Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator 𝐻-𝜆𝑊 in a gap of 𝐻

Levendorskiĭ, S.

doi:10.1090/S0002-9947-99-01994-7

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