Vadim Kaloshin (University of Maryland)
Title:
Marked length spectral determination of analytic strictly convex domains
Abstract:
M. Kac popularized a beautiful and important question
"Can you hear the shape of a drum?". Formally, for a domain
Ω ⊂ R^2 the Laplace spectrum Sp(Ω) is the collection of
the eigenvalues of the Dirichlet problem for the Laplacian
∆u + λ^2 u = 0, u = 0 on ∂Ω.
Does Sp(Ω) determine a domain Ω? In general, the answer
is negative due to examples of Gordon-Webb-Wolpert, but
the boundary in this example is neither smooth nor analytic.
The (marked) length spectrum L(Ω) is a collection of lengths of
all periodic orbits of the billiard inside Ω (marked by period).
The Laplace spectrum generically determines the length spectrum.
We show that generically (for an open dense set) the marked
length spectrum L(Ω) determines an analytic strictly convex domain.
Earlier Zelditch showed that in the class of axis-symmetric analytic
domains the Laplace spectrum generically (for a residual set) does
determine a domain. This is a joint work in progress joint with
M. Leguil and K. Zhang.