Mathematics

This work is a continuation of our previos paper \cite{Zel1}, where for the the Schrödinger operator $H=-\Delta+ V(\e)\cdot$ $(V(\e)\ge 0)$, acting in the space $L_2(\R^d)\,(d\ge 3)$, some constructive sufficient conditions for discreteness of its spectrum have been obtained on the base of well known Mazya -Shubin criterion and an optimization problem for a set function. Using a {\it capacitary strong type inequality} of David Adams, the concept of {\it base polyhedron} for the harmonic capacity and some properties of Choquet integral by this capacity, we obtain more general sufficient conditions for discreteness of the spectrum of H in terms of a repeated nonincreasing rearrangement of the function $Y(\e,\bt)=\sqrt{V(\e)}\frac{1}{|\e-\bt|^{d-2}}\sqrt{V(\bt)}$ on cubes that are going to infinity.

In this note we present upper bounds for the variational eigenvalues of the p -Laplacian on smooth domains of complete n -dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given manifold (M,g) for 1<p≤n , and upper bounds for all p>1 when we fix a metric g . To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio.

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.

An inverse spectral problem is studied for the matrix Sturm-Liouville operator on a finite interval with the general self-adjoint boundary condition. We obtain a constructive solution based on the method of spectral mappings for the considered inverse problem. The nonlinear inverse problem is reduced to a linear equation in a special Banach space of infinite matrix sequences. In addition, we apply our results to the Sturm-Liouville operator on a star-shaped graph.

We employ some results about continued fraction expansions of Herglotz-Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. In particular, this solves the corresponding inverse spectral problem through explicit formulas.

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k -th perimeter-normalised Steklov eigenvalue is 8πk , which is the best upper bound for the k -th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realising a weighted Neumann problem as a limit of Steklov problems on perforated domains. For k=1 , the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

Uniform convergence of the expansion of an absolutely continuous function for eigenfunctions of the Sturm-Liouville problem −y"+q(x)y=μy, y(0)=0, y(π)cosβ+ y ′ (π)sinβ=0, β∈(0,π) with summable potential q∈ L 1 R [0,π] is proved. This result is used to obtain more precise asymptotic formulae for eigenvalues and norming constants of this problem.

We consider a class of unbounded quasiperiodic Schrödinger-type operators on ℓ 2 ( Z d ) with monotone potentials (akin to the Maryland model) and show that the Rayleigh--Schrödinger perturbation series for these operators converges in the regime of small kinetic energies, uniformly in the spectrum. As a consequence, we obtain a new proof of Anderson localization in a more general than before class of such operators, with explicit convergent series expansions for eigenvalues and eigenvectors. This result can be restricted to an energy window if the potential is only locally monotone and one-to-one. A modification of this approach also allows the potential to be non-strictly monotone and have a flat segment, under additional restrictions on the frequency.

The elastic Neumann--Poincaré operator is a boundary integral operator associated with the Lamé system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two different points determined by Lamé parameters. We show that eigenvalues converge at a polynomial rate on smooth boundaries and the convergence rate is determined by smoothness of the boundary. We also show that they converge at an exponential rate if the boundary of the domain is real analytic.

We study the existence of eigenvalues of the generalized Friedrichs model H μ (p) , with a rank-one perturbation, depending on parameters μ>0 and p∈ T 2 , and found an absolutely convergent expansions for eigenvalues at μ(p) , the coupling constant threshold. The expansions are highly dependent on that, whether the threshold m(p) of the essential spectrum is: (i) neither an threshold eigenvalue nor a threshold resonance; (ii) a threshold resonance; (iii) an threshold eigenvalue.