03-77 David Damanik, Rowan Killip

Half-line Schrodinger Operators With No Bound States (102K, AMS-LaTeX) Feb 28, 03

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Abstract. We consider Sch\"odinger 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.

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