### Nodal count of eigenfunctions as index of instability

- Series
- Math Physics Seminar
- Time
- Monday, February 27, 2012 - 12:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Gregory Berkolaiko – Texas A&amp;M Univ. – berko@math.tamu.edu

Zeros of vibrational modes have been fascinating physicists for
several centuries. Mathematical study of zeros of eigenfunctions goes
back at least to Sturm, who showed that, in dimension d=1, the n-th
eigenfunction has n-1 zeros. Courant showed that in higher dimensions
only half of this is true, namely zero curves of the n-th eigenfunction of
the Laplace operator on a compact domain partition the domain into at
most n parts (which are called "nodal domains").
It recently transpired that the difference between this "natural" number n of
nodal domains and the actual values can be interpreted as an index of instability
of a certain energy functional with respect to suitably chosen perturbations. We
will discuss two examples of this phenomenon: (1) stability of the nodal
partitions of a domain in R^d with respect to a perturbation of the partition
boundaries and (2) stability of a graph eigenvalue with respect to a perturbation
by magnetic field. In both cases, the "nodal defect" of the eigenfunction
coincides with the Morse index of the energy functional at the corresponding
critical point.
Based on preprints arXiv:1107.3489 (joint with P.Kuchment and
U.Smilansky) and arXiv:1110.5373