Abstract
Existence and completeness of the wave operators is shown for the Stark effect Hamiltonian in one dimension with a potentialV =W″, whereW is a bounded function with four bounded derivatives. This class of potentials include some almost periodic functions and periodic functions with average zero over a period (Stark-Wannier Hamiltonians). In the last section we discuss classical particle scattering for the same class of potentials.
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Agler, J., Froese, R.: Existence of Stark ladder resonances. Commun. Math. Phys.100, 161–171 (1985)
Avron, J. E., Herbst, I. W.: Spectral and scattering theory for Schrödinger operators related to the Stark effect. Commun. Math. Phys.52, 239–254 (1977)
Ben-Artzi, M.: Remarks on Schrödinger operators with an electric field and deterministic potentials. J. Math. Anal. Appl.109, 333–339 (1985)
Bentosela, F., Carmona, R., Duclos, P., Simon, B., Souillard, B., Weder, R.: Schrödinger operators with an electric field and random or deterministic potentials. Commun. Math. Phys.88, 387–397 (1983)
Dunford, N., Schwartz, J. T.: Linear operators II. New York: Wiley 1963
Herbst, I. W.: Unitary equivalence of Stark effect Hamiltonians. Math. Z.155, 55–70 (1977)
Jensen, A., Mourre, E., Perry, P.: Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Ann. Inst. H. Poincaré, Sect. A,41, 207–225 (1984)
Mourre, E.: Link between the geometrical and the spectral transformation approaches in scattering theory. Commun. Math. Phys.68, 91–94 (1979)
Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–408 (1981)
Perry, P. A.: Scattering theory by the Enss method. Math. Rep.1, part 1, 1983
Rejto, P. A., Sinha, K.: Absolute continuity for a 1-dimensional model of the Stark-Hamiltonian. Helv. Phys. Acta49, 389–413 (1976)
Simon, B.: Phase space analysis of simple scattering systems: Extensions of some work of Enss. Duke Math. J.46, 119–168 (1979)
Simon, B.: Trace ideals and their applications. Cambridge: Cambridge University Press 1977
Yajima, K.: Spectral and scattering theory for Schrödinger operators with Stark-effect. J. Fac. Sci. Univ. Tokyo, Sec IA,26, 377–390 (1979)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III Scattering Theory. New York: Academic Press 1979
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Communicated by B. Simon
Partially supported by NSF-grant DMS-8401748
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Jensen, A. Asymptotic completeness for a new class of Stark effect Hamiltonians. Commun.Math. Phys. 107, 21–28 (1986). https://doi.org/10.1007/BF01206951
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DOI: https://doi.org/10.1007/BF01206951