We establish a spectral gap for resonances of the Laplacian of random Schottky surfaces, which is optimal according to a conjecture of Jakobson and Naud.
The equivalence of spectral convergence and Benjamini-Schramm convergence is extended from homogeneous spaces to spaces which are compact modulo isometry group. The equivalence is proven under the condition of a uniform discreteness property. It is open, which implications hold without this condition.
This paper deals with the concepts of upper and lower weakly α(Λ, sp)-continuous multifunctions. Moreover, several characterizations of upper and lower weakly α(Λ, sp)-continuous multifunctions are established.
Our main purpose is to introduce the notion of almost α(Λ, sp)-continuous multifunctions. Moreover, some characterizations of almost α(Λ, sp)-continuous multifunctions are established.
We consider 3-dim Schrödinger operators with a complex potential. We obtain new trace formulas. In order to prove these results we study analytic properties of a modified Fredholm determinant. In fact we reformulate spectral theory problems as the problems of analytic functions from Hardy spaces in upper half-plane.
This paper is devoted to establish semiclassical Weyl formulae for the Robin Laplacian on smooth domains in any dimension. Theirs proofs are reminiscent of the Born-Oppenheimer method.
Orthogonal decomposition of tensors is a generalization of the singular value decomposition of matrices. In this paper, we study the spectral theory of orthogonally decomposable tensors. For such a tensor, we give a description of its singular vector tuples as a variety in a product of projective spaces.
We describe a family of half-line continuum Schroedinger operators with purely singular continuous essential spectrum, exhibiting asymptotic strong level repulsion (known as clock behavior). This follows from the convergence of the renormalized continuum Christoffel-Darboux kernel to the sine kernel.
Variable order differential equations with non-integrable singularities are considered on spatial networks. Properties of the spectrum are established, and the solution of the inverse spectral problem is obtained.
For bounded domains, eigenvalues and eigenfunctions of double layer potentials are considered. The aim of this paper is to establish some relationships between eigenvalues, eigenfunctions and the geometry of domain boundaries.
We analyze spectral minimal $k$-partitions for the torus. In continuation with what we have obtained for thin annuli or thin strips on a cylinder (Neumann case), we get similar results for anisotropic tori.
In the present paper several bounds on multiplicities of eigenvalues of the Laplacian operator on surfaces are generalized from the case of either closed surface or simply-connected planar domain to the case of a surface of positive genus with holes.
We provide a class of unbounded three-dimensional domains of infinite volume for which the spectrum of the associated Dirichlet Laplacian is purely discrete. The construction is based on considering tubes with asymptotically diverging twisting angle.
We consider a model of planar PT-symmetric waveguide and study the phenomenon of the eigenvalues collision under the perturbation of boundary conditions. This phenomenon was discovered numerically in previous works. The main result of this work is an analytic explanation of this phenomenon.
This note proves the following inequality: if $n=3k$ for some positive integer $k$, then for any $n$ positive definite matrices $A_1,A_2,\cdots,A_n$, \begin{equation} \frac{1}{n^3}\Big\|\sum_{j_1,j_2,j_3=1}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\| \geq \frac{(n-3)!}{n!} \Big\|\sum_{\substack{j_1,j_2,j_3=1,\\\text{$j_1$, $j_2$, $j_3$ all distinct}}}^{n}A_{j_1}A_{j_2}A_{j_3}\Big\|, \end{equation} where $\|\cdot\|$ represents the operator norm. This inequality is a special case of a recent conjecture by Recht and Ré.
The present paper gives a priori bounds on the possible non-real eigenvalues of regular indefinite Sturm-Liouville problems and obtains sufficient conditions for such problems to admit non-real eigenvalues.
We prove the spectral instability of the complex cubic oscillator $-\frac{d^2}{dx^2}+ix^3+iαx$ for non-negative values of the parameter $α$, by getting the exponential growth rate of $\|Π_n(α)\|$, where $Π_n(α)$ is the spectral projection associated with the $n$-th eigenvalue of the operator. More precisely, we show that for all non-negative $α$ \[ \lim\limits_{n\to+\infty}\frac{1}{n}\log\|Π_n(α)\| = \fracπ{\sqrt{3}}. \]
A number of topics in the qualitative spectral analysis of the Schrödinger operator $-Δ+ V$ are surveyed. In particular, some old and new results concerning the positivity and semiboundedness of this operator as well as the structure of different parts of its spectrum are considered. The attention is focused on conditions both necessary and sufficient, as well as on their sharp corollaries.