Geodesic
Wannier Functions for Quasiperiodic Finite-Gap Potentials
Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 2, pp. 234-256 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider Wannier functions of quasiperiodic $g$-gap ($g\geq1$) potentials and investigate their main properties. In particular, we discuss the problem of averaging that underlies the definition of the Wannier functions for both periodic and quasiperiodic potentials and express Bloch functions and quasimomenta in terms of hyperelliptic $\sigma$-functions. Using this approach, we derive a power series for the Wannier function for quasiperiodic potentials valid for $|x|\simeq0$, and an asymptotic expansion valid at large distances. These functions are important in a number of applied problems.
Keywords: Wannier functions, finite-gap potentials, theta functions, hyperelliptic curves. 
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E. D. Belokolos; V. Z. Ènol'skii; M. Salerno. Wannier Functions for Quasiperiodic Finite-Gap Potentials. Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 2, pp. 234-256. http://geodesic.mathdoc.fr/item/TMF_2005_144_2_a2/

[1] E. D. Belokolos, V. Z. Enolskii, M. Salerno, J. Phys. A, 37 (2004), 9685–9704 ; E-print cond-mat/0401440 | DOI | MR | Zbl

[2] N. Ercolani, M. G. Forest, D. W. McLaughlin, R. Montgomery, Duke Math. J., 55 (1987), 949–983 ; H. Flaschka, G. Forest, D. McLaughlin, Commun. Pure Appl. Math., 33:6 (1980), 739–794 | DOI | MR | Zbl | DOI | MR

[3] I. M. Krichever, Funkts. analiz i ego prilozh., 22:3 (1988), 37–52 | MR | Zbl

[4] V. M. Buchstaber, V. Z. Enolskii, D. V. Leykin, “Hyperelliptic {K}leinian functions and applications”, Solitons, {G}eometry and {T}opology: {O}n the {C}rossroad, Amer. Math. Soc. Transl. Ser. 2, 179, eds. V. M. Buchstaber, S. P. Novikov, AMS, Providence, RI, 1997, 1–34 | MR

[5] V. M. Buchstaber, V. Z. Enolskii, D. V. Leykin, Rev. Math. Math. Phys., 102:2 (1997), 1–125 | MR

[6] V. M. Bukhshtaber, V. Z. Enolskii, D. V. Leikin, Funkts. analiz i ego prilozh., 33:2 (1999), 1–15 | DOI | MR | Zbl

[7] V. M. Bukhshtaber, D. V. Leikin, Funkts. analiz i ego prilozh., 36:4 (2002), 18–34 | DOI | MR | Zbl

[8] S. Baldwin, J. Gibbons, J. Phys. A, 36 (2003), 8393–8413 ; 37 (2004), 5341–5354 ; J. D. Edelstein, M. Gómez-Reino, M. Mariño, Adv. Theor. Math. Phys., 4 (2000), 503–543 ; ; J. C. Eilbeck, V. Z. Enolskii, E. Previato, Lett. Math. Phys., 63 (2003), 5–17 ; S. Matsutani, J. Geom. Phys., 794 (2002), 1–17 ; S. Matsutani, Y.Ônishi, Rev. Math. Phys., 15 (2003), 559–628 ; Glasgow Math. J., 44 (2002), 353–364 E-print hep-th/0006113 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | MR | DOI | MR | Zbl | DOI | MR

[9] E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, N.Y., 1955 | MR | Zbl

[10] R. Johnson, J. Moser, Commun. Math. Phys., 84 (1982), 403–438 | DOI | MR | Zbl

[11] L. A. Pastur, Commun. Math. Phys., 75 (1980), 179–196 | DOI | MR | Zbl

[12] S. Kotani, “Lyapunov indices determine absolutely continuous spectra of stationary random one–dimensional {S}chrödinger operators”, Stochastic Analysis, Proc. on Stochastic Analysis. Taniguchi Symp. (Katata, Kyoto, 1982), North-Holland Math. Libr., 32, ed. K. Itô, Kinokuniya, Tokyo; North-Holland, Amsterdam, 1984, 225–247 | DOI | MR

[13] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov. Metod obratnoi zadachi, Nauka, M., 1980 | MR

[14] E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, V. B. Matveev, Algebro {G}eometric {A}pproach to {N}onlinear {I}ntegrable {E}quations, Springer, Berlin, 1994

[15] F. Gesztesy, H. Holden, {S}oliton {E}quations and {T}heir {A}lgebro-{G}eometric {S}olutions. $(1+1)$-{D}imensional {C}ontinuous {M}odels, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl

[16] J. Guckenheimer, J. Moser, S. E. Newhouse, “Dynamical Systems”, {C}.{I}.{M}.{E}. {L}ectures (Bressanone, Italy, June 1978), Progress in Mathematics, 8, Birkhäuser, Basel–Stuttgart, 1980, 233–289 ; В. А. Марченко, И. В. Островский, Матем. сб., 97 (1975), 493–554 | MR

[17] H. Weyl, Math. Ann., 77 (1916), 313–352 | DOI | MR | Zbl

[18] Y. Ônishi, “Determinant expressions for hyperelliptic functions”, with an Appendix by Shigeki Matsutani: Connection of The formula of Cantor and of Brioschi–Kiepert type, Proc. Edinburgh Math. Soc., 2005 (to appear) | MR

[19] O. Bolza, Amer. J. Math., 17 (1895), 11–36 | DOI | MR

[20] H. F. Baker, Abel's Theorem and the Allied Theory of Theta Functions, Cambridge Univ. Press, Cambridge, 1897 ; reprinted, 1995; H. F. Baker, Amer. J. Math., 20 (1898), 301–384 | MR | Zbl | DOI | MR | Zbl

[21] H. F. Baker, Multiple Periodic Functions, Cambridge Univ. Press, Cambridge, 1907

[22] A. R. Its, V. B. Matveev, TMF, 23 (1975), 51–68 | MR

[23] B. A. Dubrovin, UMN, 36 (1981), 11–80 | MR | Zbl

[24] A. Krazer, Lehrbuch der {T}hetafunktionen, Teubner, Leipzig, 1903 ; Reprinted, AMS Chelsea Publishing, 1998 | MR | Zbl

[25] E. D. Belokolos, V. Z. Enolskii, J. Math. Sci., 106:6 (2001), 3395–3486 ; 108:3 (2002), 295–374 | DOI | MR | Zbl | DOI | MR | Zbl

[26] L. Gavrilov, A. M. Perelomov, J. Math. Phys., 40 (1999), 6339–6352 | DOI | MR | Zbl

[27] I. G. Makdonald, Simmetricheskie funktsii i mnogochleny Kholla, Mir, M., 1984 | MR

[28] N. Seiberg, E. Witten, Nucl. Phys. B, 426:2 (1994), 19–52 ; Erratum, 430, 485–486 ; ; 431, no. 3, 1994 ; E-print hep-th/9407087E-print hep-th/9407087 | DOI | MR | Zbl | MR | Zbl | DOI | MR | Zbl

[29] A. Gorskii, I. Krichever, A. Marshakov, A. Mironov, A. Morosov, Phys. Lett. B, 355 (1995), 466–474 | DOI | MR

[30] P. Griffitth, J. Harris, Principles of {A}lgebraic {G}eometry, Wiley, N.Y., 1978 | MR