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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator $H - \lambda W$ in a gap of $H$
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by S. Z. Levendorskiĭ
Trans. Amer. Math. Soc. 351 (1999), 857-899
DOI: https://doi.org/10.1090/S0002-9947-99-01994-7

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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Bibliographic Information
  • S. Z. Levendorskiĭ
  • Affiliation: Rostov Institute of National Economy, Engels’a 69, 344798, Rostov-on-Don, Russia
  • Email: leven@ns.rnd.runnet.ru
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 857-899
  • MSC (1991): Primary 35P20
  • DOI: https://doi.org/10.1090/S0002-9947-99-01994-7
  • MathSciNet review: 1433122