Accurate Spectral Asymptotics for periodic operators

Ivrii, Victor

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 $A(x,D)-tW(x,x)$ 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.

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