An inverse spectral problem for the Hermite operator

In this seminar, generalizing to the non smooth case already existing results, we prove that, for any convex planar set Ω, the first non-trivial Neumann eigenvalue μ1(Ω) of the Hermite operator is greater than or equal to 1. Furthermore, and this is our main result, under some additional assumptions on Ω, we show that μ1(Ω) = 1 if and only if Ω is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.