Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schrödinger operators

Abstract

We prove the complete asymptotic expansion of the integrated density of states of a Schrödinger operator $H=-\Delta+b$ acting in $\Bbb{R}^d$ when the potential $b$ is either smooth periodic, or generic quasi-periodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions.

  • [BarPar] Go to document G. Barbatis and L. Parnovski, "Bethe-Sommerfeld conjecture for pseudodifferential perturbation," Comm. Partial Differential Equations, vol. 34, iss. 4-6, pp. 383-418, 2009.
    @article {BarPar, MRKEY = {2530702},
      AUTHOR = {Barbatis, G. and Parnovski, L.},
      TITLE = {Bethe-{S}ommerfeld conjecture for pseudodifferential perturbation},
      JOURNAL = {Comm. Partial Differential Equations},
      FJOURNAL = {Communications in Partial Differential Equations},
      VOLUME = {34},
      YEAR = {2009},
      NUMBER = {4-6},
      PAGES = {383--418},
      ISSN = {0360-5302},
      CODEN = {CPDIDZ},
      MRCLASS = {35P20 (35B10 35J10 35S05 47G30 58J40)},
      MRNUMBER = {2530702},
      MRREVIEWER = {Viorel Iftimie},
      DOI = {10.1080/03605300902769006},
      ZBLNUMBER = {1185.35349},
      }
  • [HelMoh] Go to document B. Helffer and A. Mohamed, "Asymptotic of the density of states for the Schrödinger operator with periodic electric potential," Duke Math. J., vol. 92, iss. 1, pp. 1-60, 1998.
    @article {HelMoh, MRKEY = {1609321},
      AUTHOR = {Helffer, Bernard and Mohamed, Abderemane},
      TITLE = {Asymptotic of the density of states for the {S}chrödinger operator with periodic electric potential},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {92},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {1--60},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {35P20 (35J10 47F05 47N50 81Q10)},
      MRNUMBER = {1609321},
      MRREVIEWER = {Timo Weidl},
      DOI = {10.1215/S0012-7094-98-09201-8},
      ZBLNUMBER = {0951.35104},
     }
  • [HitPol] Go to document M. Hitrik and I. Polterovich, "Regularized traces and Taylor expansions for the heat semigroup," J. London Math. Soc., vol. 68, iss. 2, pp. 402-418, 2003.
    @article {HitPol, MRKEY = {1994690},
      AUTHOR = {Hitrik, Michael and Polterovich, Iosif},
      TITLE = {Regularized traces and {T}aylor expansions for the heat semigroup},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {68},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {402--418},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {35J10 (35P25 58J35 58J37)},
      MRNUMBER = {1994690},
      DOI = {10.1112/S0024610703004538},
      ZBLNUMBER = {1167.35335},
      }
  • [HitPol1] M. Hitrik and I. Polterovich, "Resolvent expansions and trace regularizations for Schrödinger operators," in Advances in Differential Equations and Mathematical Physics, Providence, RI: Amer. Math. Soc., 2003, vol. 327, pp. 161-173.
    @incollection {HitPol1, MRKEY = {1991539},
      AUTHOR = {Hitrik, Michael and Polterovich, Iosif},
      TITLE = {Resolvent expansions and trace regularizations for {S}chrödinger operators},
      BOOKTITLE = {Advances in Differential Equations and Mathematical Physics} , VENUE={{B}irmingham, {AL},
      2002},
      SERIES = {Contemp. Math.},
      VOLUME = {327},
      PAGES = {161--173},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2003},
      MRCLASS = {35J10 (35P25 58J37)},
      MRNUMBER = {1991539},
      MRREVIEWER = {Viorel Iftimie},
      ZBLNUMBER = {1109.35030},
     }
  • [Kar] Go to document Y. E. Karpeshina, "On the density of states for the periodic Schrödinger operator," Ark. Mat., vol. 38, iss. 1, pp. 111-137, 2000.
    @article {Kar, MRKEY = {1749362},
      AUTHOR = {Karpeshina, Yulia E.},
      TITLE = {On the density of states for the periodic {S}chrödinger operator},
      JOURNAL = {Ark. Mat.},
      FJOURNAL = {Arkiv för Matematik},
      VOLUME = {38},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {111--137},
      ISSN = {0004-2080},
      CODEN = {AKMTAJ},
      MRCLASS = {47F05 (35J10 35P20 47N20 81Q10)},
      MRNUMBER = {1749362},
      MRREVIEWER = {Rainer Hempel},
      DOI = {10.1007/BF02384494},
      ZBLNUMBER = {1021.35027},
     }
  • [Kar1] Y. E. Karpeshina, Perturbation theory for the Schrödinger operator with a periodic potential, New York: Springer-Verlag, 1997, vol. 1663.
    @book {Kar1, MRKEY = {1472485},
      AUTHOR = {Karpeshina, Yulia E.},
      TITLE = {Perturbation theory for the {S}chrödinger operator with a periodic potential},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1663},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1997},
      PAGES = {viii+352},
      ISBN = {3-540-63136-4},
      MRCLASS = {35-02 (35J10 35P05 47A55 47N50 81Q10 81Q15)},
      MRNUMBER = {1472485},
      MRREVIEWER = {Rainer Hempel},
      ZBLNUMBER = {0883.35002},
     }
  • [Kat] T. Kato, Perturbation Theory for Linear Operators, Second ed., New York: Springer-Verlag, 1976, vol. 132.
    @book {Kat, MRKEY = {0407617},
      AUTHOR = {Kato, Tosio},
      TITLE = {Perturbation Theory for Linear Operators},
      EDITION = {Second},
      SERIES = {Grundlehren Math. Wiss.},
      VOLUME={132},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1976},
      PAGES = {xxi+619},
      MRCLASS = {47-XX},
      MRNUMBER = {0407617},
      ZBLNUMBER = {0342.47009},
      }
  • [KorPush] Go to document E. Korotyaev and A. Pushnitski, "On the high-energy asymptotics of the integrated density of states," Bull. London Math. Soc., vol. 35, iss. 6, pp. 770-776, 2003.
    @article {KorPush, MRKEY = {2000023},
      AUTHOR = {Korotyaev, E. and Pushnitski, A.},
      TITLE = {On the high-energy asymptotics of the integrated density of states},
      JOURNAL = {Bull. London Math. Soc.},
      FJOURNAL = {The Bulletin of the London Mathematical Society},
      VOLUME = {35},
      YEAR = {2003},
      NUMBER = {6},
      PAGES = {770--776},
      ISSN = {0024-6093},
      CODEN = {LMSBBT},
      MRCLASS = {35J10 (35P20 47F05 47N50 81Q10)},
      MRNUMBER = {2000023},
      MRREVIEWER = {B. Hellwig},
      DOI = {10.1112/S0024609303002467},
      ZBLNUMBER = {1075.35542},
      }
  • [Nai] M. A. Nauimark, Normed Rings, Groningen: P. Noordhoff N. V., 1959.
    @book {Nai, MRKEY = {0110956},
      AUTHOR = {Na{\u\i}mark, M. A.},
      TITLE = {Normed Rings},
      PUBLISHER = {P. Noordhoff N. V.},
      ADDRESS = {Groningen},
      YEAR = {1959},
      PAGES = {xvi+560},
      MRCLASS = {46.00},
      MRNUMBER = {0110956},
      ZBLNUMBER = {0089.10102},
      }
  • [Par] Go to document L. Parnovski, "Bethe-Sommerfeld conjecture," Ann. Henri Poincaré, vol. 9, iss. 3, pp. 457-508, 2008.
    @article {Par, MRKEY = {2419769},
      AUTHOR = {Parnovski, Leonid},
      TITLE = {Bethe-{S}ommerfeld conjecture},
      JOURNAL = {Ann. Henri Poincaré},
      FJOURNAL = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      VOLUME = {9},
      YEAR = {2008},
      NUMBER = {3},
      PAGES = {457--508},
      ISSN = {1424-0637},
      MRCLASS = {81Q10 (35J10 47F05 47N50)},
      MRNUMBER = {2419769},
      MRREVIEWER = {Alberto Parmeggiani},
      DOI = {10.1007/s00023-008-0364-x},
      ZBLNUMBER = {1201.81054},
      }
  • [ParSht] Go to document L. Parnovski and R. Shterenberg, "Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schrödinger operator," Invent. Math., vol. 176, iss. 2, pp. 275-323, 2009.
    @article {ParSht, MRKEY = {2495765},
      AUTHOR = {Parnovski, Leonid and Shterenberg, Roman},
      TITLE = {Asymptotic expansion of the integrated density of states of a two-dimensional periodic {S}chrödinger operator},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {176},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {275--323},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {35J10 (35B10 35P20 47N50 81Q10)},
      MRNUMBER = {2495765},
      MRREVIEWER = {Viorel Iftimie},
      DOI = {10.1007/s00222-008-0164-4},
      ZBLNUMBER = {1171.35092},
     }
  • [ParSob] Go to document L. Parnovski and A. V. Sobolev, "Bethe-Sommerfeld conjecture for periodic operators with strong perturbations," Invent. Math., vol. 181, iss. 3, pp. 467-540, 2010.
    @article {ParSob, MRKEY = {2660451},
      AUTHOR = {Parnovski, Leonid and Sobolev, Alexander V.},
      TITLE = {Bethe-{S}ommerfeld conjecture for periodic operators with strong perturbations},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {181},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {467--540},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {35P20 (47A55 47G30 81Q10)},
      MRNUMBER = {2660451},
      DOI = {10.1007/s00222-010-0251-1},
      ZBLNUMBER = {1200.47067},
      }
  • [ReeSim] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV. Analysis of Operators, New York: Academic Press, 1978.
    @book {ReeSim, MRKEY = {0493422},
      AUTHOR = {Reed, M. and Simon, B.},
      TITLE = {Methods of Modern Mathematical Physics IV. Analysis of Operators},
      PUBLISHER = {Academic Press},
      ADDRESS={New York},
      YEAR = {1978},
      MRNUMBER = {0493422}
    }
  • [Sav] A. V. Savin, Asymptotic expansion of the density of states for one-dimensional Schrödinger and Dirac operators with almost periodic and random potentials, 1988.
    @misc{Sav,
      author={Savin, A. V.},
      TITLE={Asymptotic expansion of the density of states for one-dimensional {S}chrödinger and {D}irac operators with almost periodic and random potentials},
      NOTE={Sb. Nauchn. Tr., I.F.T.P., Moscow (Russian)},
      YEAR={1988},
     }
  • [Shu0] Go to document M. A. Shubin, "Almost periodic functions and partial differential operators," Uspehi Mat. Nauk, vol. 33, pp. 3-47, 1978.
    @article {Shu0, MRKEY = {0636410},
      AUTHOR = {Shubin, M. A.},
      TITLE = {Almost periodic functions and partial differential operators},
      JOURNAL = {Uspehi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      VOLUME = {33},
      YEAR = {1978},
      NOTE = {translated in {\it Russian Math. Surveys} {\bf 33} (1978), 1--52},
      PAGES = {3--47},
      ISSN = {0042-1316},
      MRCLASS = {47G05 (35S05 46L10 58G10)},
      MRNUMBER = {0636410},
      MRREVIEWER = {V. S. Rabinovic},
      ZBLNUMBER = {0408.47039},
      DOI = {10.1070/RM1978v033n02ABEH002303},
     }
  • [Shu] Go to document M. A. Shubin, "Spectral theory and the index of elliptic operators with almost-periodic coefficients," Uspekhi Mat. Nauk, vol. 34, pp. 95-135, 1979.
    @article {Shu, MRKEY = {0535710},
      AUTHOR = {Shubin, M. A.},
      TITLE = {Spectral theory and the index of elliptic operators with almost-periodic coefficients},
      JOURNAL = {Uspekhi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      VOLUME = {34},
      YEAR = {1979},
      NOTE = {translated in {\it Russian Math. Surveys} \textbf{34} (1979), 109--157},
      PAGES = {95--135},
      ISSN = {0042-1316},
      MRCLASS = {35P05 (34B30 58G12 58G25)},
      MRNUMBER = {0535710},
      MRREVIEWER = {P. A. Mishnaevski{\u\i}},
      DOI = {10.1070/RM1979v034n02ABEH002908},
      ZBLNUMBER = {0448.47032},
     }
  • [ShuSche] Go to document D. Shenk and M. Shubin, "Asymptotic expansion of the state density and the spectral function of a Hill operator," Math. USSR Sbornik, vol. 56, pp. 473-490, 1987.
    @article{ShuSche,
      author={Shenk, D. and Shubin, M.},
      TITLE= {Asymptotic expansion of the state density and the spectral function of a {H}ill operator},
      JOURNAL={Math. USSR Sbornik},
      VOLUME={56},
      YEAR={1987},
      PAGES={473--490},
      DOI = {10.1070/SM1987v056n02ABEH003047},
      ZBLNUMBER = {0624.34018},
     }
  • [Skr] M. Skriganov, "Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators," Proc. Steklov Math. Inst., vol. 171, 1984.
    @article{Skr,
      author={Skriganov, M.},
      TITLE={Geometrical and arithmetical methods in the spectral theory of the multi-dimensional periodic operators},
      JOURNAL={Proc. Steklov Math. Inst.},
      VOLUME={171},
      YEAR={1984},
     }
  • [Sob] Go to document A. V. Sobolev, "Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one," Rev. Mat. Iberoam., vol. 22, iss. 1, pp. 55-92, 2006.
    @article {Sob, MRKEY = {2267313},
      AUTHOR = {Sobolev, Alexander V.},
      TITLE = {Asymptotics of the integrated density of states for periodic elliptic pseudo-differential operators in dimension one},
      JOURNAL = {Rev. Mat. Iberoam.},
      FJOURNAL = {Revista Mathemática Iberoamericana},
      VOLUME = {22},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {55--92},
      ISSN = {0213-2230},
      MRCLASS = {35S05 (35P20 47A55 47G10)},
      MRNUMBER = {2267313},
      MRREVIEWER = {Yu. V. Egorov},
      DOI = {10.4171/RMI/449},
      ZBLNUMBER = {1121.35149},
      }
  • [Sob1] Go to document A. V. Sobolev, "Integrated density of states for the periodic Schrödinger operator in dimension two," Ann. Henri Poincaré, vol. 6, iss. 1, pp. 31-84, 2005.
    @article {Sob1, MRKEY = {2119355},
      AUTHOR = {Sobolev, Alexander V.},
      TITLE = {Integrated density of states for the periodic {S}chrödinger operator in dimension two},
      JOURNAL = {Ann. Henri Poincaré},
      FJOURNAL = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      VOLUME = {6},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {31--84},
      ISSN = {1424-0637},
      MRCLASS = {35J10 (35B10 35P20 47N50 81Q10)},
      MRNUMBER = {2119355},
      MRREVIEWER = {M. Znojil},
      DOI = {10.1007/s00023-005-0198-8},
      ZBLNUMBER = {1065.81051},
     }
  • [Vel] Go to document O. A. Veliev, "The spectrum of multidimensional periodic operators," Teor. Funkts., Funkts. Anal. i Prilozhen., vol. 49, pp. 17-34, 1988.
    @article {Vel, MRKEY = {0963619},
      AUTHOR = {Veliev, O. A.},
      TITLE = {The spectrum of multidimensional periodic operators},
      JOURNAL = {Teor. Funkts., Funkts. Anal. i Prilozhen.},
      FJOURNAL = {Khar$'$kovskiĭOrdena Trudovogo Krasnogo Znameni GosudarstvennyĭUniversitet imeni A. M. Gor$'$kogo. Teoriya Funktsi\u\i, Funktsional$'$nyĭAnaliz i ikh Prilozheniya},
      YEAR = {1988},
      VOLUME = {49},
      PAGES = {17--34},
      ISSN = {0321-4427},
      MRCLASS = {35P20 (47F05 81C10)},
      MRNUMBER = {0963619},
      DOI = {10.1007/BF02216095},
      ZBLNUMBER = {0664.47005},
      NOTE={translated in \emph{J. Math. Sci.} \textbf{4} (1990), 1045--1058},
     }

Authors

Leonid Parnovski

Department of Mathematics
University College London
Gower Street
London WC1E 6BT
United Kingdom

Roman Shterenberg

Department of Mathematics
University of Alabama at Birmingham
1300 University Boulevard
Birmingham AL 35294-1170