“Tunneling for a Semi-classical Schrödinger operator with symmetries”
CANCELLED Venerdì 21 Aprile 2023, ore 15:00 - Aula 2BC60 - Michel Rouleux (Centre de Physique Théorique, CNRS, Aix-Marseille-Université et Université de Toulon)
Abstract
We consider a semi-classical quantum particle in a magnetic field $A$ and an external potential $V$. The underlying classical system, symplectic structures and their symmetries play an important role in investigating tunneling properties. Our main results concern the operator $P_A(x,hD_x)=(hD_x-\mu A(x))^2+V(x)$ on $L^2({\bf R}^d)$ when $V$ has two non degenerate potential wells symmetric with respect to an hyperplane. We study the low-lying eigenvalues of $P_A(x,hD_x)$ and estimate their splitting with an exponential accuracy. This splitting is related with the decay of eigenfunctions in the classically forbidden region, and bounded from above by Agmon estimates. In different settings we improve standard Agmon estimates by including the correction due to the magnetic field, and refined estimates are given with respect to the small parameter $\mu$.
