Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

In this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

On the Neumann Laplacian in nonuniformly collapsing strips

Consider the Neumann Laplacian in the region below the graph of [Formula: see text], for a positive smooth function [Formula: see text] with both [Formula: see text] and [Formula: see text] bounded. As [Formula: see text] such region collapses to [Formula: see text] and an effective operator is found, which has Robin boundary conditions at [Formula: see text]. Then, we recover (under suitable assumptions in the case of unbounded [Formula: see text]) such effective operators through uniformly collapsing regions; in such approach, we have (roughly) got norm resolvent convergence for [Formula: see text] diverging less than exponentially.

Approximations of Metric Graphs by Thick Graphs and Their Laplacians

The main purpose of this article is two-fold: first, to justify the choice of Kirchhoff vertex conditions on a metric graph as they appear naturally as a limit of Neumann Laplacians on a family of open sets shrinking to the metric graph (“thick graphs”) in a self-contained presentation; second, to show that the metric graph example is close to a physically more realistic model where the edges have a thin, but positive thickness. The tool used is a generalization of norm resolvent convergence to the case when the underlying spaces vary. Finally, we give some hints about how to extend these convergence results to some mild non-linear operators.

Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.