Course: Studies of double well Schrödinger operators with magnetic fields

This mini-course offered by Søren Dyhr will take place in person in the lab itself at the EPSEB. However, it will also be transmitted by videoconference and on the b-lab's youtube page. If you wish to get access to the videoconference please send us an e-mail.

There is no registration fee for this course and registration is only required for people attending at the videoconference.

Title: Studies of double well Schrödinger operators with magnetic fields.

Summary: The (1 dimensional) harmonic oscillator is a simple example of a quantum mechanical system. Connecting two wells with quadratic behaviour and placing them on a circle leads to more complicated behaviour and lets magnetic fields play a role. It is natural to ask how the eigenvalues and eigenfunctions of this operator behave (in a semiclassical limit).
I will describe different approximation techniques that can be applied in such situations. Locally, wells can be approximated quadratically for rough information on eigenvalues, and via a WKB construction pointwise approximate eigenfunctions can be constructed to arbitrary asymptotic precision on compact intervals. The local results can be extended to related global results, and by applying some tricks one can calculate an interaction matrix and obtain fine splitting results on the eigenvalues of the global operator. Information about the shape of eigenfunctions can also be obtained using the WKB approximations.
This will be an exposition of work done in my master's thesis, the strategy for obtaining the fine splitting of eigenvalues is based on a 2015 article by Bonnaillie-Noël, Hérau, and Raymond.