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

Abstract:

We obtain the leading asymptotic behaviour as $\varepsilon \to 0 +$ of the exponentially small imaginary part of the "eigenvalue" of the perturbed nonself-adjoint problem comprising $y''(x) + (\lambda + \varepsilon {x^2})y(x) = 0$ with a linear homogeneous boundary condition at $x = 0$ and an "outgoing wave" condition as $x \to + \infty$. 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 $m(\lambda )$ for the corresponding formally self-adjoint problem with ${L^2}(0,\infty )$ boundary condition.