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Asymptotic of the density of states for the Schrödinger operator with periodic electric potential. (English) Zbl 0951.35104

Let \(\Gamma\) be a lattice on \(\mathbb{R}^n\), \(V\) be a \(C^\infty\) \(\Gamma\)-periodic potential and \(P_V\) be the Schrödinger operator on \(L^2(\mathbb{R}^n)\), \(n\geq 2: P_V= \sum_{1\leq j\leq n} D^2_j+ V\). The authors prove that for the integrated density of states \(N(\mu, P_V)\), we have for any fixed \(\varepsilon> 0\), the following asymptotic expansion as \(\mu\to\infty\): \[ N(\mu, P_V)= a_n \mu^{n/2}+ a_{n-2} \mu^{(n- 2)/2}+ O(1)+ O(\mu^{(n- 3+\varepsilon)/2}) \] with \[ a_n= (2\pi)^{-n}|S^{n-1}|/n> a_{n- 2}= -{na_n\over 2} |K|^{-1} \int_K V(x) dx \] (\(K\) a periodic cell). They also recover a proof of the Béthe-Sommerfeld conjecture for \(n\leq 4\).
The method is related to the one used by Hörmander for the study of the counting function of the eigenvalues of an elliptic operator and needs a periodic pseudodifferential calculus.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
47F05 General theory of partial differential operators
47N50 Applications of operator theory in the physical sciences
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35J10 Schrödinger operator, Schrödinger equation
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[1] R. Beals, Characterization of pseudodifferential operators and applications , Duke Math. J. 44 (1977), no. 1, 45-57. · Zbl 0353.35088
[2] P. M. Bleher, Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems , Duke Math. J. 70 (1993), no. 3, 655-682. · Zbl 0803.11052
[3] K. H. Boĭ matov and A. G. Kostjučenko, Asymptotic behavior of Riesz means of the spectral function of an elliptic operator , Dokl. Akad. Nauk SSSR 241 (1978), no. 3, 517-520 (Russian). · Zbl 0455.47043
[4] J.-M. Bony and J.-Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander , Bull. Soc. Math. France 122 (1994), no. 1, 77-118. · Zbl 0798.35172
[5] A. Calderón and R. Vaillancourt, On the boundedness of pseudo-differential operators , J. Math. Soc. Japan 23 (1971), 374-378. · Zbl 0214.39004
[6] Y. Colin de Verdière, Nombre de points entiers dans une famille homothétique de domains de \(\mathbf R\) , Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 559-575. · Zbl 0409.58011
[7] B. Dahlberg and E. Trubowitz, A remark on two dimensional periodic potentials , Comment. Math. Helv. 57 (1982), no. 1, 130-134. · Zbl 0539.35059
[8] A. Grigis and A. Mohamed, Finitude des lacunes dans le spectre de l’opérateur de Schrödinger et de celui de Dirac avec des potentiels électrique et magnétique périodiques , J. Math. Kyoto Univ. 33 (1993), no. 4, 1071-1096. · Zbl 0814.47056
[9] A. Grigis and A. Mohamed, Résultats de finitude pour les lacunes spectrales , Séminaire sur les Équations aux Dérivées Partielles, 1992-1993, École Polytechnique, Palaiseau, 1993, Exp. No. XXIII, 7. · Zbl 0872.35074
[10] B. Helffer and D. Robert, Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles , J. Funct. Anal. 53 (1983), no. 3, 246-268. · Zbl 0524.35103
[11] B. Helffer and J. Sjöstrand, Équation de Schrödinger avec champ magnétique et équation de Harper , Schrödinger Operators (Sønderborg, 1988) eds. H. Holden and A. Jensen, Lecture Notes in Phys., vol. 345, Springer-Verlag, Berlin, 1989, pp. 118-197. · Zbl 0699.35189
[12] B. Helffer and J. Sjöstrand, On diamagnetism and de Haas-van Alphen effect , Ann. Inst. H. Poincaré Phys. Théor. 52 (1990), no. 4, 303-375. · Zbl 0715.35070
[13] L. Hörmander, The spectral function of an elliptic operator , Acta Math. 121 (1968), 193-218. · Zbl 0164.13201
[14] L. Hörmander, The Analysis of Linear Partial Differential Operators. I , Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. · Zbl 0521.35001
[15] L. Hörmander, The analysis of linear partial differential operators. III , Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. · Zbl 0601.35001
[16] P. Kuchment, Floquet Theory for Partial Differential Equations , Operator Theory: Advances and Applications, vol. 60, Birkhäuser Verlag, Basel, 1993. · Zbl 0789.35002
[17] A. Mohamed, Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential , J. Math. Phys. 38 (1997), no. 8, 4023-4051. · Zbl 0883.47045
[18] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of operators , Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978. · Zbl 0401.47001
[19] D. Robert, Autour de l’approximation semi-classique , Progr. Math., vol. 68, Birkhäuser Boston Inc., Boston, MA, 1987. · Zbl 0621.35001
[20] R. T. Seeley, Complex powers of an elliptic operator , Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 288-307. · Zbl 0159.15504
[21] M. Shubin, Spectral theory and the index of elliptic operators with almost-periodic coefficients , Russian Math. Surveys 34 (1979), 109-157. · Zbl 0448.47032
[22] J. Sjöstrand, Microlocal analysis for the periodic magnetic Schrödinger equation and related questions , Microlocal analysis and applications (Montecatini Terme, 1989), Lecture Notes in Math., vol. 1495, Springer, Berlin, 1991, pp. 237-332. · Zbl 0761.35090
[23] M. M. Skriganov, Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators , Proc. Steklov Inst. Math. (1987), no. 2(171), vi+121. · Zbl 0615.47004
[24] M. M. Skriganov, The spectrum band structure of the three-dimensional Schrödinger operator with periodic potential , Invent. Math. 80 (1985), no. 1, 107-121. · Zbl 0578.47003
[25] O. A. Veliev, The spectrum of multidimensional periodic operators , Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. (1988), no. 49, 17-34, 123, (in Russian) English trans. in J. Soviet Math. 49, (1990), 1045-1058. · Zbl 0664.47005
[26] A. V. Volovoy, Improved two-term asymptotics for the eigenvalue distribution function of an elliptic operator on a compact manifold , Comm. Partial Differential Equations 15 (1990), no. 11, 1509-1563. · Zbl 0724.35081
[27] A. V. Volovoy, Verification of the Hamilton flow conditions associated with Weyl’s conjecture , Ann. Global Anal. Geom. 8 (1990), no. 2, 127-136. · Zbl 0721.58048
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