Asymptotic dynamics and spectral analysis for the one-dimensional Schrödinger operator with accelerating potential. (Russian) Zbl 0599.47013

Buslaeva, M. V.

Probl. Mat. Fiz. 11, 67-77 (1986).

The one-dimensional Schrödinger operator H is considered on a semi-axis with a potential allowing the asymptotic estimate \(-vx^{2\alpha}\leq v(x)\leq -v_+x^{2\alpha}\), \(0<v_+\), \(1/3<\alpha <1\). The intertwining operator is constructed explicitly. It allows the asymptotic, for large time intervals, identification of the unitary group generated by the operator H with the shift in the function space on the axis. This result leads directly to the unitary equivalence of H and the differentiation operator on the axis. The conditions of existence of ordinary and generalized wave operators for a pair of different operators of the type H are obtained.

Reviewer: O.Dumbrajs

MSC:

47A40	Scattering theory of linear operators
47F05	General theory of partial differential operators

Keywords:

one-dimensional Schrödinger operator; asymptotic estimate; intertwining operator; unitary group; existence of ordinary and generalized wave operators