A singular perturbation method. Part II

Abstract

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