On the Bethe-Sommerfeld conjecture

Leonid Parnovski; Alexander V. Sobolev

Journées équations aux dérivées partielles (2000)

  • page 1-13
  • ISSN: 0752-0360

Abstract

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We consider the operator in ℝ d , d ≥ 2 , of the form 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 8 l > d + 3 . Previously the finiteness of the number of gaps was known for 4 l > d + 1 . Various approaches to this problem are discussed.

How to cite

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Parnovski, Leonid, and Sobolev, Alexander V.. "On the Bethe-Sommerfeld conjecture." Journées équations aux dérivées partielles (2000): 1-13. <http://eudml.org/doc/93394>.

@article{Parnovski2000,
abstract = {We consider the operator in $\mathbb \{R\}^d, d\ge 2$, of the form $H = (-\Delta )^l+V, l&gt;0$ with a function $V$ periodic with respect to a lattice in $\mathbb \{R\}^d$. We prove that the number of gaps in the spectrum of $H$ is finite if $8l&gt;d+3$. Previously the finiteness of the number of gaps was known for $4l&gt;d+1$. Various approaches to this problem are discussed.},
author = {Parnovski, Leonid, Sobolev, Alexander V.},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-13},
publisher = {Université de Nantes},
title = {On the Bethe-Sommerfeld conjecture},
url = {http://eudml.org/doc/93394},
year = {2000},
}

TY - JOUR
AU - Parnovski, Leonid
AU - Sobolev, Alexander V.
TI - On the Bethe-Sommerfeld conjecture
JO - Journées équations aux dérivées partielles
PY - 2000
PB - Université de Nantes
SP - 1
EP - 13
AB - We consider the operator in $\mathbb {R}^d, d\ge 2$, of the form $H = (-\Delta )^l+V, l&gt;0$ with a function $V$ periodic with respect to a lattice in $\mathbb {R}^d$. We prove that the number of gaps in the spectrum of $H$ is finite if $8l&gt;d+3$. Previously the finiteness of the number of gaps was known for $4l&gt;d+1$. Various approaches to this problem are discussed.
LA - eng
UR - http://eudml.org/doc/93394
ER -

References

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