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.