On the Bethe-Sommerfeld conjecture

Parnovski, Leonid; Sobolev, Alexander V.

Parnovski, Leonid ; Sobolev, Alexander V.

Journées équations aux dérivées partielles (2000), article no. 17, 13 p.

Résumé

We consider the operator in $ℝ^{d}, d \geq 2$ , 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.

MR Zbl

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     author = {Parnovski, Leonid and Sobolev, Alexander V.},
     title = {On the {Bethe-Sommerfeld} conjecture},
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     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {17},
     pages = {1--13},
     publisher = {Universit\'e de Nantes},
     year = {2000},
     mrnumber = {2002i:35137},
     zbl = {01808707},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2000____A17_0/}
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Parnovski, Leonid; Sobolev, Alexander V. On the Bethe-Sommerfeld conjecture. Journées équations aux dérivées partielles (2000), article  no. 17, 13 p. http://www.numdam.org/item/JEDP_2000____A17_0/

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