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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