Related papers: On a minimax principle in spectral gaps

We study the phenomenon of an eigenvalue emerging from essential spectrum of a Schroedinger operator perturbed by a fast oscillating compactly supported potential. We prove the sufficient conditions for the existence and absence of such…

We develop an analytic perturbation theory for eigenvalues with finite multiplicities, embedded into the essential spectrum of a self-adjoint operator $H$. We assume the existence of another self-adjoint operator $A$ for which the family…

We consider the problem of variation of spectral subspaces for linear self-adjoint operators under off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of…

In this work, we examine spectral properties of Markov transition operators corresponding to Gaussian perturbations of discrete time dynamical systems on the circle. We develop a method for calculating asymptotic expressions for eigenvalues…

Spectral gaps play a fundamental role in many areas of mathematics, computer science, and physics. In quantum mechanics, the spectral gap of Schr\"odinger operators has a long history of study due to its physical relevance, while in quantum…

In [Camano, Lackner, Monk, SIAM J. Math. Anal., Vol. 49, No. 6, pp. 4376-4401 (2017)] it was suggested to use Stekloff eigenvalues for Maxwell equations as target signature for nondestructive testing via inverse scattering. The authors…

For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite…

I present an example of a discrete Schr"odinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than…

We consider the adjacency operator $A$ of the Linial-Meshulam model $X(d,n,p)$ for random $d-$dimensional simplicial complexes on $n$ vertices, where each $d-$cell is added independently with probability $p\in[0,1]$ to the complete…

In this paper we extend classical criteria for determining lower bounds for the least point of the essential spectrum of second-order elliptic differential operators on domains $\Omega\subset\R^n$ allowing for degeneracy of the coefficients…

We study the problem of minimizing or maximizing the fundamental spectral gap of Schr\"odinger operators on metric graphs with either a convex potential or a ``single-well'' potential on an appropriate specified subset. (In the case of…

We consider eigenvalue problems for general elliptic operators of arbitrary order subject to homogeneous boundary conditions on open subsets of the euclidean N-dimensional space. We prove stability results for the dependence of the…

We consider conditions under which an embedded eigenvalue of a self-adjoint operator remains embedded under small perturbations. In the case of a simple eigenvalue embedded in continuous spectrum of multiplicity m < \infty we show that in…

We study the spectral properties of Schr\"{o}dinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for the resolvent, construct a spectral…

This paper surveys the main results obtained during the period 1992-1999 on three aspects mentioned at the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e., the first non-trivial…

Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics. For each self-adjoint, trace-class operator $O$ we define a set $\Lambda_n\subset \mathbb{R}$, and we show that it converges to…

For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the…

The essential spectral radius of a sub-Markovian process is defined as the infimum of the spectral radiuses of all local perturbations of the process. When the family of rescaled processes satisfies sample path large deviation principle,…

We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the…

We introduce the biharmonic Steklov problem on differential forms by considering suitable boundary conditions. We characterize its smallest eigenvalue and prove elementary properties of the spectrum. We obtain various estimates for the…