Session S33 - Spectral Geometry

Wednesday, July 21, 16:00 ~ 16:20 UTC-3

Analytic centrally symmetric plane domains are spectrally determined in that class

Steve Zelditch

Northwestern, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The purpose of my talk is to sketch the proof of the statement in the title. To date, there are only 4 classes of plane domains known to be determined by their Dirichlet (or, Neumann) eigenvalues in some specified class of domains: Discs (Kac, 1965) and general ellipses of small eccentricity (Hezari-Z, 2019), among all smooth plane domains; up-down symmetric analytic domains (Z, '09) among all analytic domains satisfying a finite number of conditions; and extremal domains for certain spectral invariants (Marvizi-Melrose, Watanabe). My talk adds a new class: centrally symmetric domains. The proof uses wave invariants in a similar way to that for up-down symmetric domain. Aside from the necessary modifications, the proof contains two new additions. First, it is proved that the finite number of constraints on the analytic domains gives an open-dense set of convex analytic domains when they are assumed convex (and a residual set in general). Second, we exhibit a new duality among billiard maps of domains with a bouncing ball (2-link) orbit. Namely, there are two non-isometric domains whose billiard maps have symplectically equivalent billiard maps around bouncing ball orbits (more precisely, the same Birkhoff normal form). These domains are not isospectral: they can be distinguished by a Maslov index invariant. Joint work with Hamid Hezari.

Joint work with Hamid Hezari, UCI, USA..

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