turkmath.org

In this talk, we consider the Schrödinger operators with particular class of singular potentials, namely circular and spherical shell potentials. Such interactions are very typical even in quantum mechanics textbooks and they are usually solved by partial wave analysis. For the low energy sector, this problem can actually be solved directly by writing the interaction term as a projection operator on some appropriate sequence. However, this naive approach raises some mathematical questions and the formal expressions for Hamiltonian needs to be defined. For this reason, we first derive a Kreia type of formula for the resolvent operator associated with delta potentials supported by a circle/sphere. Then we show that  there exists a proper self-adjoint operator associated with the initial formal expression for the Hamiltonian. We then discuss the spectral properties of such potentials, in particular, the bound state spectrum. Finally, we consider the small deformations of support of delta potentials in the normal directions. We show that the first order change in the bound state energies under small deformations of circle/sphere has a simple geometric interpretation. This is a joint work with Sema Seymen and O. Teoman Turgut..

Link to the seminar is:

https://teams.microsoft.com/l/meetup-join/19%3a0ae91f7b86a24e8fa1d818a6f74b22a1%40thread.tacv2/1653379367658?context=%7b%22Tid%22%3a%22066690f2-a8a6-4889-852e-124371dcbd6f%22%2c%22Oid%22%3a%22a49a1949-9464-4e2b-861d-221a6985322e%22%7d