Asymptotic completeness for a new class of Stark effect Hamiltonians. (English) Zbl 0606.34020

Existence and completeness of the ordinary wave operators is shown for the Stark effect Hamiltonian \(H=-\frac{d^ 2}{dx^ 2}+x+V(x)\) in one dimension with a potential \(V(x)=W''(x)\), where W is a real-valued bounded function with four bounded derivatives. This class of potentials includes some almost-periodic functions and periodic functions with average zero over a period (Stark-Wannier Hamiltonian). The proofs use commutator computations. In the last section asymptotic completeness in the classical scattering theory is shown for the same class of potentials.


34L99 Ordinary differential operators
47A40 Scattering theory of linear operators
47E05 General theory of ordinary differential operators
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[1] Agler, J., Froese, R.: Existence of Stark ladder resonances. Commun. Math. Phys.100, 161-171 (1985) · Zbl 0651.47006 · doi:10.1007/BF01212445
[2] 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) · Zbl 0351.47007 · doi:10.1007/BF01609485
[3] Ben-Artzi, M.: Remarks on Schrödinger operators with an electric field and deterministic potentials. J. Math. Anal. Appl.109, 333-339 (1985) · Zbl 0579.34018 · doi:10.1016/0022-247X(85)90154-4
[4] 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) · Zbl 0531.60061 · doi:10.1007/BF01213215
[5] Dunford, N., Schwartz, J. T.: Linear operators II. New York: Wiley 1963 · Zbl 0128.34803
[6] Herbst, I. W.: Unitary equivalence of Stark effect Hamiltonians. Math. Z.155, 55-70 (1977) · doi:10.1007/BF01322607
[7] 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) · Zbl 0561.47007
[8] Mourre, E.: Link between the geometrical and the spectral transformation approaches in scattering theory. Commun. Math. Phys.68, 91-94 (1979) · Zbl 0429.47006 · doi:10.1007/BF01562544
[9] Mourre, E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391-408 (1981) · Zbl 0489.47010 · doi:10.1007/BF01942331
[10] Perry, P. A.: Scattering theory by the Enss method. Math. Rep.1, part 1, 1983 · Zbl 0529.35004
[11] Rejto, P. A., Sinha, K.: Absolute continuity for a 1-dimensional model of the Stark-Hamiltonian. Helv. Phys. Acta49, 389-413 (1976)
[12] Simon, B.: Phase space analysis of simple scattering systems: Extensions of some work of Enss. Duke Math. J.46, 119-168 (1979) · Zbl 0402.35076 · doi:10.1215/S0012-7094-79-04607-6
[13] Simon, B.: Trace ideals and their applications. Cambridge: Cambridge University Press 1977
[14] Yajima, K.: Spectral and scattering theory for Schrödinger operators with Stark-effect. J. Fac. Sci. Univ. Tokyo, Sec IA,26, 377-390 (1979) · Zbl 0429.35027
[15] Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III Scattering Theory. New York: Academic Press 1979 · Zbl 0405.47007
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