Singular Schrodinger operators in one dimension

Details Singular Schrodinger operators in one dimension Singular Schrodinger operators in one dimension

2011-12-17

arxiv.org/abs/1112.4041v1

Mathematics

Spectral Theory

23 pages

Scientific paper

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.

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