Symmetry of Bound and Antibound States in the Semiclassical Limit

(2) HV def = −∂ x + V (x), on R, or on [0,∞) with Dirichlet or Neumann boundary conditions. The resonances or scattering poles of HV are defined as the poles of the meromorphic continuation of the resolvent, RV (λ) = (HV − λ ), from Imλ > 0, to C. Except for the poles at λ for which λ are eigenvalues of HV , RV (λ) is bounded on L 2 for Imλ > 0. Its Schwartz kernel, that is the Green function, continues meromorphically across the continuous spectrum corresponding to R. Its poles are the resonances of HV . An illustration based on the numerical codes of [4] is given in Fig.1. The poles on the positive imaginary axis correspond to the bound states ofHV , and the poles on the negative are called antibound states. Note that they appear to be exactly symmetric with the bound states. In this note we prove a simple theorem inspired by numerical experiments using [4]: