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


Authors: Neal Brazel, Fiona Lawless and Alastair Wood
Journal: Proc. Amer. Math. Soc. 114 (1992), 1025-1032
MSC: Primary 34B20; Secondary 33C10, 34E20, 34L40
DOI: https://doi.org/10.1090/S0002-9939-1992-1086323-X
MathSciNet review: 1086323
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1086323-X
Keywords: Exponential asymptotics, singular perturbation, spectral theory, Titchmarsh-Weyl coefficient, parabolic cylinder functions
Article copyright: © Copyright 1992 American Mathematical Society