Half-line Schrodinger Operators With No Bound States

Abstract

We consider Schödinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $\Delta +V$ has no spectrum outside of the interval $[-2,2]$, then it has purely absolutely continuous spectrum. In the continuum case we show that if both $-\Delta +V$ and $-\Delta -V$ have no spectrum outside $[0,\infty )$, then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.

34 pages

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