MCS-MP Seminar: Jacob Schach Møller

In this talk we discuss the structure of the pure point spectrum of one-body Schroedinger operators on the cubic metric graph in dimension at least 3.

The cubic metric graph in d dimensions is the d dimensional mesh consisting of vertices at Z^d together with all the edges between nearest neighbours. The free metric Laplacian acts on each edge as a one-dimensional Laplacian, and we get a self-adjoint realization by imposing Kirchoff boundary conditions at the vertices.

The spectrum of the metric Laplacian is the positive half-line with eigenvalues of infinite multiplicity embedded at the Dirichlet spectrum (j x Pi)^2, for j=1,2,.... When the metric Laplacian is perturbed by a potential, new embedded eigenvalues may appear away from the Dirichlet spectrum. The goal of the talk is to establish that eigenvalues of the perturbed metric Laplacian can only arise within a distance of the Dirichlet spectrum that scales with an L^p norm of the potential, uniformly in energy.

The talk is based on joint work with Evgeny Korotyaev and Morten Grud Rasmussen