On the eigenvalues of a polyharmonic matrix operator near diffraction planes.

In this study, we consider the d-dimensional polyharmonic matrix operator H (l , V) u = (− Δ) l u + V (x) u , where (−Δ)^l is a diagonal s×s matrix, whose diagonal elements are the scalar polyharmonic operators, V is the operator of multiplication by a symmetric s×s matrix, V(x) is periodic with respect to an arbitrary lattice and s ≥ 2, x = (x 1 , x 2 , ⋯ , x d) ∈ ℝ d , d ≥ 2 , 1 2 < l < 1. We obtain the high energy asymptotics for the eigenvalues of this operator which lies near the diffraction planes. [ABSTRACT FROM AUTHOR]