We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under certain genericity condition, we show that the number of nodal domains of the n-th eigenfunction is bounded below by n− l, where l is the number of links that distinguish the graph from a tree. Our results apply to operators on bo...

An ion moving in a constant magnetic field is considered as the generic example of two-dimensional topological (Chern-Simons) theory. The eigenfunctions for the generic particle are determined by solving Schrödinger’s equation in the secular and non-secular forms of the theory, corresponding to the inclusion and non-inclusion respectively in the Lagrangian of the particle’s kinetic energy. A no...

We consider the zeros on the boundary ∂Ω of a Neumann eigenfunction φλj of a real analytic plane domain Ω. We prove that the number of its boundary zeros is O(λj) where −∆φλj = λ2jφλj . We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is O(λj). It follows that the number of nodal lines of φλj (components of the nodal set) which touch the b...

Stress intensities of a singular near-tip field around the vertex of a triple junction wedge in multilevel thin film packages are calculated using the two-state M-integral. For this calculation the existence of a simple auxiliary field in the sense of the M-integral associated with every eigenfunction for the triple junction vertex is first verified numerically. The auxiliary field is then empl...