Minisymposia Abstracts

Speaker: Yaroslav Vorobets
Texas A&M University
Title: Laplacians isospectral under multiple boundary conditions
Abstract: Two domains in a plane are called isospectral if the Dirichlet Laplacians in these domains have the same spectrum. The isospectral domains need not be congruent (i.e., in general, one cannot hear the shape of a drum). The first counterexample of that kind was constructed by Gordon, Webb, and Wolpert in 1992. The domains were polygons, and it was noted that the Neumann Laplacians in them also had the same spectrum. More recently, it was observed that Laplacians in the same polygons with mixed boundary conditions (Dirichlet on some sides, Neumann on the others) can be isospectral as well. Two examples were found by Driscoll and Gottlieb (numerically) and another one by Band and Parzanchevski.
In the talk, all the above examples will be placed into a family of 16 pairs of isospectral Laplacians corresponding to various choices of mixed boundary conditions for the two polygons. Isospectrality is established in a unified way using transplantation of eigenfuctions.
The construction extends to other known examples of isospectral polygons. They admit multiple isospectrality too.

Speaker: Ruth Gornet
University of Texas Arlington
Title: The eta invariant on Nilmanifolds
Abstract: In joint work with Ken Richardson of TCU, we discuss recent results on the eta invariant on two-step nilmanifolds

Speaker: Hamid Hezari
Title: Spectral rigidity of the ellipse
Abstract: This is a joint work with Steve Zelditch. We prove that ellipses are infinitesimally spectrally rigid among smooth domains with the symmetries of the ellipse. Spectral rigidity of the ellipse has been expected for a long time and is a kind of model problem in inverse spectral theory. Ellipses are special since their billiard flows and maps are completely integrable. It was conjectured by G. D. Birkhoff that the ellipse is the only convex smooth plane domain with completely integrable billiards. Our results are somewhat analogous to the spectral rigidity of flat tori or the sphere in the Riemannian setting. The main novel step in the proof is the Hadamard variational formula for the wave trace. It is of independent interest and has applications to spectral rigidity beyond the setting of ellipses. The main advance over prior results is that the domains are allowed to be smooth rather than real analytic.

Speaker: Gregory Berkolaiko
Texas A&M University
Title: Nodal count of eigenfunctions and stability of nodal partitions
Abstract: In this talk we address the nodal count (i.e., the number of nodal domains) of eigenfunctions of Schroedinger operators with Dirichlet boundary conditions in bounded domains (also called billiards). According to the classical Sturm theorem, in dimension one the n-th eigenfunction has exactly n nodal domains. In higher dimensions the situation is less definite: the Nodal Theorem of Courant asserts that, in any dimension, the number of nodal domains of the n-th eigenfunction cannot exceed n. However, in most cases the ``nodal deficiency'' -- the difference between n and the number of nodal domains of the n-th eigenfunction -- is non-zero. Moreover, examples are known of eigenfunctions with an arbitrarily large index n that have just two nodal domains. One can say that the nature of the nodal deficiency had not been understood.
We show that, under some genericity conditions, the nodal deficiency is connected to stability of the partition of the billiard by the zeros of the eigenfunction. More precisely, we define an "energy" functional on an infinite dimensional variety of partitions of the billiard. We show that the critical points of the functional correspond exactly to the partitions induced by the eigenfunctions while the Morse indices of the critical points coincide with the nodal deficiencies.

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