The absolutely continuous spectrum of Jacobi matrices

Pages 125-171 from Volume 174 (2011), Issue 1 by Christian Remling

Abstract

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schrödinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case.
The main theme is the following: It is extremely difficult to produce absolutely continuous spectrum in one space dimension and thus its existence has strong implications.

  • [ArD1] Go to document N. Aronszajn and W. F. Donoghue, "On exponential representations of analytic functions in the upper half-plane with positive imaginary part," J. Analyse Math., vol. 5, pp. 321-388, 1956/1957.
    @article{ArD1,
      author={Aronszajn, N. and Donoghue, W. F.},
      TITLE = {On exponential representations of analytic functions in the upper half-plane with positive imaginary part},
      JOURNAL={J. Analyse Math.},
      VOLUME={5},
      YEAR={1956/1957},
      PAGES={321--388},
      ZBLNUMBER = {0138.29502},
      DOI={10.1007/BF02937349},
      }
  • [ArD2] Go to document N. Aronszajn and W. F. Donoghue, "A supplement to the paper on exponential representations of analytic functions in the upper half-plane with positive imaginary part," J. Analyse Math., vol. 12, pp. 113-127, 1964.
    @article {ArD2, MRKEY = {0168769},
      AUTHOR = {Aronszajn, N. and Donoghue, W. F.},
      TITLE = {A supplement to the paper on exponential representations of analytic functions in the upper half-plane with positive imaginary part},
      JOURNAL = {J. Analyse Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {12},
      YEAR = {1964},
      PAGES = {113--127},
      ISSN = {0021-7670},
      MRCLASS = {30.65},
      MRNUMBER = {0168769},
      MRREVIEWER = {P. L. Duren},
      ZBLNUMBER = {0138.29601},
      DOI = {10.1007/BF02807431},
      }
  • [BP1] Go to document S. V. Breimesser and D. B. Pearson, "Asymptotic value distribution for solutions of the Schrödinger equation," Math. Phys. Anal. Geom., vol. 3, iss. 4, pp. 385-403 (2001), 2000.
    @article {BP1, MRKEY = {1845358},
      AUTHOR = {Breimesser, S. V. and Pearson, D. B.},
      TITLE = {Asymptotic value distribution for solutions of the {S}chrödinger equation},
      JOURNAL = {Math. Phys. Anal. Geom.},
      FJOURNAL = {Mathematical Physics, Analysis and Geometry. An International Journal Devoted to the Theory and Applications of Analysis and Geometry to Physics},
      VOLUME = {3},
      YEAR = {2000},
      NUMBER = {4},
      PAGES = {385--403 (2001)},
      ISSN = {1385-0172},
      CODEN = {MPAGFO},
      MRCLASS = {34B20 (34L40 47E05 81Q10)},
      MRNUMBER = {1845358},
      MRREVIEWER = {Christian Remling},
      ZBLNUMBER = {1016.47033},
      DOI = {10.1023/A:1011420706256},
      }
  • [BP2] Go to document S. V. Breimesser and D. B. Pearson, "Geometrical aspects of spectral theory and value distribution for Herglotz functions," Math. Phys. Anal. Geom., vol. 6, iss. 1, pp. 29-57, 2003.
    @article {BP2, MRKEY = {1962701},
      AUTHOR = {Breimesser, S. V. and Pearson, D. B.},
      TITLE = {Geometrical aspects of spectral theory and value distribution for {H}erglotz functions},
      JOURNAL = {Math. Phys. Anal. Geom.},
      FJOURNAL = {Mathematical Physics, Analysis and Geometry. An International Journal Devoted to the Theory and Applications of Analysis and Geometry to Physics},
      VOLUME = {6},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {29--57},
      ISSN = {1385-0172},
      CODEN = {MPAGFO},
      MRCLASS = {47E05 (34L05 35P20 81Q10)},
      MRNUMBER = {1962701},
      MRREVIEWER = {Horst Behncke},
      DOI = {10.1023/A:1022410108020},
      ZBLNUMBER = {1027.47038},
      }
  • [BGHT] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, "Algebro-geometric quasi-periodic finite-gap solutions of the Toda and Kac-van Moerbeke hierarchies," Mem. Amer. Math. Soc., vol. 135, iss. 641, p. x, 1998.
    @article {BGHT, MRKEY = {1432141},
      AUTHOR = {Bulla, W. and Gesztesy, F. and Holden, H. and Teschl, G.},
      TITLE = {Algebro-geometric quasi-periodic finite-gap solutions of the {T}oda and {K}ac-van {M}oerbeke hierarchies},
      JOURNAL = {Mem. Amer. Math. Soc.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {135},
      YEAR = {1998},
      NUMBER = {641},
      PAGES = {x+79},
      ISSN = {0065-9266},
      CODEN = {MAMCAU},
      MRCLASS = {58F07 (35Q55 39A12)},
      MRNUMBER = {1432141},
      MRREVIEWER = {Mihai Pascu},
      ZBLNUMBER = {0906.35099},
      }
  • [Craig] Go to document W. Craig, "The trace formula for Schrödinger operators on the line," Comm. Math. Phys., vol. 126, iss. 2, pp. 379-407, 1989.
    @article {Craig, MRKEY = {1027503},
      AUTHOR = {Craig, Walter},
      TITLE = {The trace formula for {S}chrödinger operators on the line},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {126},
      YEAR = {1989},
      NUMBER = {2},
      PAGES = {379--407},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {47E05 (34B25 47A10 58F19 82A42)},
      MRNUMBER = {1027503},
      MRREVIEWER = {Russell A. Johnson},
      ZBLNUMBER = {0681.34026},
      DOI= {10.1007/BF02125131},
      }
  • [DKS] Go to document D. Damanik, R. Killip, and B. Simon, "Perturbations of orthogonal polynomials with periodic recursion coefficients," Ann. of Math., vol. 171, iss. 3, pp. 1931-2010, 2010.
    @article {DKS, MRKEY = {2680401},
      AUTHOR = {Damanik, David and Killip, Rowan and Simon, Barry},
      TITLE = {Perturbations of orthogonal polynomials with periodic recursion coefficients},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {171},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {1931--2010},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {47B36 (42C10 47A10)},
      MRNUMBER = {2680401},
      MRREVIEWER = {Leonid B. Golinski{\u\i}},
      DOI = {10.4007/annals.2010.171.1931},
      ZBLNUMBER = {1194.47031},
      }
  • [dRMS] Go to document R. Del Rio, N. Makarov, and B. Simon, "Operators with singular continuous spectrum. II. Rank one operators," Comm. Math. Phys., vol. 165, iss. 1, pp. 59-67, 1994.
    @article {dRMS, MRKEY = {1298942},
      AUTHOR = {Del Rio, R. and Makarov, N. and Simon, B.},
      TITLE = {Operators with singular continuous spectrum. {II}. {R}ank one operators},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {165},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {59--67},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {47-02 (34L05 47A55 47B25 81Q10)},
      MRNUMBER = {1298942},
      MRREVIEWER = {Horst Behncke},
      ZBLNUMBER = {1055.47500},
      DOI = {10.1007/BF02099737},
      }
  • [Den] Go to document S. A. Denisov, "On Rakhmanov’s theorem for Jacobi matrices," Proc. Amer. Math. Soc., vol. 132, iss. 3, pp. 847-852, 2004.
    @article {Den, MRKEY = {2019964},
      AUTHOR = {Denisov, Sergey A.},
      TITLE = {On {R}akhmanov's theorem for {J}acobi matrices},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {132},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {847--852},
      ISSN = {0002-9939},
      CODEN = {PAMYAR},
      MRCLASS = {47B36 (39A70 42C05)},
      MRNUMBER = {2019964},
      DOI = {10.1090/S0002-9939-03-07157-0},
      ZBLNUMBER = {1050.47024},
      }
  • [Dom] Go to document J. Dombrowski, "Quasitriangular matrices," Proc. Amer. Math. Soc., vol. 69, iss. 1, pp. 95-96, 1978.
    @article {Dom, MRKEY = {0467373},
      AUTHOR = {Dombrowski, J.},
      TITLE = {Quasitriangular matrices},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {69},
      YEAR = {1978},
      NUMBER = {1},
      PAGES = {95--96},
      ISSN = {0002-9939},
      MRCLASS = {47B15 (47A65)},
      MRNUMBER = {0467373},
      MRREVIEWER = {P. R. Halmos},
      DOI = {10.2307/2043198},
      ZBLNUMBER = {0379.47013},
      }
  • [Dub] Go to document B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, "Non-linear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties," Uspekhi Mat. Nauk, vol. 31, iss. 1, pp. 55-136, 1976.
    @article {Dub, MRKEY = {0427869},
      AUTHOR = {Dubrovin, B. A. and Matveev, V. B. and Novikov, S. P.},
      TITLE = {Non-linear equations of {K}orteweg-de {V}ries type, finite-band linear operators and {A}belian varieties},
      JOURNAL = {Uspekhi Mat. Nauk},
      FJOURNAL = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk},
      VOLUME = {31},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {55--136},
      ISSN = {0042-1316},
      MRCLASS = {35Q99 (34B25 35P25 58F05)},
      MRNUMBER = {0427869},
      MRREVIEWER = {Shunichi Tanaka},
      ZBLNUMBER = {1110.32008},
      URL = {http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=3642 &option_lang=eng},
     }
  • [FHS] R. Froese, D. Hasler, and W. Spitzer, "Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs," J. Funct. Anal., vol. 230, iss. 1, pp. 184-221, 2006.
    @article {FHS, MRKEY = {2184188},
      AUTHOR = {Froese, Richard and Hasler, David and Spitzer, Wolfgang},
      TITLE = {Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete {S}chrödinger operators on graphs},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {230},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {184--221},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {81Q10 (47B39 47N50 82B20)},
      MRNUMBER = {2184188},
      MRREVIEWER = {H. Hogreve},
      ZBLNUMBER = {1094.35104},
     }
  • [GeoI] Go to document V. Georgescu and A. Iftimovici, "Localizations at infinity and essential spectrum of quantum Hamiltonians. I. General theory," Rev. Math. Phys., vol. 18, iss. 4, pp. 417-483, 2006.
    @article {GeoI, MRKEY = {2245367},
      AUTHOR = {Georgescu, Vladimir and Iftimovici, Andrei},
      TITLE = {Localizations at infinity and essential spectrum of quantum {H}amiltonians. {I}. {G}eneral theory},
      JOURNAL = {Rev. Math. Phys.},
      FJOURNAL = {Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics},
      VOLUME = {18},
      YEAR = {2006},
      NUMBER = {4},
      PAGES = {417--483},
      ISSN = {0129-055X},
      MRCLASS = {81Q10 (46L60 47A10 47B25 47F05 47N50 81R15)},
      MRNUMBER = {2245367},
      MRREVIEWER = {Serge C. Richard},
      DOI = {10.1142/S0129055X06002693},
      ZBLNUMBER = {1109.47004},
      }
  • [GerC] Go to document J. S. Geronimo and K. M. Case, "Scattering theory and polynomials orthogonal on the real line," Trans. Amer. Math. Soc., vol. 258, iss. 2, pp. 467-494, 1980.
    @article {GerC, MRKEY = {0558185},
      AUTHOR = {Geronimo, J. S. and Case, K. M.},
      TITLE = {Scattering theory and polynomials orthogonal on the real line},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {258},
      YEAR = {1980},
      NUMBER = {2},
      PAGES = {467--494},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {81F99 (30C10 42C05)},
      MRNUMBER = {0558185},
      MRREVIEWER = {C. L. Parihar},
      DOI = {10.2307/1998068},
      ZBLNUMBER = {0436.42018},
      }
  • [GKT] F. Gesztesy, M. Krishna, and G. Teschl, "On isospectral sets of Jacobi operators," Comm. Math. Phys., vol. 181, iss. 3, pp. 631-645, 1996.
    @article {GKT, MRKEY = {1414303},
      AUTHOR = {Gesztesy, F. and Krishna, M. and Teschl, G.},
      TITLE = {On isospectral sets of {J}acobi operators},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {181},
      YEAR = {1996},
      NUMBER = {3},
      PAGES = {631--645},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {47B37 (47B25)},
      MRNUMBER = {1414303},
      MRREVIEWER = {Mahmoud M. Kutkut},
      ZBLNUMBER = {0881.58069},
      }
  • [GYud] Go to document F. Gesztesy and P. Yuditskii, "Spectral properties of a class of reflectionless Schrödinger operators," J. Funct. Anal., vol. 241, iss. 2, pp. 486-527, 2006.
    @article {GYud, MRKEY = {2271928},
      AUTHOR = {Gesztesy, Fritz and Yuditskii, Peter},
      TITLE = {Spectral properties of a class of reflectionless {S}chrödinger operators},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {241},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {486--527},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {34L40 (47A10 47E05)},
      MRNUMBER = {2271928},
      MRREVIEWER = {Plamen Dzhakov},
      DOI = {10.1016/j.jfa.2006.08.006},
      ZBLNUMBER = {pre05116346},
     }
  • [Gor] Go to document Y. A. Gordon, "Pure point spectrum under $1$-parameter perturbations and instability of Anderson localization," Comm. Math. Phys., vol. 164, iss. 3, pp. 489-505, 1994.
    @article {Gor, MRKEY = {1291242},
      AUTHOR = {Gordon, A. Ya.},
      TITLE = {Pure point spectrum under {$1$}-parameter perturbations and instability of {A}nderson localization},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {164},
      YEAR = {1994},
      NUMBER = {3},
      PAGES = {489--505},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {47A55 (47N55)},
      MRNUMBER = {1291242},
      MRREVIEWER = {Russell A. Johnson},
      ZBLNUMBER = {0839.47002},
      DOI = {10.1007/BF02101488}
    }
  • [HKnSim] Go to document A. Hof, O. Knill, and B. Simon, "Singular continuous spectrum for palindromic Schrödinger operators," Comm. Math. Phys., vol. 174, iss. 1, pp. 149-159, 1995.
    @article {HKnSim, MRKEY = {1372804},
      AUTHOR = {Hof, A. and Knill, O. and Simon, B.},
      TITLE = {Singular continuous spectrum for palindromic {S}chrödinger operators},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {174},
      YEAR = {1995},
      NUMBER = {1},
      PAGES = {149--159},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {47B39 (47N50 81Q10)},
      MRNUMBER = {1372804},
      MRREVIEWER = {Fr{é}d{é}ric Klopp},
      ZBLNUMBER = {0839.11009},
      DOI={10.1007/BF02099468},
      DOI = {10.1007/BF02099468},
      }
  • [JitSim] Go to document S. Jitomirskaya and B. Simon, "Operators with singular continuous spectrum. III. Almost periodic Schrödinger operators," Comm. Math. Phys., vol. 165, iss. 1, pp. 201-205, 1994.
    @article {JitSim, MRKEY = {1298948},
      AUTHOR = {Jitomirskaya, S. and Simon, B.},
      TITLE = {Operators with singular continuous spectrum. {III}. {A}lmost periodic {S}chrödinger operators},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {165},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {201--205},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {47-02 (34L05 47E05 81Q10)},
      MRNUMBER = {1298948},
      MRREVIEWER = {Horst Behncke},
      ZBLNUMBER = {0830.34074},
      DOI={10.1007/BF02099743},
      }
  • [KLS] Go to document A. Kiselev, Y. Last, and B. Simon, "Modified Prüfer and EFGP transforms and the spectral analysis of one-dimensional Schrödinger operators," Comm. Math. Phys., vol. 194, iss. 1, pp. 1-45, 1998.
    @article {KLS, MRKEY = {1628290},
      AUTHOR = {Kiselev, Alexander and Last, Yoram and Simon, Barry},
      TITLE = {Modified {P}rüfer and {EFGP} transforms and the spectral analysis of one-dimensional {S}chrödinger operators},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {194},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {1--45},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {34L40 (39A70 47B39 47B80 47E05 81Q10)},
      MRNUMBER = {1628290},
      MRREVIEWER = {G{ü}nter Stolz},
      ZBLNUMBER = {0912.34074},
      DOI = {10.1007/s002200050346},
      }
  • [Kotac] Go to document S. Kotani, "Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators," in Stochastic Analysis, Amsterdam, 1984, pp. 225-247.
    @inproceedings {Kotac, MRKEY = {0780760},
      AUTHOR = {Kotani, Shinichi},
      TITLE = {Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional {S}chrödinger operators},
      BOOKTITLE = {Stochastic Analysis},
      VENUE={{K}atata/{K}yoto, 1982},
      SERIES = {North-Holland Math. Library},
      VOLUME = {32},
      PAGES = {225--247},
      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
      YEAR = {1984},
      MRCLASS = {60H25 (35P05 81C20 82A57)},
      MRNUMBER = {0780760},
      MRREVIEWER = {W. Kirsch},
      ZBLNUMBER = {0549.60058},
      DOI={10.1016/S0924-6509(08)70395-7},
      }
  • [Kot] S. Kotani, "One-dimensional random Schrödinger operators and Herglotz functions," in Probabilistic Methods in Mathematical Physics, Boston, MA, 1987, pp. 219-250.
    @inproceedings {Kot, MRKEY = {0933826},
      AUTHOR = {Kotani, Shinichi},
      TITLE = {One-dimensional random {S}chrödinger operators and {H}erglotz functions},
      BOOKTITLE = {Probabilistic Methods in Mathematical Physics},
      VENUE={{K}atata/{K}yoto, 1985},
      PAGES = {219--250},
      PUBLISHER = {Academic Press},
      ADDRESS = {Boston, MA},
      YEAR = {1987},
      MRCLASS = {60H25 (34B99 47E05 82A42)},
      MRNUMBER = {0933826},
      MRREVIEWER = {Helge Holden},
      ZBLNUMBER = {0646.60071},
     }
  • [Kotfin] Go to document S. Kotani, "Jacobi matrices with random potentials taking finitely many values," Rev. Math. Phys., vol. 1, iss. 1, pp. 129-133, 1989.
    @article {Kotfin, MRKEY = {1041533},
      AUTHOR = {Kotani, Shinichi},
      TITLE = {Jacobi matrices with random potentials taking finitely many values},
      JOURNAL = {Rev. Math. Phys.},
      FJOURNAL = {Reviews in Mathematical Physics. A Journal for Both Review and Original Research Papers in the Field of Mathematical Physics},
      VOLUME = {1},
      YEAR = {1989},
      NUMBER = {1},
      PAGES = {129--133},
      ISSN = {0129-055X},
      MRCLASS = {81Q10 (47B99 82B44)},
      MRNUMBER = {1041533},
      MRREVIEWER = {Dimitri Petritis},
      DOI = {10.1142/S0129055X89000067},
      ZBLNUMBER = {0713.60074},
      }
  • [Kot97] Go to document S. Kotani, "Generalized Floquet theory for stationary Schrödinger operators in one dimension," Chaos Solitons Fractals, vol. 8, iss. 11, pp. 1817-1854, 1997.
    @article {Kot97, MRKEY = {1477262},
      AUTHOR = {Kotani, Shinichi},
      TITLE = {Generalized {F}loquet theory for stationary {S}chrödinger operators in one dimension},
      JOURNAL = {Chaos Solitons Fractals},
      FJOURNAL = {Chaos, Solitons and Fractals},
      VOLUME = {8},
      YEAR = {1997},
      NUMBER = {11},
      PAGES = {1817--1854},
      ISSN = {0960-0779},
      CODEN = {CSFOEH},
      MRCLASS = {34L40 (34A55 34L05 47B80 60H25 82B44)},
      MRNUMBER = {1477262},
      MRREVIEWER = {Nariyuki Minami},
      ZBLNUMBER = {0936.34074},
      DOI = {10.1016/S0960-0779(97)00042-8},
      }
  • [Krantz] S. G. Krantz, Complex Analysis: The Geometric Viewpoint, Second ed., Washington, DC: Mathematical Association of America, 2004, vol. 23.
    @book {Krantz, MRKEY = {2047863},
      AUTHOR = {Krantz, Steven G.},
      TITLE = {Complex Analysis: The Geometric Viewpoint},
      SERIES = {Carus Math. Monogr.},
      VOLUME = {23},
      EDITION = {Second},
      PUBLISHER = {Mathematical Association of America},
      ADDRESS = {Washington, DC},
      YEAR = {2004},
      PAGES = {xviii+219},
      ISBN = {0-88385-035-4},
      MRCLASS = {30-01 (32-01)},
      MRNUMBER = {2047863},
      ZBLNUMBER = {1051.30001},
      }
  • [KrRem] Go to document T. Kriecherbauer and C. Remling, "Finite gap potentials and WKB asymptotics for one-dimensional Schrödinger operators," Comm. Math. Phys., vol. 223, iss. 2, pp. 409-435, 2001.
    @article {KrRem, MRKEY = {1864439},
      AUTHOR = {Kriecherbauer, Thomas and Remling, Christian},
      TITLE = {Finite gap potentials and {WKB} asymptotics for one-dimensional {S}chrödinger operators},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {223},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {409--435},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {81Q20 (34E20 34L40 81Q05 81Q10)},
      MRNUMBER = {1864439},
      MRREVIEWER = {Alexander A. Kiselev},
      ZBLNUMBER = {1017.81015},
      DOI = {10.1007/s002200100550},
      }
  • [KruRem] Go to document D. Krutikov and C. Remling, "Schrödinger operators with sparse potentials: asymptotics of the Fourier transform of the spectral measure," Comm. Math. Phys., vol. 223, iss. 3, pp. 509-532, 2001.
    @article {KruRem, MRKEY = {1866165},
      AUTHOR = {Krutikov, Denis and Remling, Christian},
      TITLE = {Schrödinger operators with sparse potentials: asymptotics of the {F}ourier transform of the spectral measure},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {223},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {509--532},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {81Q10 (47B39 47N50)},
      MRNUMBER = {1866165},
      MRREVIEWER = {Pavel B. Kurasov},
      DOI = {10.1007/s002200100552},
      ZBLNUMBER = {1161.81378},
     }
  • [Kur] P. Kurka, Topological and Symbolic Dynamics, Paris: Société Mathématique de France, 2003, vol. 11.
    @book {Kur, MRKEY = {2041676},
      AUTHOR = {Kurka, Petr},
      TITLE = {Topological and Symbolic Dynamics},
      SERIES = {Cours Spécialisés [Specialized Courses]},
      VOLUME = {11},
      PUBLISHER = {Société Mathématique de France},
      ADDRESS = {Paris},
      YEAR = {2003},
      PAGES = {xii+315},
      ISBN = {2-85629-143-0},
      MRCLASS = {37Bxx (54H20)},
      MRNUMBER = {2041676},
      MRREVIEWER = {Michael Hurley},
      ZBLNUMBER = {1038.37011},
     }
  • [LS] Go to document Y. Last and B. Simon, "Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators," Invent. Math., vol. 135, iss. 2, pp. 329-367, 1999.
    @article {LS, MRKEY = {1666767},
      AUTHOR = {Last, Yoram and Simon, Barry},
      TITLE = {Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional {S}chrödinger operators},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {135},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {329--367},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {47B39 (34L05 34L40 47E05 81Q10)},
      MRNUMBER = {1666767},
      MRREVIEWER = {G{ü}nter Stolz},
      DOI = {10.1007/s002220050288},
      ZBLNUMBER = {0931.34066},
     }
  • [LS2] Go to document Y. Last and B. Simon, "The essential spectrum of Schrödinger, Jacobi, and CMV operators," J. Anal. Math., vol. 98, pp. 183-220, 2006.
    @article {LS2, MRKEY = {2254485},
      AUTHOR = {Last, Yoram and Simon, Barry},
      TITLE = {The essential spectrum of {S}chrödinger, {J}acobi, and {CMV} operators},
      JOURNAL = {J. Anal. Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {98},
      YEAR = {2006},
      PAGES = {183--220},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {47A10 (34L40 35P05 47B36 47E05 47F05 81Q10)},
      MRNUMBER = {2254485},
      MRREVIEWER = {Jaouad Sahbani},
      DOI = {10.1007/BF02790275},
      ZBLNUMBER = {1145.34052},
     }
  • [McK] Go to document H. P. McKean, "Variation on a theme of Jacobi," Comm. Pure Appl. Math., vol. 38, iss. 5, pp. 669-678, 1985.
    @article {McK, MRKEY = {0803254},
      AUTHOR = {McKean, H. P.},
      TITLE = {Variation on a theme of {J}acobi},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {38},
      YEAR = {1985},
      NUMBER = {5},
      PAGES = {669--678},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {14K20 (14H40 32G20 34B25 58F07)},
      MRNUMBER = {0803254},
      MRREVIEWER = {H. H. Martens},
      DOI = {10.1002/cpa.3160380514},
      ZBLNUMBER = {0591.34025},
      }
  • [Mol] S. A. Molchanov, "Multiscale averaging for ordinary differential equations. Applications to the spectral theory of one-dimensional Schrödinger operator with sparse potentials," in Homogenization, World Sci. Publ., River Edge, NJ, 1999, vol. 50, pp. 316-397.
    @incollection {Mol, MRKEY = {1792693},
      AUTHOR = {Molchanov, S. A.},
      TITLE = {Multiscale averaging for ordinary differential equations. {A}pplications to the spectral theory of one-dimensional {S}chrödinger operator with sparse potentials},
      BOOKTITLE = {Homogenization},
      SERIES = {Ser. Adv. Math. Appl. Sci.},
      VOLUME = {50},
      PAGES = {316--397},
      PUBLISHER = {World Sci. Publ., River Edge, NJ},
      YEAR = {1999},
      MRCLASS = {34L40 (34E13 81Q10)},
      MRNUMBER = {1792693},
      MRREVIEWER = {Christian Remling},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {1051.30001},
      }
  • [Mum] D. Mumford, Tata Lectures on Theta. II: Jacobian theta functions and differential equations (with the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman, and H. Umemura), Boston, MA: Birkhäuser, 1984, vol. 43.
    @book {Mum, MRKEY = {0742776},
      AUTHOR = {Mumford, David},
      TITLE = {Tata Lectures on Theta. {\rm II}: {J}acobian theta functions and differential equations {\rm (with the collaboration of {C. M}usili, {M. N}ori, {E. P}reviato, {M. S}tillman, and {H. U}memura)}},
      SERIES = {Progr. Math.},
      VOLUME = {43},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1984},
      PAGES = {xiv+272},
      ISBN = {0-8176-3110-0},
      MRCLASS = {14K25 (14H40 32G20)},
      MRNUMBER = {0742776},
      MRREVIEWER = {M. Kh. Gizatullin},
      ZBLNUMBER = {0549.14014},
      }
  • [Psc] Go to document D. B. Pearson, "Singular continuous measures in scattering theory," Comm. Math. Phys., vol. 60, iss. 1, pp. 13-36, 1978.
    @article {Psc, MRKEY = {0484145},
      AUTHOR = {Pearson, D. B.},
      TITLE = {Singular continuous measures in scattering theory},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {60},
      YEAR = {1978},
      NUMBER = {1},
      PAGES = {13--36},
      ISSN = {0010-3616},
      MRCLASS = {81.47},
      MRNUMBER = {0484145},
      MRREVIEWER = {H. Wakita},
      ZBLNUMBER = {0451.47013},
      DOI = {10.1007/BF01609472},
     }
  • [P] Go to document D. B. Pearson, "Value distribution and spectral analysis of differential operators," J. Phys. A, vol. 26, iss. 16, pp. 4067-4080, 1993.
    @article {P, MRKEY = {1236597},
      AUTHOR = {Pearson, D. B.},
      TITLE = {Value distribution and spectral analysis of differential operators},
      JOURNAL = {J. Phys. A},
      FJOURNAL = {Journal of Physics. A. Mathematical and General},
      VOLUME = {26},
      YEAR = {1993},
      NUMBER = {16},
      PAGES = {4067--4080},
      ISSN = {0305-4470},
      CODEN = {JPHAC5},
      MRCLASS = {34L05 (34B20 34L40 47E05)},
      MRNUMBER = {1236597},
      MRREVIEWER = {J. K. Shaw},
      ZBLNUMBER = {0795.34071},
      DOI = {10.1088/0305-4470/26/16/022},
     }
  • [Pearvd] Go to document D. B. Pearson, "Value distribution and spectral theory," Proc. London Math. Soc., vol. 68, iss. 1, pp. 127-144, 1994.
    @article {Pearvd, MRKEY = {1243838},
      AUTHOR = {Pearson, D. B.},
      TITLE = {Value distribution and spectral theory},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {68},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {127--144},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {47A11 (34L05 47E05)},
      MRNUMBER = {1243838},
      MRREVIEWER = {Lawrence E. Thomas},
      DOI = {10.1112/plms/s3-68.1.127},
      ZBLNUMBER = {0809.34037},
      }
  • [Rakh] E. A. Rakhmanov, "The asymptotic behavior of the ratio of orthogonal polynomials. II," Mat. Sb., vol. 118(160), iss. 1, pp. 104-117, 143, 1982.
    @article {Rakh, MRKEY = {0654647},
      AUTHOR = {Rakhmanov, E. A.},
      TITLE = {The asymptotic behavior of the ratio of orthogonal polynomials. {II}},
      JOURNAL = {Mat. Sb.},
      FJOURNAL = {MatematicheskiĭSbornik. Novaya Seriya},
      VOLUME = {118(160)},
      YEAR = {1982},
      NUMBER = {1},
      PAGES = {104--117, 143},
      ISSN = {0368-8666},
      MRCLASS = {42C05},
      MRNUMBER = {0654647},
      MRREVIEWER = {F. M{ó}ricz},
      ZBLNUMBER = {0509.30028},
     }
  • [Remsparse] Go to document C. Remling, "A probabilistic approach to one-dimensional Schrödinger operators with sparse potentials," Comm. Math. Phys., vol. 185, iss. 2, pp. 313-323, 1997.
    @article {Remsparse, MRKEY = {1463044},
      AUTHOR = {Remling, Christian},
      TITLE = {A probabilistic approach to one-dimensional {S}chrödinger operators with sparse potentials},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {185},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {313--323},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {34L40 (47E05 81Q10)},
      MRNUMBER = {1463044},
      MRREVIEWER = {G{ü}nter Stolz},
      DOI = {10.1007/s002200050092},
      }
  • [Remac] Go to document C. Remling, "The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials," Comm. Math. Phys., vol. 193, iss. 1, pp. 151-170, 1998.
    @article {Remac, MRKEY = {1620313},
      AUTHOR = {Remling, Christian},
      TITLE = {The absolutely continuous spectrum of one-dimensional {S}chrödinger operators with decaying potentials},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {193},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {151--170},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {34L40 (47E05 81Q05)},
      MRNUMBER = {1620313},
      MRREVIEWER = {Alexander A. Kiselev},
      DOI = {10.1007/s002200050322},
      ZBLNUMBER = {0908.34067},
     }
  • [Remsc] Go to document C. Remling, "Embedded singular continuous spectrum for one-dimensional Schrödinger operators," Trans. Amer. Math. Soc., vol. 351, iss. 6, pp. 2479-2497, 1999.
    @article {Remsc, MRKEY = {1665336},
      AUTHOR = {Remling, Christian},
      TITLE = {Embedded singular continuous spectrum for one-dimensional {S}chrödinger operators},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {351},
      YEAR = {1999},
      NUMBER = {6},
      PAGES = {2479--2497},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {34L40 (34L05 47E05 81Q10)},
      MRNUMBER = {1665336},
      MRREVIEWER = {Alexander A. Kiselev},
      DOI = {10.1090/S0002-9947-99-02495-2},
      ZBLNUMBER = {0918.34074},
     }
  • [RemWeyl] Go to document C. Remling, Weyl theory, manuscript (not intended for publication).
    @misc{RemWeyl,
      author = {Remling, Christian},
      TITLE = {Weyl theory, manuscript (not intended for publication)},
      URL={www.math.ou.edu/~cremling},
     }
  • [Siegel] C. L. Siegel, Topics in Complex Function Theory, Vol. 2, New York: Wiley, 1971.
    @book{Siegel,
      author={Siegel, C. L.},
      TITLE={Topics in Complex Function Theory, Vol.~2},
      PUBLISHER={Wiley},
      ADDRESS={New York},
      YEAR={1971},
      MRNUMBER={1008931},
      ZBLNUMBER={0635.30003},
      }
  • [Simsc] Go to document B. Simon, "Operators with singular continuous spectrum. I. General operators," Ann. of Math., vol. 141, iss. 1, pp. 131-145, 1995.
    @article {Simsc, MRKEY = {1314033},
      AUTHOR = {Simon, Barry},
      TITLE = {Operators with singular continuous spectrum. {I}. {G}eneral operators},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {141},
      YEAR = {1995},
      NUMBER = {1},
      PAGES = {131--145},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {47B15 (47A10 47B25 47F05 81Q10)},
      MRNUMBER = {1314033},
      MRREVIEWER = {Horst Behncke},
      DOI = {10.2307/2118629},
      ZBLNUMBER = {0851.47003},
     }
  • [Simmom] Go to document B. Simon, "The classical moment problem as a self-adjoint finite difference operator," Adv. Math., vol. 137, iss. 1, pp. 82-203, 1998.
    @article {Simmom, MRKEY = {1627806},
      AUTHOR = {Simon, Barry},
      TITLE = {The classical moment problem as a self-adjoint finite difference operator},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {137},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {82--203},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {47A57 (39A70 44A60 47B39)},
      MRNUMBER = {1627806},
      DOI = {10.1006/aima.1998.1728},
      ZBLNUMBER = {0910.44004},
      }
  • [Simopuc2] B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2: Spectral Theory, Providence, RI: Amer. Math. Soc., 2005, vol. 54.
    @book {Simopuc2, MRKEY = {2105089},
      AUTHOR = {Simon, Barry},
      TITLE = {Orthogonal Polynomials on the Unit Circle. {P}art 2: {S}pectral Theory},
      SERIES = {Amer. Math. Soc. Colloq. Publ.},
      VOLUME = {54},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      PAGES = {i--xxii and 467--1044},
      ISBN = {0-8218-3675-7},
      MRCLASS = {42-02 (30C85 33C45 42C05 47B36 47N50)},
      MRNUMBER = {2105089},
      MRREVIEWER = {P. L. Duren},
      ZBLNUMBER = {1082.42021},
      }
  • [SimSp] B. Simon and T. Spencer, "Trace class perturbations and the absence of absolutely continuous spectra," Comm. Math. Phys., vol. 125, iss. 1, pp. 113-125, 1989.
    @article {SimSp, MRKEY = {1017742},
      AUTHOR = {Simon, Barry and Spencer, Thomas},
      TITLE = {Trace class perturbations and the absence of absolutely continuous spectra},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {125},
      YEAR = {1989},
      NUMBER = {1},
      PAGES = {113--125},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {81Q10 (47A10 47B39)},
      MRNUMBER = {1017742},
      MRREVIEWER = {Helge Holden},
      ZBLNUMBER = {0684.47010},
      }
  • [SY] Go to document M. Sodin and P. Yuditskii, "Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions," J. Geom. Anal., vol. 7, iss. 3, pp. 387-435, 1997.
    @article {SY, MRKEY = {1674798},
      AUTHOR = {Sodin, Mikhail and Yuditskii, Peter},
      TITLE = {Almost periodic {J}acobi matrices with homogeneous spectrum, infinite-dimensional {J}acobi inversion, and {H}ardy spaces of character-automorphic functions},
      JOURNAL = {J. Geom. Anal.},
      FJOURNAL = {The Journal of Geometric Analysis},
      VOLUME = {7},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {387--435},
      ISSN = {1050-6926},
      MRCLASS = {47B36 (30F35 37K10 37K20 46E15 46E22 47A10)},
      MRNUMBER = {1674798},
      MRREVIEWER = {Vladimir S. Matveev},
      ZBLNUMBER = {1041.47502},
      DOI = {10.1007/BF02921627},
      }
  • [Sto] Go to document G. Stolz, "Spectral theory for slowly oscillating potentials. I. Jacobi matrices," Manuscripta Math., vol. 84, iss. 3-4, pp. 245-260, 1994.
    @article {Sto, MRKEY = {1291120},
      AUTHOR = {Stolz, G{ü}nter},
      TITLE = {Spectral theory for slowly oscillating potentials. {I}. {J}acobi matrices},
      JOURNAL = {Manuscripta Math.},
      FJOURNAL = {Manuscripta Mathematica},
      VOLUME = {84},
      YEAR = {1994},
      NUMBER = {3-4},
      PAGES = {245--260},
      ISSN = {0025-2611},
      CODEN = {MSMHB2},
      MRCLASS = {47B39 (39A12 47A55 47N50 81Q10)},
      MRNUMBER = {1291120},
      MRREVIEWER = {Russell A. Johnson},
      DOI = {10.1007/BF02567456},
      ZBLNUMBER = {0811.34074},
     }
  • [Teschl] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Providence, RI: Amer. Math. Soc., 2000, vol. 72.
    @book {Teschl, MRKEY = {1711536},
      AUTHOR = {Teschl, Gerald},
      TITLE = {Jacobi Operators and Completely Integrable Nonlinear Lattices},
      SERIES = {Math. Surveys Monogr.},
      VOLUME = {72},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2000},
      PAGES = {xvii+351},
      ISBN = {0-8218-1940-2},
      MRCLASS = {39A70 (35Q58 37K60 39A12 47B36 82B23)},
      MRNUMBER = {1711536},
      MRREVIEWER = {Malcolm Adams},
      ZBLNUMBER = {1056.39029},
      }
  • [vMoer] Go to document P. van Moerbeke, "The spectrum of Jacobi matrices," Invent. Math., vol. 37, iss. 1, pp. 45-81, 1976.
    @article {vMoer, MRKEY = {0650253},
      AUTHOR = {van Moerbeke, Pierre},
      TITLE = {The spectrum of {J}acobi matrices},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {37},
      YEAR = {1976},
      NUMBER = {1},
      PAGES = {45--81},
      ISSN = {0020-9910},
      MRCLASS = {58F05 (14K99 34B25 35Q99)},
      MRNUMBER = {0650253},
      MRREVIEWER = {J. S. Joel},
      DOI = {10.1007/BF01418827},
      ZBLNUMBER = {0361.15010},
      }
  • [vMoerMum] Go to document P. van Moerbeke and D. Mumford, "The spectrum of difference operators and algebraic curves," Acta Math., vol. 143, iss. 1-2, pp. 93-154, 1979.
    @article {vMoerMum, MRKEY = {0533894},
      AUTHOR = {van Moerbeke, Pierre and Mumford, David},
      TITLE = {The spectrum of difference operators and algebraic curves},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {143},
      YEAR = {1979},
      NUMBER = {1-2},
      PAGES = {93--154},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {58F07 (14H45 34C05)},
      MRNUMBER = {0533894},
      MRREVIEWER = {Mark Adler},
      DOI = {10.1007/BF02392090},
      ZBLNUMBER = {0502.58032},
      }
  • [VolYud] Go to document A. Volberg and P. Yuditskii, "On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals or a Cantor set of positive length," Comm. Math. Phys., vol. 226, iss. 3, pp. 567-605, 2002.
    @article {VolYud, MRKEY = {1896882},
      AUTHOR = {Volberg, A. and Yuditskii, P.},
      TITLE = {On the inverse scattering problem for {J}acobi matrices with the spectrum on an interval, a finite system of intervals or a {C}antor set of positive length},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {226},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {567--605},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {47B36 (34L25 39A70 81Q10 81U40)},
      MRNUMBER = {1896882},
      MRREVIEWER = {Gerald Teschl},
      DOI = {10.1007/s002200200623},
      ZBLNUMBER = {1019.34082},
      }
  • [Wal] P. Walters, An Introduction to Ergodic Theory, New York: Springer-Verlag, 1982, vol. 79.
    @book {Wal, MRKEY = {0648108},
      AUTHOR = {Walters, Peter},
      TITLE = {An Introduction to Ergodic Theory},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {79},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {ix+250},
      ISBN = {0-387-90599-5},
      MRCLASS = {28Dxx (54H20 58F11)},
      MRNUMBER = {0648108},
      MRREVIEWER = {M. A. Akcoglu},
      ZBLNUMBER = {0475.28009},
      }
  • [Zla] Go to document A. Zlatovs, "Sparse potentials with fractional Hausdorff dimension," J. Funct. Anal., vol. 207, iss. 1, pp. 216-252, 2004.
    @article {Zla, MRKEY = {2027640},
      AUTHOR = {Zlato{š},
      Andrej},
      TITLE = {Sparse potentials with fractional {H}ausdorff dimension},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {207},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {216--252},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {39A70 (34L40 47B36 47N50 81Q10)},
      MRNUMBER = {2027640},
      MRREVIEWER = {Denis Krutikov},
      DOI = {10.1016/S0022-1236(03)00180-0},
      ZBLNUMBER = {1038.47026},
      }

Keywords
, ,
DOI
http://doi.org/10.4007/annals.2011.174.1.4
MR
2811596
Zbl
05960698
Milestones
Received: 7 June 2007
Revised: 11 August 2010
Accepted: 6 August 2010
Published online: 1 July 2011

Authors

Christian Remling

Department of Mathematics
University of Oklahoma
Norman, OK 73019-0315