Absence of bound states in a class of time-dependent Hamiltonians (Q1329337)

This paper proves the absence of bound state solutions for the time- dependent one-dimensional Schrödinger equation. The potential \(V(x,t)\) satisfies (1) \(\partial V/ \partial x = O(x^{-\alpha})\) at infinity, with \(\alpha > 1\); (2) \(x \partial V/ \partial x \leq-(1/2) \alpha (\alpha + 1) x^{-2}\) for large \(x\), resp. \(\leq 0\) for small \(x\) (repulsivity); (3) smoothness conditions. In addition, in case \(V\) is time-periodic, it is shown that the monodromy operator has purely absolutely continuous spectrum.