Tunnel Effect for Kramers–Fokker–Planck Type Operators

Abstract

We consider operators of Kramers-Fokker-Planck type in the semi-classical limit such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions we establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues. RésuméOn considère des opérateurs du type de Kramers-Fokker-Planck dans la limite semi-classique tels que l'exposant du maxwellien associé soit une fonction de Morse avec deux minima et un point selle. Sous des hypothèses supplémentaires convenables onétablit un développement asymptotique complet de l'écart exponentiellement petit entre les deux premières valeurs propres.1

Lecture notes : Spectral properties of non-self-adjoint operators

“…Now this result becomes a direct application of Theorem 11.6 to A := −P/h and we do not need any bounds on the resolvent in the region z > h ω. In [45,46] similar results were obtained for more general operators, for which we do not necessarily have any bound on the resolvent beyond a strip, and the proof was to use microlocal coercivity outside a compact set in slightly weighted L 2 -spaces. Again Theorem 11.6 would give some simplifications.…”

“…We also let P denote the graph closure of P from S(R 2n ) which coincides with the maximal extension of P in L 2 (see [47,41,45]). We have P ≥ 0 and the spectrum of P is contained in the right half plane.…”

“…(Near infinity this last assumption has to be modified slightly and we refer to [45] for the details.) The following result from [45] is very close to the main result of [48] and generalizes Theorem 5.1:…”

“…With Hérau and C.Stolk [48] we made a study in the semi-classical limit and studied small eigenvalues modulo O(h ∞ ). More recently with Hérau and M. Hitrik [45] we have made a precise study of the exponential decay of µ(h) when V has two local minima (and in that case µ(h) turns out to be real). This involves tunneling, i.e.…”

Tunnel Effect for Kramers-Fokker-Planck Type Operators: Return to Equilibrium and Applications

“…The principal purpose of the present paper is to apply the spectral results of [12] to obtain a precise information concerning the large time behavior of the heat semigroup generated by the semiclassical Kramers-Fokker-Planck operator P = y · h∂ x − V ′ (x) · h∂ y + γ 2 (−h∂ y + y) · (h∂ y + y) , x, y ∈ R n , γ > 0. (1.1) In fact, as in [12], our main result will be valid for a large class of supersymmetric second order differential operators, including (1.1).…”