Determination of the Genus of Surfaces from the Spectrum of Schrödinger Operators attached to height functions. (An inverse spectral problem for surfaces)

Abstract

Using results on inverse spectral problems, in particular the so-called new wave invariants attached to a classical equilibrium, we show that it is possible to determine the Morse index of height functions. For compact Riemannian surfaces $M\subset {\mathbb{R}}^3$ this imply that we can retrieve the topology (via the genus). Our results are independent from the choice of a metric on $M$ and can be obtained from the choice of a 'generic' height-function. For surfaces of genus zero, diffeomorphic to a 2-sphere, the method allows to detect the convexity, or the local convexity of the surface. Keywords : Micro-local analysis; Schrödinger operators; Inverse spectral problems.

3 figs, perhaps to be revised

References in corpus (2)

• A semi-classical trace formula at a non-degenerate critical level
• Some inverse spectral results for semi-classical Schrödinger operators