Around supersymmetry for semiclassical second order differential operators

Abstract

Let P ( h ) , h ∈ ] 0 , 1 ] P(h),h\in ]0,1] be a semiclassical scalar differential operator of order 2 2 . The existence of a supersymmetric structure given by a matrix G ( x ; h ) G(x;h) was exhibited by Hérau, Hitrik, and Sjöstrand (2011) under rather general assumptions. In this paper we give a sufficient condition on the coefficients of P ( h ) P(h) so that the matrix G ( x ; h ) G(x;h) enjoys some nice estimates with respect to the semiclassical parameter.