Existence of Stark ladder resonances. (English) Zbl 0651.47006

This paper deals with a Hamiltonian of the form: \[ H(k,t)u(x)=- u''(x)+q(x-it)u(x)+k(x-it)u(x) \] acting in \(L^ 2(R)\), where k, \(t\in R\) and \(q(x)=\sum^{N}_{n=-N}c_ ne^{inx}\) with \(c_ 0=0\) and \(c_ n\neq 0\) for at least one n. It is known that \(\tau_{ess}(H(k,t))\subset R-ikt\) and that \(\tau_{disc}(H(k,t))\subset \{z:\quad -ikt\leq Im(z)\leq 0\}.\) The present contribution shows that there exists \(k_ 0>0\) such that for each \(k>k_ 0\) there exists \(t\in R\) such that H(k,t) has an eigenvalue (and by translation an infinity of eigenvalues) not on the line R-ikt. The authors conjecture that \(k_ 0=0\).
Reviewer: C.A.Stuart


47A55 Perturbation theory of linear operators
47E05 General theory of ordinary differential operators
81Q15 Perturbation theories for operators and differential equations in quantum theory
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[1] Avron, J. E.: Model calculation of Stark ladder resonances. Phys. Rev. Lett.37, 1568-1571 (1976)
[2] Arvon, J. E.: On the spectrum ofp 2+V(x)+?x, withV periodic and? complex. J. Phys. A.12, 2393 (1979) · Zbl 0447.34024
[3] Avron, J. E., Grossmann, A., Gunther, L., Zak, J.: Instability of the continuous spectrum: TheN-band Stark ladders. J. Math. Phys.18, 918 (1977)
[4] Bentosela, F., Carmona, R., Duclos, P., Simon, B., Soulliard, B., Weder, R.: Commun. Math. Phys.88, 387-397 (1983) · Zbl 0531.60061
[5] Herbst, I. W., Howland, J. S.: Commun. Math. Phys.80, 23-42 (1981) · Zbl 0473.47037
[6] Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1980 · Zbl 0435.47001
[7] Nenciu, A., Nenciu, G.: E17-82-207, E17-82-208, JINR, Dubna, 1982
[8] Simon, B.: Trace ideals and their applications. Cambridge: Cambridge University Press 1979 · Zbl 0423.47001
[9] Wannier, G. H.: Wave functions and effective Hamiltonian for Bloch electrons in an electric field. Phys. Rev.117, 432 (1960) · Zbl 0091.23502
[10] Wannier, G. H.: Stark ladder in solids? A reply. Phys. Rev.181, 1364 (1969)
[11] Zak, J.: Stark ladder in solids; A reply to reply. Phys. Rev.181, 1360 (1969)
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