10 - Lectures on wave invariants

Summary

yIntroduction

These are notes of some lectures on wave invariants and quantum normal form invariants of the Laplacian Δ at a closed geodesic γ of a compact boundaryless Riemannian manifold (M, g). Our purpose in the lectures was to give a survey of some recent developments involving these invariants and their applications to inverse spectral theory, mainly following the references [G.1] [G.2] [Z.1] [Z.2] [Z.3]. Originally, the notes were intended to mirror the lectures but in the intervening time we wrote another expository account on this topic [Z.4] and also extended the methods and applications to certain plane domains which were outside the scope of the original lectures [Z.5]. These events seemed to render the original notes obsolete. In their place, we have included some related but more elementary material on wave invariants and normal forms which do not seem to have been published before and which seem to us to have some pedagogical value. This material consists first of the calculation of wave invariants on manifolds without conjugate points using a global Hadamard-Riesz parametrix. Readers who are more familiar with heat kernels than wave kernels may find this calculation an easy-toread entree into wave invariants. A short section on normal forms leads the reader into this more sophisticated – and more useful – approach to wave invariants. We illustrate this approach by putting a Sturm-Liouville operator on a finite interval (with Dirichlet boundary conditions) into normal form.