Accurate Spectral Asymptotics for periodic operators
Journées équations aux dérivées partielles (1999), article no. 5, 11 p.

Asymptotics with sharp remainder estimates are recovered for number 𝐍(τ) of eigenvalues of operator A(x,D)-tW(x,x) crossing level E as t runs from 0 to τ, τ→∞. Here A is periodic matrix operator, matrix W is positive, periodic with respect to first copy of x and decaying as second copy of x goes to infinity, E either belongs to a spectral gap of A or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.

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     title = {Accurate {Spectral} {Asymptotics} for periodic operators},
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Ivrii, Victor. Accurate Spectral Asymptotics for periodic operators. Journées équations aux dérivées partielles (1999), article  no. 5, 11 p. http://archive.numdam.org/item/JEDP_1999____A5_0/

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