Finite deficiency indices and uniform remainder in Weyl's law

We give a proof that in settings where Von Neumann deficiency indices are finite the spectral counting functions of two different self-adjoint extensions of the same symmetric operator differ by a uniformly bounded term (see also Birman-Solomjak's 'Spectral Theory of Self-adjoint operators in Hilbert Space') >. We apply this result to quantum graphs, pseudo-laplacians and surfaces with conical singularities.Comment: 7 p., references adde

Eigenvalue variations and semiclassical concentration

We show that the behaviour of analytic eigenbranches of a Schr\"odinger operator depends on the way eigenfunctions concentrate in the phase space.Comment: to be published in the proceedings of the conference 'Spectrum and Dynamics', Montr\'eal, 200

Hyperbolic triangles without embedded eigenvalues

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. In particular, we use a careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator.Comment: 65 pages, 4 figures, to appear in Annals of Mathematics http://annals.math.princeton.edu/articles/1159

Generic spectral simplicity of polygons

We study the Laplace operator with Dirichlet or Neumann boundary condition on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has simple spectrum. We also address the more general case of geodesic polygons in a constant curvature space form.Comment: length reduced to 6 pages, 1 figur