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 $Δ+ V$ 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 $-Δ+ V$ and $-Δ- V$ 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