Spectral Theory of Laplace and Schrödinger Operators (13w5059)

Geometrically sharp “isoperimetric” estimates on eigenvalues of the Laplacian and the Schrödinger operator help us understand how frequencies and energies of physical systems are constrained by the shape and size of the domain in which the physical process takes place. For example, the Faber–Krahn inequality (Rayleigh’s conjecture) tells us precisely how low the frequency of a drum can be, if the area of the drumhead is given but we are free to choose its shape. In three dimensions, the same inequality tells us the minimal energy a quantum particle can possess when it is confined to a region of given volume. The minimum is attained in each case by a domain with symmetry — the disk in two dimensions, and the ball in three dimensions. Let us describe some recent progress and notable open problems that were presented both formally and informally during the Workshop.