## Duke Mathematical Journal

### On the Bethe-Sommerfeld conjecture for the polyharmonic operator

#### Abstract

We consider in $L^2(ℝ^{d}),d≥2$, the perturbed polyharmonic operator $H=(−Δ)^{l}+V, l > 0$, with a function $V$ periodic with respect to a lattice in $ℝ^d$. We prove that the number of gaps in the spectrum of $H$ is finite if $6 l > d+2$. Previously the finiteness of the number of gaps was known for $4 l > d+1$. The proof is based on arithmetic properties of the lattice and elementary perturbation theory.

#### Article information

Source
Duke Math. J., Volume 107, Number 2 (2001), 209-238.

Dates
First available in Project Euclid: 5 August 2004

https://projecteuclid.org/euclid.dmj/1091736756

Digital Object Identifier
doi:10.1215/S0012-7094-01-10721-7

Mathematical Reviews number (MathSciNet)
MR1823047

Zentralblatt MATH identifier
1092.35025

#### Citation

Parnovski, Leonid; Sobolev, Alexander V. On the Bethe-Sommerfeld conjecture for the polyharmonic operator. Duke Math. J. 107 (2001), no. 2, 209--238. doi:10.1215/S0012-7094-01-10721-7. https://projecteuclid.org/euclid.dmj/1091736756

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