Related papers: Riesz property in the case of multiple eigenvalues

We prove two-term spectral asymptotics for the Riesz means of the eigenvalues of the Laplacian on a Lipschitz domain with Robin boundary conditions. The second term is the same as in the case of Neumann boundary conditions. This is valid…

For a broad class of polynomial potentials $V$, with an important and instructive representative being $V(x) = x^{2a} + i x^b$, $x \in \mathbb R$, $a, b \in \mathbb N$, we show that the system of spectral projections $\{P_n\}_n$ of an…

We construct a Riemannian metric on the $ 2 $-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of…

We consider the Laplacian in $\mathbb{R}^n$ perturbed by a finite number of distant perturbations those are abstract localized operators. We study the asymptotic behaviour of the discrete spectrum as the distances between perturbations tend…

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $\mathbb{R}^N$. We prove that the symmetric functions of the eigenvalues depend real analytically…

In harmonic analysis, studies of inequalities of Riesz potential in various function spaces have a very important place. Variable exponent Morrey type spaces and the examines of the boundedness of such operators on these spaces have an…

The authors study the spectral theory of self-adjoint operators that are subject to certain types of perturbations. An iterative introduction of infinitely many randomly coupled rank-one perturbations is one of our settings. Spectral…

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means…

We exploit the so called form-local subordination in the analysis of non-symmetric perturbations of unbounded self-adjoint operators with isolated simple positive eigenvalues. If the proper condition relating the size of gaps between the…

An algebraic Riccati equation for linear operators is studied, which arises in systems theory. For the case that all involved operators are unbounded, the existence of infinitely many selfadjoint solutions is shown. To this end, invariant…

In this paper, we give Lieb-Thirring type inequalities for isolated eigenvalues of $d$-dimensional non-selfadjoint Schr\"{o}dinger operators with complex-valued and dilation analytic potentials. In order to derive them, we prove that…

Consider the Hill operator $L(v) = - d^2/dx^2 + v(x) $ on $[0,\pi]$ with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough $n$ close to $n^2 $ there are one Dirichlet eigenvalue $\mu_n$ and two periodic (if $n$…

The Berezin--Li--Yau and the Kr\"oger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $\Omega$ of finite measure are bounded in terms of their semiclassical limit expressions. We show…

For relatively form-compact perturbations of non-negative selfadjoint operators, we obtain an upper bound on the number of discrete eigenvalues in half-planes separated from the positive real axis. The bound is given in terms of a partial…

In the paper the general case of a normal discrete Hausdorff operators in $L^2(\mathbb{R}^d)$ is considered. The main result states that under some natural arithmetic condition the spectrum of such an operator is rotationally invariant.…

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading…

We quantify the subcriticality of the bilaplacian in dimensions greater than four by providing explicit repulsivity/smallness conditions on complex additive perturbations under which the spectrum remains stable. Our assumptions cover…

In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace,…

We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The…

We consider abstract non-negative self-adjoint operators on $L^2(X)$ which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of…