An application of asymptotic techniques to certain problems of spectral and scattering theory of Stark-like Hamiltonians
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- by Matania Ben-Artzi PDF
- Trans. Amer. Math. Soc. 278 (1983), 817-839 Request permission
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.References
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Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 278 (1983), 817-839
- MSC: Primary 35P25; Secondary 81C10
- DOI: https://doi.org/10.1090/S0002-9947-1983-0701525-6
- MathSciNet review: 701525