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 $𝐍\left(\tau \right)$ of eigenvalues of operator $A\left(x,D\right)-tW\left(x,x\right)$ crossing level $E$ as $t$ runs from $0$ to $\tau$, $\tau \to \infty$. 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.

@article{JEDP_1999____A5_0,
author = {Ivrii, Victor},
title = {Accurate Spectral Asymptotics for periodic operators},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Universit\'e de Nantes},
year = {1999},
zbl = {01810578},
mrnumber = {2000h:35125},
language = {en},
url = {http://www.numdam.org/item/JEDP_1999____A5_0}
}

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

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