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Transactions of the American Mathematical Society

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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 PDF
Trans. Amer. Math. Soc. 351 (1999), 857-899 Request permission

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. 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