ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp) Nonclassical Sturm-Liouville problems and

Schrödinger operators on radial trees

Blow-up scaling and global behaviour of solutions of the bi-Laplace equation via pencil operators

This work has been partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258.Publicad

Singular Sturm–Liouville Theory on Manifolds

AbstractIn this paper we investigate Schrödinger operators L=−Δg+a(x) on a compact Riemannian manifold (M, g), where the potential function a(x) is assumed to be continuous, but not necessarily bounded, outside of some closed set Σ⊂M of measure zero. Under certain geometric hypotheses on Σ and growth conditions on a(x) as x→Σ, we prove that the Dirichlet extension of L is bounded from below with discrete spectrum; in many cases, a(x) is allowed to approach −∞ as x→Σ. We also consider conditions on Σ and a(x) under which the Sturm–Liouville theory of L is “singular” in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of L; in particular, this occurs when the domain of L does not depend on boundary conditions, for example, when L is essentially self-adjoint or more generally “essentially Dirichlet” (a new property that we define). The behavior of L on weighted Sobolev spaces is also discussed. In most of the paper we assume that Σ is a k-dimensional submanifold without boundary, but in the last few sections we generalize our results to stratified sets

Geometry, Dynamics and Spectrum of Operators on Discrete Spaces (online meeting)

Spectral theory is a gateway to fundamental insights in geometry and mathematical physics. In recent years the study of spectral problems in discrete spaces has gained enormous momentum. While there are some relations to continuum spaces, fascinating new phenomena have been discovered in the discrete setting throughout the last decade. The goal of the workshop was to bring together experts reporting about the recent developments in a broad variety of dynamical or geometric models and to reveal new connections and research directions

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature

Application of Helmholtz/Hodge Decomposition to Finite Element Methods for Two-Dimensional Maxwell\u27s Equations

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell\u27s equations. We begin with the introduction of Maxwell\u27s equations and a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Helmholtz/Hodge decomposition for divergence-free vector fields. In this approach an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. In Chapter 5, we further extend the new approach described in Chapter 4 to the interface problem for Maxwell\u27s equations. We use the extraction formulas and multigrid method to overcome the low regularity of the solution for the Maxwell interface problem. The theoretical results obtained in this dissertation are confirmed by numerical experiments