Exponential asymptotics for an eigenvalue of a problem involving parabolic cylinder functions

Abstract

We obtain the leading asymptotic behaviour as ε → 0 + \varepsilon \to 0 + of the exponentially small imaginary part of the "eigenvalue" of the perturbed nonself-adjoint problem comprising y ( x ) + ( λ + ε x 2 ) y ( x ) = 0 y(x) + (\lambda + \varepsilon {x^2})y(x) = 0 with a linear homogeneous boundary condition at x = 0 x = 0 and an "outgoing wave" condition as x → + ∞ x \to + \infty . The problem is a generalization of a model equation for optical tunnelling considered by Paris and Wood [10]. We show that this "eigenvalue" corresponds to a pole in the Titchmarsh-Weyl function m ( λ ) m(\lambda ) for the corresponding formally self-adjoint problem with L 2 ( 0 , ∞ ) {L^2}(0,\infty ) boundary condition.