01-337 Yu. Karpeshina

On Spectral Properties of Periodic Polyharmonic Matrix Operators. (501K, postscript) Sep 24, 01

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Abstract. We consider a matrix operator $H=(-\Delta )^l+ V $ in $R^n$, where $n\geq 2$, $l\geq 1$, $4l>n+1$, and $V$ is the operator of multiplication by a periodic in $x$ matrix $V(x)$. We study spectral properties of $H$ in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum of $H$ can have only a finite number of gaps, is proved.

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