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Transactions of the American Mathematical Society

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


Author: Matania Ben-Artzi
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
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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$   -$ {\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})\vert{x_1}\,{\vert^\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) = \vert x\vert^{ - 1}$ is included.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0701525-6
Article copyright: © Copyright 1983 American Mathematical Society

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