Singular Schrodinger operators in one dimension

We consider a class of singular Schrödinger operators $H$ that act in $L^2(0,\infty)$, each of which is constructed from a positive function $ϕ$ 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 $ϕ$. 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 $ϕ$ or of the potential. Similar comments apply to our treatment of the existence and completeness of the wave operators.

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