Index theorems for nodal count and a lateral variation principle

Our study is motivated by earlier results about nodal count of Laplacian eigenfunctions on manifolds and graphs that share the same flavor: a normalized nodal count is equal to the Morse index of a certain energy functional at the critical point given by the eigenfunction in question (the precise meaning of the "normalized count" and the "energy functional" depends on the setting of the problem). In the hindsight, in all these results, the nodal count can be understood as the spectral shift obtained by perturbing the operator in an appropriate way. This brings us to the following general result: the spectral shift can be recovered as the stability index of the eigenvalue with respect to small "lateral" variations of the perturbation. Based on joint work with Y. Canzani, G. Cox, P. Kuchment and J. Marzuola, as well as earlier work with R. Band, Y. Colin de Verdiere, H. Raz, U. Smilansky and T. Weyand.