School of Mathematical Sciences

Nodal count of eigenfunctions as index of instability menu

Nodal count of eigenfunctions as index of instability

Gregory Berkolaiko, Texas A&M
Tue, 07/06/2016 - 16:00
Seminar series: 

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 upper bound
and the actual value 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. We will also discuss some applications of the above results.

Based on arXiv:1103.1423, CMP'12 (with R.Band, H.Raz, U.Smilansky),
arXiv:1107.3489, GAFA'12 (with P.Kuchment, U.Smilansky),
arXiv:1110.5373, APDE'13
arXiv:1212.4475, PTRSA'13 to appear (with T.Weyand),
arXiv:1503.07245, JMP'15 to appear (with R.Band and T.Weyand)