Finding the nodal points on a quantum graph

TYPETheor./Math. Physics Seminar
Speaker:Dr. Ram Band
Location:Lidow Nathan Rosen (300)
Abstract:The investigation of nodal patterns on manifolds has began already in the 19th century by the pioneering work of Chladni on the nodal structures of vibrating plates. Sturm's oscillation theorem states that the n-th vibrational mode of a string has n-1 nodal points equally distributed on the string. However, finding the location of the nodal points on a general quantum graphs is far less trivial task. This is despite a quantum graph being nothing more than a structure of strings attached to each other. We manage to study the number of nodal points and their location without solving the full eigenvalue problem on the graph. This is done by defining an energy function on the space of nodal points' possible locations and examining its critical points. This work follows the approach taken by Helffer, Hoffmann-Ostenhof and Terracini to study Schroedinger operators on two-dimensional domains. Quantum graphs obey the analogue of the results by Helffer et al, and are used to further generalize them. This is a joint work with Gregory Berkolaiko, Hillel Raz and Uzy Smilansky.