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 TMF, 2000, Volume 123, Number 1, Pages 132–149 (Mi tmf592)

A crystal with a singular potential in a homogeneous electric field

A. A. Pozharskii

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We study the asymptotic behavior of solutions to the one-dimensional Schrödinger equation $-\psi"+q(x)\psi-Fx\psi=E\psi$ for large arguments. We assume that the potential $q$ is a periodic function and is absolutely integrable over the period. We show that the spectral problem for the original Schrödinger equation can be reduced to the spectral problem for a discrete system. If the potential $q$ is smooth, the transition matrix tends to the unit matrix rapidly; if $q$ is not smooth, the transition matrix tends to the unit matrix slowly, and the discrete system demonstrates random properties. This explains why the spectrum of the original equation has remained practically unexplored.

DOI: https://doi.org/10.4213/tmf592

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English version:
Theoretical and Mathematical Physics, 2000, 123:1, 524–538

Bibliographic databases:

Citation: A. A. Pozharskii, “A crystal with a singular potential in a homogeneous electric field”, TMF, 123:1 (2000), 132–149; Theoret. and Math. Phys., 123:1 (2000), 524–538

Citation in format AMSBIB
\Bibitem{Poz00} \by A.~A.~Pozharskii \paper A crystal with a singular potential in a homogeneous electric field \jour TMF \yr 2000 \vol 123 \issue 1 \pages 132--149 \mathnet{http://mi.mathnet.ru/tmf592} \crossref{https://doi.org/10.4213/tmf592} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1773216} \zmath{https://zbmath.org/?q=an:0981.34082} \transl \jour Theoret. and Math. Phys. \yr 2000 \vol 123 \issue 1 \pages 524--538 \crossref{https://doi.org/10.1007/BF02551059} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000087532100012} 

• http://mi.mathnet.ru/eng/tmf592
• https://doi.org/10.4213/tmf592
• http://mi.mathnet.ru/eng/tmf/v123/i1/p132

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This publication is cited in the following articles:
1. A. A. Pozharskii, “Semicrystal with a singular potential in an accelerating electric field”, Theoret. and Math. Phys., 146:3 (2006), 343–360
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