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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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} ÊÊÊ(1990), 1045--1058},
}