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)