Duke Mathematical Journal

On the Bethe-Sommerfeld conjecture for the polyharmonic operator

Leonid Parnovski and Alexander V. Sobolev

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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.

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

First available in Project Euclid: 5 August 2004

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Primary: 35J10: Schrödinger operator [See also 35Pxx]
Secondary: 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07] 31B30: Biharmonic and polyharmonic equations and functions 35B25: Singular perturbations 35P15: Estimation of eigenvalues, upper and lower bounds 35P20: Asymptotic distribution of eigenvalues and eigenfunctions


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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