A singular perturbation method. Part II

This paper extends the work of Part I to the partial differential equation \[ {\epsilon ^3}{\nabla ^2}\psi - g\left ( x \right )\psi = 0\] where $\epsilon$ is a small positive parameter and $g\left ( x \right )$ is a bounded function of $x$ which vanishes along simple closed surfaces in the solution domain. In particular, the eigenproblem corresponding to the case in which $g\left ( x \right )$ is positive at infinity and in which the boundary condition $\psi \to 0$ as $\left | x \right | \to \infty$ is imposed, is considered. One class of eigensolutions is extracted.