

We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\infty}(\mathbb R)$. The eigenfunctions are described in terms of the confluent hypergeometric…

We consider a wide class of the two-particle Schr\"{o}dinger operators $H_{\mu}(k)=H_{0}(k)+\mu V, \,\mu>0,$ with a fixed two-particle quasi-momentum $k$ in the $d$ -dimensional torus $\mathbb{T}^d$, associated to the Bose-Hubbard…

Dealing with one-dimensional diffusion operators, we obtain upper and lower variational formulae on the eigenvalues given by the max-min principle, generalizing the celebrated result of Chen and Wang on the spectral gap. Our inequalities…

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,\Delta^2)$ associated with fourth-order Schr\"odinger operators $H=\Delta^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim…

The threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type $$ H_\lambda=-\frac{d^2}{dx^2}+U(x)+\lambda\alpha_\lambda V(\alpha_\lambda x) $$ is considered. The potentials $U$ and $V$ are real-valued bounded…

We present a novel deep learning method for computing eigenvalues of the fractional Schr\"odinger operator. Our approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior…

We establish an upper bound on the multiplicity of eigenvalues of the Sch\"odinger-Dirichlet operator in dimension two. We give a proof based on a generalized Morse Lemma due to Cheng \cite{Ch}.

The Schroedinger equation with one and two dimensional potentials are solved in the frame work of the sl(2) Lie algebra. Eigenfunctions of the Schroedinger equation for various asymmetric double-well potentials have been determined and the…

For the direct problem, we give the asymptotic distribution of the (real and non-real) transmission eigenvalues for the Schrodinger operator on the half line. For the inverse problem, we prove that the potential can be uniquely determined…

We consider non-selfadjoint operators of the kind arising in linearized NLS and prove dispersive bounds for the time-evolution without assuming that the edges of the essential spectrum are regular. Our approach does not depend on any…

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^2)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are…

We consider the Schrodinger operator a given domain. Our goal is to study some optimization problems where an optimal (non-negative) potential V has to be determined in some suitable admissible classes and for some suitable optimization…

We consider a Schr\"odinger operator on the half-line with a Dirichlet boundary condition at the origin and show that moments of its negative eigenvalues can be estimated by the part of the potential that is larger than the critical Hardy…

In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schr\"odinger equation with quartic potentials. We consider the eigenvalue problem with a complex-valued…