Payne-Pólya-Weinberger inequalities are known to be exclusive to bounded Euclidean domains with Dirichlet boundary condition. In this paper, we discuss the corresponding inequalities on Riemannian manifolds of dimension $n \geq3$, and we prove explicit bounds in terms of geometric quantities such as scalar curvature, Yamabe constant, isoperimetric constant and conformal volume.
In our recent papers, we studied semiclassical spectral problems for the Bochner-Schrödinger operator on a manifold of bounded geometry. We survey some results of these papers in the setting of the magnetic Schrödinger operator in the Euclidean space and describe some ideas of the proofs.
We show that Weyl's law for the number and the Riesz means of negative eigenvalues of Schrödinger operators remains valid under minimal assumptions on the potential, the vector potential and the underlying domain.
We prove that a bounded linear operator $T$ is a direct sum of an invertible operator and an operator with at most countable spectrum iff $0\notin\mbox{acc}^{ω_{1}}\,σ(T),$ where $ω_{1}$ is the smallest uncountable ordinal and $\mbox{acc}^{ω_{1}}\,σ(T)$ is the $ω_{1}$-th Cantor-Bendixson derivative of $σ(T).$
Żywilla Fechner, Eszter Gselmann, László Székelyhidi
The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions.
For the radial and one-dimensional Schrödinger operator $H$ with growing potential $q(x)$ we outline a method of obtaining the trace identities - an asymptotic expansion of the Fredholm determinant $\mathrm{det}_{F}(H-λI)$ as $λ\to-\infty$. As an illustrating example, we consider Schrödinger operator with the potential $q(x)=2\cosh 2x$, associated with the modified Mathieu equation.
We consider the non-self-adjoint Sturm-Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local solvability and stability of this inverse problem, relying on the method of spectral mappings. Possible splitting of multiple eigenvalues is taken into account.
We prove a lower bound on the spacing of zeros of paraorthogonal polynomials on the unit circle, based on continuity of the underlying measure as measured by Hausdorff dimensions. We complement this with the analog of the result from arXiv:1011.3159 showing that clock spacing holds even for certain singular continuous measures.
In this paper we explore some characteristics of the quasi-Fredholm resolvent set $ρ_{qf}(T)$ of an operator $T$ defined on an infinite dimensional Banach space $X$. Moreover, in the case of Hilbert space $H$, we study the stability of the SVEP and describe the operators for which the SVEP is preserved under compact perturbations using quasi-Fredholm spectrum and $ρ_{qf}(T)$.
We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky--Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in $L_2$-norm is given.
We consider a Sturm-Liouville operator a with integrable potential $q$ on the unit interval $I=[0,1]$. We consider a Schrödinger operator with a real compactly supported potential on the half line and on the line, where this potential coincides with $q$ on the unit interval and vanishes outside $I$. We determine the relationships between eigenvalues of such operators and obtain estimates of eigenvalues in terms of potentials.
It is proved that if a bounded domain in three dimensions satisfies a certain concavity condition, then the Neumann-Poincaré operator on the boundary of the domain or its inversion in a sphere has at least one negative eigenvalue. The concavity condition is quite simple, and is satisfied if there is a point on the boundary at which the Gaussian curvature is negative.
In this paper, we give a necessary and sufficient condition for an even order three dimensional strongly symmetric circulant tensor to be positive semi-definite. In some cases, we show that this condition is also sufficient for this tensor to be sum-of-squares. Numerical tests indicate that this is also true in the other cases.
C. Remling obtained a theorem on limit set of the shift operation on a space of functions on R when the associated 1-D half line Schrödinger operators have absolutely continuous component in their spectrum. The purpose of the paper is to define a KdV flow on a certain class of functions containing algebro-geometric functions and to extend this result to the KdV flow.
For a self--adjoint Laplace operator on a finite, not necessarily compact, metric graph lower and upper bounds on each of the negative eigenvalues are derived. For compact finite metric graphs Poincaré type inequalities are given.
We consider discrete Schroedinger operator J with Wigner-von Neumann potential not belonging to l^2. We find asymptotics of orthonormal polynomials associated to J. We prove the Weyl-Titchmarsh type formula, which relates the spectral density of J to a coefficient in asymptotics of orthonormal polynomials.
I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these, can be reduced to questions about ergodic Jacobi operators. Then, I apply this to $a(n) = 1$ and $b(n) = f(n^ρ\pmod{1})$ for $ρ> 0$ not an integer, and to obtain a probabilistic version of the Denisov--Rakhmanov--Remling Theorem.
The structured pseudospectra of a matrix A are sets of complex numbers that are eigenvalues of matrices X which are near to A and have the same entries as A at a fixed set of places. The sum of multiplicities of the eigenvalues of X inside each connected component of the structured pseudospectra of A does not depend on X. This fact is known, but not so much as it should be. For this reason, we give here an elementary and detailed proof of the result.