An application of asymptotic techniques to certain problems of spectral and scattering theory of Stark-like Hamiltonians

Abstract:

Let ${L_0} = - \Delta + V({x_1}),L = {L_0} + {V_p}(x)$ be selfadjoint in ${L^2}({R^n})$. Here $V,{V_p}$ are real functions, $V({x_1})$ depends only on the first coordinate. Existence of the wave-operators ${W_ \pm } (L,{L_0}) = s \text {-} {\lim _{t \to \pm \infty }} \exp (itL)\exp ( - it{L_0})$ is proved, using the stationary phase method. For this, an asymptotic technique is applied to the study of $-{d^2}/d{t^2} + V(t)$ in ${L^2}(R)$. Its absolute continuity is proved as well as a suitable eigenfunction expansion. $V$ is a "Stark-like" potential. In particular, the cases $V({x_1}) = ( - \operatorname {sgn}{x_1})|{x_1} {|^\alpha },0 < \alpha \leqslant 2$, are included. ${V_p}$ may be taken as the sum of an ${L^2}$-function and a function satisfying growth conditions in the $+ {x_1}$ direction. ${V_p}(x) = |x|^{ - 1}$ is included.