Supersymmetric structures for second order differential operators

Abstract

Necessary and sufficient conditions are obtained for a real semiclassical partial differential operator of order two to possess a supersymmetric structure. For the operator coming from a chain of oscillators, coupled to two heat baths, we show the non-existence of a smooth supersymmetric structure, for a suitable interaction potential, provided that the temperatures of the baths are different. * In memory of Vladimir Buslaev

Tunnel effect and symmetries for non-selfadjoint operators

“…The following result, established in [14], gives a necessary and sufficient condition for (2.4) to hold. Proposition 2.1.…”

“…The purpose of this section is to describe, following [14], an example of a physically significant second order semiclassical operator, for which the supersymmetric structure may break down. Specifically, we shall be concerned with the semigroup generator for the stochastic process describing a chain of two oscillators, coupled to two heat baths, [5],…”

“…The purpose of this talk is to describe some recent results [13], [14], [4], concerning the spectral analysis of some classes of non-selfadjoint operators in the presence of symmetries.…”

“…In Section 2, we shall follow [14] and discuss some general conditions for a second order differential operator to possess a supersymmetric structure. In Section 3, we describe the results of [13], giving complete asymptotic expansions for the exponentially small eigenvalues of the Kramers-Fokker-Planck operator.…”

“…In Section 3, we describe the results of [13], giving complete asymptotic expansions for the exponentially small eigenvalues of the Kramers-Fokker-Planck operator. Section 4 is concerned with the case of the operator describing the short chain of two oscillators coupled to two heat baths, where it turns out that the supersymmetry may break down in the case when the temperatures of the baths are distinct, [14]. Finally, in Section 5, we describe the result of [4], concerning PTsymmetric quadratic differential operators with real spectrum, that are similar to selfadjoint operators.…”