Robin spectral rigidity of strictly convex domains with a reflectional symmetry

TL;DR

This paper extends spectral rigidity results for convex symmetric domains to Robin boundary conditions by combining wave and heat trace invariants, allowing Robin function deformations.

Contribution

It generalizes previous spectral rigidity results from Dirichlet and Neumann to Robin boundary conditions for symmetric convex domains.

Findings

Spectral rigidity holds for Robin boundary conditions in symmetric convex domains close to a circle.

The method combines wave trace invariants with heat trace invariants for Robin Laplacians.

Results apply to arbitrary Robin functions and their deformations within the symmetric class.

Abstract

This is a note on a recent paper of De Simoi-Kaloshin-Wei \cite{DKW}. We show that using their results combined with wave trace invariants of Guillemin-Melrose and the heat trace invariants of Zayed for the Laplacian with Robin boundary conditions, one can extend the Dirichlet/Neumann spectral rigidity results of \cite{DKW} to the case of Robin boundary conditions. We will consider the same generic subset as in \cite{DKW} of smooth strictly convex $\mathbb Z_2$-symmetric planar domains sufficiently close to a circle, however we pair them with arbitrary $\mathbb Z_2$-symmetric smooth Robin functions on the boundary and of course allow deformations of Robin functions as well.