Singular Schrodinger operators in one dimension

TL;DR

This paper studies a class of singular one-dimensional Schrödinger operators constructed from positive functions, showing that their eigenvalue asymptotics are unaffected by rapid oscillations and providing elementary analysis without potential assumptions.

Contribution

It introduces a direct, elementary approach to analyze singular Schrödinger operators, demonstrating independence of eigenvalue asymptotics from oscillatory behavior of the functions involved.

Findings

Eigenvalue asymptotics are unaffected by rapid oscillations.

Analysis does not rely on potential magnitude or derivatives.

Results apply to operators with discrete spectrum.

Abstract

We consider a class of singular Schr\"odinger operators $H$ that act in $L^2(0,\infty)$, each of which is constructed from a positive function $\phi$ on $(0,\infty)$. Our analysis is direct and elementary. In particular it does not mention the potential directly or make any assumptions about the magnitudes of the first derivatives or the existence of second derivatives of $\phi$. For a large class of $H$ that have discrete spectrum, we prove that the eigenvalue asymptotics of $H$ does not depend on rapid oscillations of $\phi$ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.