Passage to the limit with respect to a small parameter for the eigenvalues of a singularly perturbed Sturm-Liouville problem (Q2703996)

The authors study the limiting behaviour of the eigenvalues of the following problem as \(\varepsilon\to 0\): NEWLINE\[NEWLINE\varepsilon^2\psi''=[U(x,\varepsilon)+\lambda]\psi,\quad -1<x<1,\quad \psi(-1,\varepsilon)=\psi(1,\varepsilon)=0.NEWLINE\]NEWLINE In particular, they show that under certain conditions on the function \(U\) the eigenvalues \(\lambda^{(n)}(\varepsilon)\) either converge as \(\varepsilon\to 0\) to a fixed value; in which case the set of eigenvalues are referred to as the dense spectrum. Otherwise the spectrum is said to be sparse. Motivating examples are given.