Exponential Asymptotics for an Eigenvalue of a Problem Involving Parabolic Cylinder Functions

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