Mini course on Semi-classical approximation and the tunnelling effect
Spectral Methods in Mathematical Physics
30 January 09:00 - 11:00
Bernard Helffer - Université de Nantes
In 1982-1983, the so-called symmetric double well problem was rigorously analyzed in any dimension in the semi-classical context by B. Simon (announced in 1982) and Helffer-Sjöstrand (1983).
This involves semi-classical Agmon estimates, WKB constructions and a very fine analysis of the so-called tunneling effect in order to establish the splitting between the lowest eigenvalues.
The strategy followed by Helffer-Sjöstrand appears to be quite efficient in many other contexts. We will focus on the Robin problem for the Laplacian in a domain where in some asymptotic regime for the Robin parameter and some specific symmetric domain (ellipse like domain) a double well appears.
We will also discuss the case of the magnetic Laplacian with large constant magnetic field in a similar domain. The optimal result in this case is still open.
The results correspond to contributions by various subsets of the following set of authors : Bonnaillie, Fournais, Helffer, Hérau, Kachmar, Morame, Pankrashkin, Popoff, Raymond, Sjöstrand.
University of Zurich, UZH