On the Bethe-sommerfeld Conjecture for the Polyharmonic Operator

It is a general property of elliptic differential operators with periodic coefficients, that their spectra are formed by union of closed intervals called spectral bands (see [12], [14]) possibly separated by gaps. One of the challenging questions of the spectral theory of periodic operators is to find out whether or not the number of gaps in the spectrum of a given operator is finite. The statement asserting the finiteness is usually referred to as the Bethe-Sommerfeld conjecture, after H. Bethe and A. Sommerfeld who raised this issue in the 30’s for the Schrödinger operator H = −∆+V with a periodic electric potential V in dimension three. It is convenient to rephrase this problem in quantitative terms by introducing the multiplicity of overlapping m(λ)(see [17]) which, by definition, is equal to the number of bands containing a given point λ ∈ R. Then, naturally, the Bethe-Sommerfeld conjecture holds iff m(λ) ≥ 1 for all sufficiently large positive λ. The aim of the present paper is to justify the Bethe-Sommerfeld conjecture for the polyharmonic operator with a self-adjoint perturbation V periodic with respect to some lattice Γ ⊂ R, d ≥ 2: H = H0 + V, H0 = (−∆), l > 0.