Qualitative Properties of Solutions to elliptic Equations with Emphasis on Quantum Mechanics and Membrane

One of the central objects in quantum mechanics is the Schrödinger equation. An important issue is to understand local and global properties of solutions of this equation. In particular, we want to investigate the behaviour of solutions near zeros. Due to the importance of Coulombic systems (i.e., the case of one (or more) nucleus with N electrons) in physical applications, special emphasis will be given to this case. Moreover, ground state wave functions of N-electron atoms and spherical averaged atomic electron densities of atoms will be considered. This constitutes the first part of our proposal. The second part deals with membrane problems. We want to treat several questions (and extensions thereof) concerning the famous nodal line conjecture.