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**The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential.**
*(English)*
Zbl 0578.47003

It is well known that the spectrum of a one-dimensional Schrödinger operator with a periodic potential is a purely absolutely continuous band spectrum, and generically there are infinitely many gaps [M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV, Ch. VIII.16 (1978; Zbl 0401.47001)]. In higher dimensions a particle may have enough leeway to evade a periodic obstacle, and there is the famous Bethe-Sommerfeld conjecture (pertaining, naturally, to dimensions 2 and 3) that such operators have only finitely many gaps. This long-standing conjecture was proved for bounded potentials with rational period lattice in the plane by the author [Sov. Math., Dokl. 20, 956-959 (1979; Zbl 0435.35028)] and for arbitrary planar lattices by V. N. Popov and the author in [Zap. Nauchn. Semin. Leningrad Otd. Mat. Inst. Steklova 109, 131-133 (1983; Zbl 0492.47024)] and also by E. J. Dahlberg and E. Trubowitz [Commentat. Math. Helv. 57, 130-134 (1982; Zbl 0539.35059)]. See also the author’s paper [Proc. Steklov Inst. Math. 158, 177-189 (1983; Zbl 0525.47034)] for further informations.

In the present paper the conjecture is proved for a Schrödinger operator H with a continuous periodic potential V with arbitrary lattice \(T\subset {\mathbb{R}}^ 3\). It turns out that the number of gaps is primarily determined by the geometric and arithmetic properties of T rather than by the analytic behaviour of V. To describe the author’s main estimate we need the following notation and results. For \(k\in {\mathbb{R}}^ 3\) \[ (*)\quad H(k):=(1/i)\nabla -k)^ 2+V \] is a self-adjoint operator with a purely discrete spectrum. Let \(\{E_ j(k)\}\) (j\(\in {\mathbb{N}})\) be a numbering of its eigenvalues in non-decreasing order, counting multiplicities. For \(j\in {\mathbb{N}}\), \(E_ j(.)\) is continuous and periodic with period lattice \(\Gamma\) dual to T. Moreover, \[ \sigma (H)=\cup_{j\in {\mathbb{N}}}\Delta_ j\quad where\quad \Delta_ j:=R(E_ j(.)). \] Given \(\lambda\in {\mathbb{R}}\), let M(H,\(\lambda)\) be the multiplicity of the covering of \(\lambda\), i.e. the number of j such that \(\lambda \in \Delta_ j\). The author shows that there is a number \(c_ T>0\) with \[ (*)\quad M(H,\lambda)>c_ T\lambda^{1/2}\quad as\quad \lambda \to \infty. \] As a consequence, H has only finitely many gaps. An inspection of \(c_ T\) reveals that there are no gaps at all when \(\| V\|_{\infty}\) is sufficiently small.

The derivation of (*) is very intricate, a central role being played by the number \(N_{\Gamma}(\lambda,k)\) of lattice points of \(\Gamma\) in a ball with radius \(\lambda^{1/2}\) and centre \(k\in {\mathbb{R}}^ 3\), which was estimated in an elegant way by Dahlberg and Trubowitz [loc. cit.]. The author’s presentation is reasonably self-contained. Occasionally reference is made to a paper of his which has meanwhile appeared in Trudy Mat. Inst. Steklov. 171, (1985; Zbl 0567.47004)]. An estimate analogous to (*) for dimensions \(n\geq 4\) seems to have been established so far for special lattices only [cf. the author’s paper in Math. USSR Izv. 22, 619- 645 (1984; Zbl 0545.35027)].

{Reviewer’s remark. For smooth V in \({\mathbb{R}}^ 3\) another proof of the Bethe-Sommerfeld conjecture has been given by O. A. Veliev in Sov. Math. Dokl. 27, 234-237 (1983; Zbl 0541.47017).}

In the present paper the conjecture is proved for a Schrödinger operator H with a continuous periodic potential V with arbitrary lattice \(T\subset {\mathbb{R}}^ 3\). It turns out that the number of gaps is primarily determined by the geometric and arithmetic properties of T rather than by the analytic behaviour of V. To describe the author’s main estimate we need the following notation and results. For \(k\in {\mathbb{R}}^ 3\) \[ (*)\quad H(k):=(1/i)\nabla -k)^ 2+V \] is a self-adjoint operator with a purely discrete spectrum. Let \(\{E_ j(k)\}\) (j\(\in {\mathbb{N}})\) be a numbering of its eigenvalues in non-decreasing order, counting multiplicities. For \(j\in {\mathbb{N}}\), \(E_ j(.)\) is continuous and periodic with period lattice \(\Gamma\) dual to T. Moreover, \[ \sigma (H)=\cup_{j\in {\mathbb{N}}}\Delta_ j\quad where\quad \Delta_ j:=R(E_ j(.)). \] Given \(\lambda\in {\mathbb{R}}\), let M(H,\(\lambda)\) be the multiplicity of the covering of \(\lambda\), i.e. the number of j such that \(\lambda \in \Delta_ j\). The author shows that there is a number \(c_ T>0\) with \[ (*)\quad M(H,\lambda)>c_ T\lambda^{1/2}\quad as\quad \lambda \to \infty. \] As a consequence, H has only finitely many gaps. An inspection of \(c_ T\) reveals that there are no gaps at all when \(\| V\|_{\infty}\) is sufficiently small.

The derivation of (*) is very intricate, a central role being played by the number \(N_{\Gamma}(\lambda,k)\) of lattice points of \(\Gamma\) in a ball with radius \(\lambda^{1/2}\) and centre \(k\in {\mathbb{R}}^ 3\), which was estimated in an elegant way by Dahlberg and Trubowitz [loc. cit.]. The author’s presentation is reasonably self-contained. Occasionally reference is made to a paper of his which has meanwhile appeared in Trudy Mat. Inst. Steklov. 171, (1985; Zbl 0567.47004)]. An estimate analogous to (*) for dimensions \(n\geq 4\) seems to have been established so far for special lattices only [cf. the author’s paper in Math. USSR Izv. 22, 619- 645 (1984; Zbl 0545.35027)].

{Reviewer’s remark. For smooth V in \({\mathbb{R}}^ 3\) another proof of the Bethe-Sommerfeld conjecture has been given by O. A. Veliev in Sov. Math. Dokl. 27, 234-237 (1983; Zbl 0541.47017).}

Reviewer: H.Kalf

### MSC:

47A10 | Spectrum, resolvent |

47F05 | General theory of partial differential operators |

35J10 | Schrödinger operator, Schrödinger equation |

### Keywords:

spectrum band structure; three-dimensional Schrödinger operator; with periodic potential; spectrum of a one-dimensional Schrödinger operator with a; periodic potential; purely absolutely continuous band spectrum; Bethe-Sommerfeld conjecture; number of gaps; period lattice; lattice points; spectrum of a one-dimensional Schrödinger operator with a periodic potential### Citations:

Zbl 0401.47001; Zbl 0435.35028; Zbl 0492.47024; Zbl 0539.35059; Zbl 0525.47034; Zbl 0567.47004; Zbl 0545.35027; Zbl 0541.47017### References:

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[2] | Callaway, J.: Energy hand theory. New York: Academic Press 1964 · Zbl 0121.23303 |

[3] | Cassels, J.W.S.: An introduction to the geometry of numbers. Berlin-Göttingen-Heidelberg: Springer 1959 · Zbl 0086.26203 |

[4] | Dahlberg, B.E.J., Trubowitz, E.: A remark on two dimensional periodic potentials. Comment. Math. Helvetici57, (N 1), 130-134 (1982) · Zbl 0539.35059 |

[5] | Eastham, M.S.P.: The spectral theory of periodic differential equations. Edinburgh: Scottish Academic Press 1973 · Zbl 0287.34016 |

[6] | Landau, E.: Vorlesungen über Zahlentheorie. Bd. 2. Leipzig: Hirzel 1927 |

[7] | Popov, V.N., Skriganov, M.M.: A remark on the spectrum structure of the two dimensional Schrödinger operator with periodic potential. Notes Sci. Sem. Steklov Math. Inst. (Leningrad Branch)109, 131-133 (1981) · Zbl 0492.47024 |

[8] | Reed, M., Simon, B.: Methods in modern mathematical physics. Vol. IV: Analysis of operators. New York: Academic Press 1978 · Zbl 0401.47001 |

[9] | Skriganov, M.M.: Prof of the Bethe-Sommerfeld hypothesis in dimension two. Dokl. Akad. Nauk SSSR248 (N 1), 39-42 (1979) · Zbl 0435.35028 |

[10] | Skriganov, M.M.: General properties of the spectrum of differential and pseudodifferential operators with periodic coefficients and some problems of the geometry of numbers. Dokl. Akad. Nauk SSSR256 (N 1), 47-51 (1981) · Zbl 0483.35059 |

[11] | Skriganov, M.M.: On the spectrum of the multi-dimensional operators with periodic coefficients. Func. Anal. i Pril.16 (N 4), 88-89 (1982) |

[12] | Skriganov, M.M.: The multi-dimensional Schrödinger operator with periodic potential. Izvestiya Akad. Nauk SSSR (Ser. Math.)47, (N 3), 659-687 (1983) |

[13] | Skriganov, M.M.: Geometrical and arithmetical methods in the spectral theory of the multidimensional periodic operators. Proc. Steklov Math. Inst. Vol. 171. Moscow-Leningrad: Nauka 1984 (to appear) · Zbl 0567.47004 |

[14] | Skriganov, M.M.: Proof of the Bethe-Sommerfeld hypothesis in dimension three. LOMI Preprints: P-6-84 (1984) |

[15] | Sommerfeld, A., Bethe, H.: Elektronentheorie der Metalle. Handbuch der Physik. 2nd. ed. Bd. XXIV/2. Berlin: Springer 1933 |

[16] | Yakovlev, N.N.: Asymptotic estimates of the density of latticek-packing andk-covering and the spectrum structure of the Schrödinger operator with periodic potential. Doklady Akad. Nauk SSSR276 (N 1), 54-57 (1984) |

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