The Morphology of nodal sets and Spectral Theory

<p>The nodal sets are the zero sets of solutions of the Schrödinger equation in quantum physics, and of other wave equations in diverse fields of physics and mathematics. For problems in two dimensions which are invariant under time reversal, the wave functions are real, and the nodel sets form a mesh of lines which are the subject of the present study. The nodal sets appear in a large varieties of forms and shapes. In separable problems, the nodal sets consist of intersecting lines which form a grid. Otherwise, the nodal lines are rarely intersecting, and they display a complex morphology of patterns. The nodal sets form a skeleton which determines and reflects many properties of the wave function. Bacause of its relevance to various fields of mathematics and physics, several properties were investigated, but many important features are still not understood. In the present project, we would like to combine the skills of mathematicians and physicists to address some of these elusive problems. In particular we shall investigate the following topics:</p>

<p>i) Universal properties of nodal lines. Here, the study of random waves ensembles will be used to extract various statistical measures which characterize the nodal sets.</p>

<p>ii) The applicability of percolation theory for the description of nodal lines and the corresponding nodal domains.</p>

<p>iii) To what extent the information on nodal sets can be inverted to determine the wave operator parameters (boundary conditions, metric, potentials).</p>