Silent Orbits and Cancellations in the Wave Trace

TL;DR

This paper demonstrates that the wave trace of certain convex domains can be arbitrarily smooth near some points, revealing limitations in inverse spectral problems and showing that the Laplace spectrum and length spectrum are fundamentally different.

Contribution

It constructs families of domains with silent periodic billiard orbits, showing the wave trace can be made arbitrarily smooth and challenging the Poisson relation's strictness.

Findings

Existence of silent periodic billiard orbits in convex domains.

Wave trace smoothness can be arbitrarily increased near certain points.

Limitations in using wave trace for inverse spectral problems.

Abstract

This paper shows that the wave trace of a bounded and strictly convex planar domain may be arbitrarily smooth in a neighborhood of some point in the length spectrum. In other words, the Poisson relation, which asserts that the singular support of the wave trace is contained in the closure of $\pm$ the length spectrum, can almost be made into a strict inclusion. To do so, we construct large families of domains for which there exist multiple periodic billiard orbits having the same length but different Maslov indices. Using the microlocal Balian-Bloch-Zelditch parametrix for wave invariants developed in our previous paper, we solve a large system of equations for the boundary curvature jets, which leads to the required cancellations. We call such periodic orbits silent, since they are undetectable from the ostensibly audible wave trace. Such cancellations show that there are potential limitations in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different mathematical objects, at least insofar as the wave trace is concerned.