Event № 111
Zeros of vibrational modes have been fascinating physicists forseveral centuries. Mathematical study of zeros of eigenfunctions goesback at least to Sturm, who showed that, in dimension d=1, the n-theigenfunction has n-1 zeros. Courant showed that in higher dimensionsonly half of this is true, namely zero curves of the n-th eigenfunction ofthe Laplace operator on a compact domain partition the domain into atmost n parts (which are called "nodal domains").
It recently transpired (first on graphs with a subsequentgeneralization to manifolds) that the difference between this upperbound and the actual value can be interpreted as an index ofinstability of a certain energy functional with respect to suitablychosen perturbations. We will discuss two examples of thisphenomenon: (1) stability of the nodal partitions with respect to aperturbation of the partition boundaries and (2) stability of aneigenvalue with respect to a perturbation by magnetic field. In bothcases, the "nodal defect" of the eigenfunction coincides with theMorse index of the energy functional at the corresponding criticalpoint.
Based on joint work with R. Band, P.Kuchment, H. Raz, U.Smilansky andT. Weyand.