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We obtain the leading asymptotic behaviour as e -O+ of the exponentially small imaginary part of the "eigenvalue" of the perturbed nonself-adjoint problem comprising y"(x) + (A + ex2) y(x) = 0 with a linear homogeneous boundary condition at x = 0 and an "outgoing wave" condition as x -* +oo . 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(A) for the corresponding formally self-adjoint problem with L2(O, oc) boundary condition. INTRODUCTION Let e > 0 and h by a positive constant. We consider the problem of finding the imaginary part of the eigenvalue )i of the differential equation (1.1) y"(x) + (A + ex2)y(x) = 0 on (0, ox) with y'(O) + hy(O) = 0, together with the condition that y(x) has a controlling behaviour exp(ip(x)) as x -+oo for some positive real-valued function p(x). There is a unique function p(x) with this property. We determine it by the Liouville-Green approximation. This seeks the controlling behaviour of the solution as x -* +oo via the substitution y(x) = eiP(x), which yields the result p(x) = !e1/2x2 for this problem. We may write this boundary condition in a Wronskian format more familiar in spectral theory as lim W[y(X), ee112X2] = 0. X-+oo It is easy to show that the unperturbed eigenvalue is at A = -h2 and that the perturbed eigenvalue problem is not self-adjoint because of the form of the Received by the editors October 7, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 34B20, 34B25, 34E20; Secondary 33A15, 33A70.

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