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Exponential asymptotics and special theory for optical tunnelling

Lawless, Fiona (1993) Exponential asymptotics and special theory for optical tunnelling. PhD thesis, Dublin City University.

Mathematically this thesis involves an investigation of the non-self-adjomt Sturm- Liouville problem comprising the differential equation, y"(x) + (A + ex2)y(x) = 0 with a linear homogeneous boundary condition at x = 0 and an “outgoing wave ” condition as x —»• oo, in a number of different settings. The purpose of such an investigation is to obtain an accurate estimate for the imaginary part of the eigenvalue A. Physically, this singular eigenproblem arises m the mathematical modelling of radiation losses m bent fibre-optic waveguides, with the imaginary part of the desired eigenvalue providing a measure of the magnitude of loss due to bending. The imaginary part of the desired eigenvalue turns out to be of much smaller order [0(e _ i —► 0+] than the perturbation of the real part [0 (e),£ —»• 0+]. To overcome the resulting computational difficulties we appeal to the area of exponential asymptotics and become involved m the smoothing of Stokes discontmumties. A number of exponentially improved approximations are required for proper estimation of Im \ and these are obtained either directly from the literature or by application of recent results. The non-self-adjomt nature of the above tunnelling problem results from the unusual condition at infinity. While we investigate this problem directly, using special functions and variational techniques, and obtain an accurate estimate for imaginary part of the desired eigenvalue, an alternative setting is also found. This more abstract approach involves the theory of “resonance poles” in quantum mechanics. We show that under certain conditions, satisfied by the tunnelling problem being considered, the “eigenvalue” of a non-self-adjomt problem corresponds to a pole in the Titchmarsh-Weyl function m(A) for a related but formally self-adjomt problem.
Item Type:Thesis (PhD)
Date of Award:1993
Supervisor(s):Wood, Alastair
Uncontrolled Keywords:Non-self-adjoint Sturm-Liouville problem
Subjects:Mathematics > Differential equations
DCU Faculties and Centres:DCU Faculties and Schools > Faculty of Science and Health > School of Mathematical Sciences
Use License:This item is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 3.0 License. View License
ID Code:18952
Deposited On:23 Aug 2013 10:49 by Celine Campbell . Last Modified 07 Oct 2013 14:48

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