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

Neal Brazel, Fiona Lawless and Alastair Wood
Proceedings of the American Mathematical Society
Vol. 114, No. 4 (Apr., 1992), pp. 1025-1032
DOI: 10.2307/2159623
Stable URL: http://www.jstor.org/stable/2159623
Page Count: 8
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Exponential Asymptotics for an Eigenvalue of a Problem Involving Parabolic Cylinder Functions
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Abstract

We obtain the leading asymptotic behaviour as ε → 0+ of the exponentially small imaginary part of the "eigenvalue" of the perturbed non-self-adjoint problem comprising y''(x) + (λ + ε x2)y(x) = 0 with a linear homogeneous boundary condition at x = 0 and an "outgoing wave" condition as x → +∞. 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) for the corresponding formally self-adjoint problem with L2(0, ∞) boundary condition.

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