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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotic formulae with remainder estimates
for eigenvalue branches of the Schrödinger
operator $H-\lambda W$ in a gap of $H$


Author: S. Z. Levendorskii
Journal: Trans. Amer. Math. Soc. 351 (1999), 857-899
MSC (1991): Primary 35P20
MathSciNet review: 1433122
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Abstract: The Floquet theory provides a decomposition of a periodic
Schrödinger operator into a direct integral, over a torus, of operators on a basic period cell. In this paper, it is proved that the same transform establishes a unitary equivalence between a multiplier by a decaying potential and a pseudo-differential operator on the torus, with an operator-valued symbol. A formula for the symbol is given. As applications, precise remainder estimates and two-term asymptotic formulas for spectral problems for a perturbed periodic Schrödinger operator are obtained.


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Additional Information

S. Z. Levendorskii
Affiliation: Rostov Institute of National Economy, Engels’a 69, 344798, Rostov-on-Don, Russia
Email: leven@ns.rnd.runnet.ru

DOI: http://dx.doi.org/10.1090/S0002-9947-99-01994-7
PII: S 0002-9947(99)01994-7
Received by editor(s): May 15, 1995
Received by editor(s) in revised form: December 9, 1995
Additional Notes: The author was supported in part by ISF grant RNH 000
Article copyright: © Copyright 1999 American Mathematical Society