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

@incollection{JEDP_2000____A17_0,
     author = {Parnovski, Leonid and Sobolev, Alexander V.},
     title = {On the {Bethe-Sommerfeld} conjecture},
     booktitle = {},
     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://archive.numdam.org/item/JEDP_2000____A17_0/}
}

TY  - JOUR
AU  - Parnovski, Leonid
AU  - Sobolev, Alexander V.
TI  - On the Bethe-Sommerfeld conjecture
JO  - Journées équations aux dérivées partielles
PY  - 2000
SP  - 1
EP  - 13
PB  - Université de Nantes
UR  - http://archive.numdam.org/item/JEDP_2000____A17_0/
LA  - en
ID  - JEDP_2000____A17_0
ER  -

%0 Journal Article
%A Parnovski, Leonid
%A Sobolev, Alexander V.
%T On the Bethe-Sommerfeld conjecture
%J Journées équations aux dérivées partielles
%D 2000
%P 1-13
%I Université de Nantes
%U http://archive.numdam.org/item/JEDP_2000____A17_0/
%G en
%F JEDP_2000____A17_0

Parnovski, Leonid; Sobolev, Alexander V. On the Bethe-Sommerfeld conjecture. Journées équations aux dérivées partielles (2000), article  no. 17, 13 p. http://archive.numdam.org/item/JEDP_2000____A17_0/

Bibliographie
Cité par

[1] V. Bentkus, F. Götze, On the lattice point problem for ellipsoids, Acta Arithm. 80 (1997), 101-125. | MR | Zbl

[2] B.E.J. Dahlberg, E. Trubowitz, A remark on two dimensional periodic potentials, Comment. Math. Helvetici 57 (1982), 130-134. | MR | Zbl

[3] P. Erdős, P.M. Gruber, J. Hammer, Lattice points, Longman 1987.

[4] F. Götze, Lattice point problems and the central limit theorem in Eucledian spaces, Documenta Mathematica, Extra Volume ICM 1998, III, 5245-255. | Zbl

[5] E.L Green, Spectral theory of Laplace-Beltrami operators with periodic metrics, J. Diff. Eq. 133 (1997), 15-29. | MR | Zbl

[6] B. Helffer, A. Mohamed, Asymptotics of the density of states for the Schrödinger operator with periodic electric potential, Duke Math. J. 92 (1998), 1-60. | MR | Zbl

[7] V. Jarnik, Sur les points à coordonnées entières dans les ellipsoïdes à plusieurs dimensions, Bull. Intern. Acad. Sci. Bohéme (1928).

[8] Yu. E. Karpeshina, Analytic perturbation theory for a periodic potential, Izv. Akad. Nauk SSSR Ser. Mat., 53 (1989), No 1, 45-65 ; English transl. : Math. USSR Izv., 34 (1990), No 1, 43-63. | Zbl

[9] Yu. E. Karpeshina, Perturbation theory for the Schrödinger operator with a periodic potential, Lecture Notes in Math. vol 1663, Springer Berlin 1997. | Zbl

[10] D.G. Kendall, R.A. Rankin, On the number of points of a given lattice in a random hypersphere, Quart. J. Math. Oxford 2, 4 (1953), 178-189. | MR | Zbl

[11] E. Krätzel, W.G. Nowak, Lattice points in large convex bodies, II, Acta Arithmetica, 62 (1992), 285-295. | MR | Zbl

[12] P. Kuchment, Floquet theory for partial differential equations, Birkhäuser, Basel, 1993. | MR | Zbl

[13] E. Landau, Zur analytischen Zahlentheorie der definiten quadratischen Formen (Über die Gitterpunkte in einem mehrdimensionalen Ellipsoid), Sitzber. Preuss. Akad. Wiss. 31 (1915), 458-476. | JFM

[14] E. Landau, Über die Gitterpunkte in einem mehrdimensionalen Ellipsoid, Math. Z. 21 (1924), 126-132. | JFM

[15] A. Mohamed, Asymptotics of the density of states for the Schrödinger operator with periodic electromagnetic potential, J. Math. Phys. 38 (1997), 4023-4051. | MR | Zbl

[16] L. Parnovski, A.V. Sobolev, On the Bethe-Sommerfeld conjecture for the polyharmonic operator, to appear in Duke Math. J. | Zbl

[17] L. Parnovski, A.V. Sobolev, Perturbation theory and the Bethe-Sommerfeld conjecture, CMAIA report No : 2000-05, 2000.

[18] V.N. Popov, M. Skriganov, A remark on the spectral structure of the two dimensional Schrödinger operator with a periodic potential, Zap. Nauchn. Sem. LOMI AN SSSR 109 (1981), 131-133 (Russian). | MR | Zbl

[19] M. Reed, B. Simon, Methods of modern mathematical physics, IV, Academic Press, New York, 1975. | Zbl

[20] M. Skriganov, Finiteness of the number of gaps in the spectrum of the multi-dimensional polyharmonic operator with a periodic potential, Mat. Sb. 113 (155) (1980), 131-145 ; Engl. transl. : Math. USSR Sb. 41 (1982). | Zbl

[21] M. Skriganov, Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators, Proc. Steklov Math. Inst. Vol. 171, 1984. | Zbl

[22] M. Skriganov, The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential, Inv. Math. 80 (1985), 107-121. | MR | Zbl

[23] O.A. Veliev, Asymptotic formulas for the eigenvalues of a periodic Schrödinger operator and the Bethe-Sommerfeld conjecture, Funkt. Anal. i Prilozhen. 21 (1987), 1-15 (in Russian) ; Engl. transl. : Functional Anal. Appl. 21 (1987), 87-99. | Zbl

[24] N.N. Yakovlev, Asymptotic estimates of the densities of lattice k-packings and k-coverings, and the structure of the spectrum of the Schrödinger operator with a periodic potential, Dokl. Akad. Nauk SSSR, 276 (1984), No 1 ; English transl. : Soviet Math. Dokl., 29 (1984), No 3, 457-460. | Zbl

[25] N.N. Yakovlev, On spectra of multi-dimensional pseudo-differential periodic operators, Vestn. Mosk. Univ. Ser. 1 Mat, Mekh, No 3 (1985), 80-81 (Russian). | MR | Zbl