Related papers: Spectral inequalities for Schr\"odinger equations …
This paper deals with the $L_p$-spectrum of Schr\"odinger operators on the hyperbolic plane. We establish Lieb-Thirring type inequalities for discrete eigenvalues and study their dependence on $p$. Some bounds on individual eigenvalues are…
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…
We investigate spectral properties of limit-periodic Schr\"odinger operators in $\ell^2(\Z)$. Our goal is to exhibit as rich a spectral picture as possible. We regard limit-periodic potentials as generated by continuous sampling along the…
In this paper we obtain minimal support properties of solutions of Schr\"odinger equations. We improve previously known conditions on the potential for which the measure of the support of solutions cannot be too small. We also use these…
We investigate approximate null-controllability for semi-discrete heat equations on the lattice $h\mathbb{Z}^d$ with a potential. By establishing spectral inequalities for the discrete Schr{\"o}dinger operator $P_h = -\Delta_h + V$ on…
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…
We study the spectral zeta functions associated to the radial Schr\"odinger problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the quantum Wronskian equation, we provide results such as closed-form evaluations for…
We investigate the spectral properties of Sturm-Liouville operators with measure potentials. We obtain two-sided estimates for the spectral distribution function of the eigenvalues. As a corollary, we derive a criterion for the discreteness…
Schr\"odinger operator on half-line with complex potential and the corresponding evolution are studied within perturbation theoretic approach. The total number of eigenvalues and spectral singularities is effectively evaluated. Wave…
Schr\"odinger operators with potentials generated by primitive substitutions are simple models for one dimensional quasi-crystals. We review recent results on their spectral properties. These include in particular an algorithmically…
An explicit solution of the spectral problem of the non-local Schr\"odinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of…
We establish an observation inequality for the Schr\"odinger equation on $\mathbf{R}^d$, uniform in the Planck constant $\hbar\in[0,1]$. The proof is based on the pseudometric introduced in [F. Golse, T. Paul, Arch. Rational Mech. Anal. 223…
In this article we consider asymptotics for the spectral function of Schr\"odinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order…
We introduce a notion of spectral singularity that applies for a general class of nonlinear Schreodinger operators involving a confined nonlinearity. The presence of the nonlinearity does not break the parity-reflection symmetry of spectral…
In the first part of the paper we consider the Schr\"odinger operator $ -\Delta-V(x),\quad V>0. $ We discuss the relation between the behavior of $V$ at the infinity and the properties of the negative spectrum of $H$. After that, we…
We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum…
The present article delves into the investigation of observability inequalities pertaining to backward stochastic evolution equations. We employ a combination of spectral inequalities, interpolation inequalities, and the telegraph series…
The spectral potential is the dynamical generalization of the Kohn-Sham potential. It targets, in principle exactly, the spectral function in addition to the electronic density. Here we examine the spectral potential in one of the simplest…
We analyze spectral properties of the operator $H=\frac{\partial^2}{\partial x^2} -\frac{\partial^2}{\partial y^2} +\omega^2y^2-\lambda y^2V(x y)$ in $L^2(\mathbb{R}^2)$, where $\omega\ne 0$ and $V\ge 0$ is a compactly supported and…
We consider decaying oscillatory perturbations of periodic Schr\"odinger operators on the half line. More precisely, the perturbations we study satisfy a generalized bounded variation condition at infinity and an $L^p$ decay condition. We…