Nodal count, Morse index, and all that
Nodal patterns of vibrating membranes have been observed and discussed since Renaissance times. Nowadays it is well known that they are formed by nodal lines of eigenfunctions of an appropriate operator (in this talk, Dirichlet Laplacian), splitting the whole membrane into nodal domains. In spite of this long history, many things about nodal lines (surfaces) and domains are still not understood well and are subjects of active investigations. In this talk we will survey the status of understanding of the so-called "nodal count," i.e. the number $\nu_n$ of nodal domains of the $n$th eigenfunction. The standard Sturm ODE theorem says that in 1D $\nu_n=n$. This, however, is incorrect in higher dimensions, and all but finitely many eigenfunctions develop a positive "nodal deficiency" $n-\nu_n$. The talk will contain a survey of some open problems and results obtained in the last decade concerning the nodal deficiency.
Password: “arizona” (all lower case)