Tunnel effect for Kramers-Fokker-Planck type operators (Q2427380)

This work is a continuation of [\textit{F. Hérau, J. Sjöstrand} and \textit{C. C. Stolk}, Commun. Partial Differ. Equations 30, No. 5--6, 689--760 (2005; Zbl 1083.35149)]. The authors consider operators of Kramers-Fokker-Planck type of the form \[ P=y\cdot h\partial_x - V' (x)\cdot h \partial_y +\frac{\gamma}{2} (-h\partial_y+y) \cdot (h\partial_y+y) \qquad x,y\in \mathbb{R}^n \] in the semi-classical limit \(h\rightarrow 0\) corresponding to low temperature, such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions they establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues.