Daniel Grieser (Oldenburg)
Inverse and direct spectral problems for triangles and thin singular domains
We study the spectrum of the Laplace operator with Dirichlet boundary conditions on Euclidean triangles. I will discuss two results: The first result, joint with S. Maronna, is a new proof of the fact that a triangle is (among the set of all triangles) uniquely determined by the spectrum. The only previously known proof of this uses wave invariants. The study of these is technically difficult. Our new proof uses heat invariants and is technically simpler, and also involves a curious and interesting – and apparently new – geometric fact about triangles. The second result, joint with R. Melrose, that I will discuss is a description of the full asymptotic behavior of the eigenvalues when the triangle degenerates into a line. The techniques extend to thin domains with a singular fibre. The degeneration may happen in various ways. More precisely, there are two parameters describing the degeneration, and we give a complete asymptotic expansion in terms of both parameters. This involves a rather intricate and unexpected blow-up of the parameter space, which will be explained in the talk.