Abstract
We study a semiclassical quantization of Poincaré maps arising in scattering problems with fractal hyperbolic trapped sets. The main application is the proof of a fractal Weyl upper bound for the number of resonances/scattering poles in small domains near the real axis. This result includes the case of several convex (hard) obstacles satisfying a no-eclipse condition.
-
[BoCh]
J. Bony and J. Chemin, "Espaces fonctionnels associés au calcul de Weyl-Hörmander," Bull. Soc. Math. France, vol. 122, iss. 1, pp. 77-118, 1994.
@article {BoCh, MRKEY = {1259109},
AUTHOR = {Bony, Jean-Michel and Chemin, Jean-Yves},
TITLE = {Espaces fonctionnels associés au calcul de {W}eyl-{H}örmander},
JOURNAL = {Bull. Soc. Math. France},
FJOURNAL = {Bulletin de la Société Mathématique de France},
VOLUME = {122},
YEAR = {1994},
NUMBER = {1},
PAGES = {77--118},
ISSN = {0037-9484},
CODEN = {BSMFAA},
MRCLASS = {35S05 (46E35 47G30 58G15)},
MRNUMBER = {1259109},
MRREVIEWER = {William Margulies},
URL = {http://smf4.emath.fr/Publications/Bulletin/122/html/smf_bull_122_77-118.html},
ZBLNUMBER = {0798.35172},
} -
[NBu] N. Burq, "Contrôle de l’équation des plaques en présence d’obstacle strictement convexes," Mém. Soc. Math. France, vol. 55, pp. 3-126, 1993.
@article {NBu, MRKEY = {1254820},
AUTHOR = {Burq, Nicolas},
TITLE = {Contrôle de l'équation des plaques en présence d'obstacle strictement convexes},
JOURNAL = {Mém. Soc. Math. France},
FJOURNAL = {Mémoires de la Société Mathématique de France. Nouvelle Série},
VOLUME = {55},
YEAR = {1993},
PAGES = {3--126},
ISSN = {0037-9484},
MRCLASS = {93B05 (35B37 35J15 73K10 73K50 93C20)},
MRNUMBER = {1254820},
MRREVIEWER = {J{ó}zef Joachim Telega},
ZBLNUMBER = {0930.93007},
} -
[Chr05]
H. Christianson, "Growth and zeros of the zeta function for hyperbolic rational maps," Canad. J. Math., vol. 59, iss. 2, pp. 311-331, 2007.
@article {Chr05, MRKEY = {2310619},
AUTHOR = {Christianson, Hans},
TITLE = {Growth and zeros of the zeta function for hyperbolic rational maps},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
VOLUME = {59},
YEAR = {2007},
NUMBER = {2},
PAGES = {311--331},
ISSN = {0008-414X},
CODEN = {CJMAAB},
MRCLASS = {37C30 (37D20 37F10 47B38)},
MRNUMBER = {2310619},
MRREVIEWER = {Thomas Michael Jordan},
DOI = {10.4153/CJM-2007-013-4},
ZBLNUMBER = {1116.37032},
} -
[Da-Dy]
K. Datchev and S. Dyatlov, "Fractal Weyl laws for asymptotically hyperbolic manifolds," Geom. Funct. Anal., vol. 23, iss. 4, pp. 1145-1206, 2013.
@article {Da-Dy, MRKEY = {3077910},
AUTHOR = {Datchev, Kiril and Dyatlov, Semyon},
TITLE = {Fractal {W}eyl laws for asymptotically hyperbolic manifolds},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {23},
YEAR = {2013},
NUMBER = {4},
PAGES = {1145--1206},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {Preliminary Data},
MRNUMBER = {3077910},
DOI = {10.1007/s00039-013-0225-8},
ZBLNUMBER = {06197801},
} -
[DatVas10]
K. Datchev and A. Vasy, "Propagation through trapped sets and semiclassical resolvent estimates," Ann. Inst. Fourier $($Grenoble$)$, vol. 62, iss. 6, pp. 2347-2377 (2013), 2012.
@article {DatVas10, MRKEY = {3060760},
AUTHOR = {Datchev, Kiril and Vasy, Andr{á}s},
TITLE = {Propagation through trapped sets and semiclassical resolvent estimates},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {62},
YEAR = {2012},
NUMBER = {6},
PAGES = {2347--2377 (2013)},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {58J47 (35P15)},
MRNUMBER = {3060760},
DOI = {10.5802/aif.2751},
ZBLNUMBER = {06159914},
} -
[DiSj]
M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, Cambridge: Cambridge Univ. Press, 1999, vol. 268.
@book {DiSj, MRKEY = {1735654},
AUTHOR = {Dimassi, Mouez and Sj{ö}strand, Johannes},
TITLE = {Spectral Asymptotics in the Semi-Classical Limit},
SERIES = {London Math. Soc. Lecture Note Ser.},
VOLUME = {268},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1999},
PAGES = {xii+227},
ISBN = {0-521-66544-2},
MRCLASS = {35P20 (35S05 47F05 47G30 58J37 58J40 81Q20)},
MRNUMBER = {1735654},
MRREVIEWER = {George D. Raikov},
DOI = {10.1017/CBO9780511662195},
ZBLNUMBER = {0926.35002},
} -
[EberMainWunn10]
A. Eberspächer, J. Main, and G. Wunner, "Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems," Phys. Rev. E, vol. 82, iss. 4, p. 046201, 2010.
@article {EberMainWunn10, MRKEY = {2788043},
AUTHOR = {Ebersp{ä}cher, Alexander and Main, J{ö}rg and Wunner, G{ü}nter},
TITLE = {Fractal {W}eyl law for three-dimensional chaotic hard-sphere scattering systems},
JOURNAL = {Phys. Rev. E},
FJOURNAL = {Physical Review E. Statistical, Nonlinear, and Soft Matter Physics},
VOLUME = {82},
YEAR = {2010},
NUMBER = {4},
PAGES = {046201, 12},
ISSN = {1539-3755},
CODEN = {PLEEE8},
MRCLASS = {37D50 (81Q50)},
MRNUMBER = {2788043},
MRREVIEWER = {Kamales Kar},
DOI = {10.1103/PhysRevE.82.046201},
} -
[ErmaShepe10]
L. Ermann and D. L. Shepelyansky, "Ulam method and fractal Weyl law for Perron-Frobenius operators," Eur. Phys. J., vol. B75, pp. 299-304, 2010.
@article{ErmaShepe10,
author={Ermann, L. and Shepelyansky, D. L.},
TITLE={Ulam method and fractal {W}eyl law for {P}erron-{F}robenius operators},
JOURNAL={Eur. Phys. J.},
VOLUME={B75},
PAGES={299--304},
YEAR={2010},
ZBLNUMBER = {1202.81074},
DOI = {10.1140/epjb/e2010-00144-0},
} -
[ermchepshep]
L. Ermann, A. D. Chepelianskii, and D. L. Shepelyansky, "Fractal Weyl law for Linux Kernel architecture," Eur. Phys. J., vol. B79, pp. 115-120, 2011.
@article{ermchepshep,
author={Ermann, L. and Chepelianskii, A. D. and Shepelyansky, D. L.},
TITLE={Fractal {W}eyl law for {L}inux {K}ernel architecture},
JOURNAL={Eur. Phys. J.},
VOLUME={B79},
YEAR={2011},
PAGES={115--120},
DOI = {10.1140/epjb/e2010-10774-7},
} -
[FGF]
F. G. Friedlander, "The wave front set of the solution of a simple initial-boundary value problem with glancing rays," Math. Proc. Cambridge Philos. Soc., vol. 79, iss. 1, pp. 145-159, 1976.
@article {FGF, MRKEY = {0393861},
AUTHOR = {Friedlander, F. G.},
TITLE = {The wave front set of the solution of a simple initial-boundary value problem with glancing rays},
JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
VOLUME = {79},
YEAR = {1976},
NUMBER = {1},
PAGES = {145--159},
ISSN = {0305-0041},
MRCLASS = {35L20},
MRNUMBER = {0393861},
MRREVIEWER = {H. Kumano-go},
DOI = {10.1017/S0305004100052166},
ZBLNUMBER = {0319.35053},
} -
[GaRi]
P. Gaspard and S. A. Rice, "Semiclassical quantization of the scattering from a classically chaotic repellor," J. Chem. Phys., vol. 90, iss. 4, pp. 2242-2254, 1989.
@article {GaRi, MRKEY = {0980393},
AUTHOR = {Gaspard, Pierre and Rice, Stuart A.},
TITLE = {Semiclassical quantization of the scattering from a classically chaotic repellor},
JOURNAL = {J. Chem. Phys.},
FJOURNAL = {The Journal of Chemical Physics},
VOLUME = {90},
YEAR = {1989},
NUMBER = {4},
PAGES = {2242--2254},
ISSN = {0021-9606},
CODEN = {JCPSA6},
MRCLASS = {81F99 (58F13 58F22 70K50)},
MRNUMBER = {0980393},
MRREVIEWER = {Muthusamy Lakshmanan},
DOI = {10.1063/1.456018},
} -
[Ge] C. Gérard, "Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes," Mém. Soc. Math. France, vol. 31, pp. 1-146, 1988.
@article {Ge, MRKEY = {0998698},
AUTHOR = {G{é}rard, C.},
TITLE = {Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes},
JOURNAL = {Mém. Soc. Math. France},
FJOURNAL = {Mémoires de la Société Mathématique de France. Nouvelle Série},
VOLUME = {31},
YEAR = {1988},
PAGES = {1--146},
ISSN = {0037-9484},
MRCLASS = {35P25 (35A27 35J05)},
MRNUMBER = {0998698},
MRREVIEWER = {Rainer Picard},
ZBLNUMBER = {0654.35081},
} -
[GeSj]
C. Gérard and J. Sjöstrand, "Semiclassical resonances generated by a closed trajectory of hyperbolic type," Comm. Math. Phys., vol. 108, iss. 3, pp. 391-421, 1987.
@article {GeSj, MRKEY = {0874901},
AUTHOR = {G{é}rard, C. and Sj{ö}strand, J.},
TITLE = {Semiclassical resonances generated by a closed trajectory of hyperbolic type},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {108},
YEAR = {1987},
NUMBER = {3},
PAGES = {391--421},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58G15 (35A27 35P99 58F05)},
MRNUMBER = {0874901},
MRREVIEWER = {Alejandro Uribe},
DOI = {10.1007/BF01212317},
ZBLNUMBER = {0637.35027},
} -
[GLZ]
L. Guillopé, K. K. Lin, and M. Zworski, "The Selberg zeta function for convex co-compact Schottky groups," Comm. Math. Phys., vol. 245, iss. 1, pp. 149-176, 2004.
@article {GLZ, MRKEY = {2036371},
AUTHOR = {Guillop{é},
Laurent and Lin, Kevin K. and Zworski, Maciej},
TITLE = {The {S}elberg zeta function for convex co-compact {S}chottky groups},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {245},
YEAR = {2004},
NUMBER = {1},
PAGES = {149--176},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {11M36 (58J52)},
MRNUMBER = {2036371},
MRREVIEWER = {Shin-ya Koyama},
DOI = {10.1007/s00220-003-1007-1},
ZBLNUMBER = {1075.11059},
} -
[He-Sj1] B. Helffer and J. Sjöstrand, "Semiclassical analysis for Harper’s equation. III. Cantor structure of the spectrum," Mém. Soc. Math. France, vol. 39, pp. 1-124, 1989.
@article {He-Sj1, MRKEY = {1041490},
AUTHOR = {Helffer, B. and Sj{ö}strand, J.},
TITLE = {Semiclassical analysis for {H}arper's equation. {III}. {C}antor structure of the spectrum},
JOURNAL = {Mém. Soc. Math. France},
FJOURNAL = {Mémoires de la Société Mathématique de France. Nouvelle Série},
VOLUME = {39},
YEAR = {1989},
PAGES = {1--124},
ISSN = {0037-9484},
MRCLASS = {35A27 (35J10 35P05 47B39 47G30 47N50 58G07 81Q20)},
MRNUMBER = {1041490},
ZBLNUMBER = {0725.34099},
} -
[Hor1]
L. Hörmander, The Analysis of Linear Partial Differential Operators. I: Distribution Theory and Fourier Analysis, New York: Springer-Verlag, 1983, vol. 256.
@book{Hor1, MRKEY={0717035},
AUTHOR = {H{ö}rmander, Lars},
TITLE = {The Analysis of Linear Partial Differential Operators. {I}: Distribution Theory and Fourier Analysis},
SERIES = {Grundlehren Math. Wiss.},
VOLUME = {256},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1983},
PAGES = {ix+391},
ISBN = {3-540-12104-8},
MRCLASS = {35-02 (42B10 46Fxx)},
MRNUMBER = {0717035},
MRREVIEWER = {L. Cattabriga},
ZBLNUMBER = {0521.35001},
DOI = {10.1007/978-3-642-96750-4},
} -
[Hor2]
L. Hörmander, The analysis of linear partial differential operators. III. Pseudo-differential operators, New York: Springer-Verlag, 1985, vol. 274.
@book{Hor2, MRKEY={0781536},
AUTHOR = {H{ö}rmander, Lars},
TITLE = {The analysis of linear partial differential operators. {III}. Pseudo-differential operators},
SERIES = {Grundlehren Math. Wiss.},
VOLUME = {274},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1985},
PAGES = {viii+525},
ISBN = {3-540-13828-5},
MRCLASS = {35-02 (35Sxx 47G05 58G15)},
MRNUMBER = {0781536},
MRREVIEWER = {Min You Qi},
ZBLNUMBER = {0601.35001},
DOI = {10.1007/978-3-540-49938-1},
} -
[Ik]
M. Ikawa, "Decay of solutions of the wave equation in the exterior of several convex bodies," Ann. Inst. Fourier $($Grenoble$)$, vol. 38, iss. 2, pp. 113-146, 1988.
@article {Ik, MRKEY = {0949013},
AUTHOR = {Ikawa, Mitsuru},
TITLE = {Decay of solutions of the wave equation in the exterior of several convex bodies},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {38},
YEAR = {1988},
NUMBER = {2},
PAGES = {113--146},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {35B40 (35L05 35P25)},
MRNUMBER = {0949013},
MRREVIEWER = {N. D. Kazarinoff},
DOI = {10.5802/aif.1137},
ZBLNUMBER = {0636.35045},
} -
[JN]
D. Jakobson and F. Naud, "Lower bounds for resonances of infinite-area Riemann surfaces," Anal. PDE, vol. 3, iss. 2, pp. 207-225, 2010.
@article {JN, MRKEY = {2657455},
AUTHOR = {Jakobson, Dmitry and Naud, Fr{é}d{é}ric},
TITLE = {Lower bounds for resonances of infinite-area {R}iemann surfaces},
JOURNAL = {Anal. PDE},
FJOURNAL = {Analysis \& PDE},
VOLUME = {3},
YEAR = {2010},
NUMBER = {2},
PAGES = {207--225},
ISSN = {1948-206X},
MRCLASS = {58J51 (11F72 30F35 35J05 35P20 47A40 58J50)},
MRNUMBER = {2657455},
MRREVIEWER = {Dieter H. Mayer},
DOI = {10.2140/apde.2010.3.207},
ZBLNUMBER = {1243.11064},
} -
[KaHa] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge: Cambridge Univ. Press, 1995, vol. 54.
@book {KaHa, MRKEY = {1326374},
AUTHOR = {Katok, Anatole and Hasselblatt, Boris},
TITLE = {Introduction to the Modern Theory of Dynamical Systems},
SERIES = {Encycl. Math. Appl.},
VOLUME = {54},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1995},
PAGES = {xviii+802},
ISBN = {0-521-34187-6},
MRCLASS = {58Fxx (34Cxx 34Dxx 58-01 58F11 58F15)},
MRNUMBER = {1326374},
MRREVIEWER = {Edoh Amiran},
ZBLNUMBER = {0878.58020},
} -
[PUSZ]
A. Potzuweit, T. Weich, S. Barkhofen, U. Kuhl, H. J. Stöckmann, and M. Zworski, "Weyl asymptotics: from closed to open systems," Phys. Rev. E, vol. 86, p. 066205, 2012.
@article{PUSZ,
author={Potzuweit, A. and Weich, T. and Barkhofen, S and Kuhl, U. and J. Stöckmann, H. and Zworski, M.},
TITLE={Weyl asymptotics: from closed to open systems},
JOURNAL={Phys. Rev. E},
VOLUME={86},
YEAR={2012},
PAGES = {066205, 8},
DOI = {10.1103/PhysRevE.86.066205},
} -
[Lin02]
K. K. Lin, "Numerical study of quantum resonances in chaotic scattering," J. Comput. Phys., vol. 176, iss. 2, pp. 295-329, 2002.
@article {Lin02, MRKEY = {1894769},
AUTHOR = {Lin, Kevin K.},
TITLE = {Numerical study of quantum resonances in chaotic scattering},
JOURNAL = {J. Comput. Phys.},
FJOURNAL = {Journal of Computational Physics},
VOLUME = {176},
YEAR = {2002},
NUMBER = {2},
PAGES = {295--329},
ISSN = {0021-9991},
CODEN = {JCTPAH},
MRCLASS = {81Q50 (81-04 81Q20)},
MRNUMBER = {1894769},
DOI = {10.1006/jcph.2001.6986},
ZBLNUMBER = {1021.81021},
} -
[LSZ]
W. Lu, S. Sridhar, and M. Zworski, "Fractal Weyl laws for chaotic open systems," Phys. Rev. Lett., vol. 91, p. 154101, 2003.
@article{LSZ,
author={Lu, W. and Sridhar, S. and Zworski, M.},
TITLE={Fractal {W}eyl laws for chaotic open systems},
JOURNAL={Phys. Rev. Lett.},
YEAR={2003},
VOLUME={91},
PAGES={154101, 4},
DOI = {10.1103/PhysRevLett.91.154101},
} -
[Mel]
R. B. Melrose, "Equivalence of glancing hypersurfaces," Invent. Math., vol. 37, iss. 3, pp. 165-191, 1976.
@article {Mel, MRKEY = {0436225},
AUTHOR = {Melrose, R. B.},
TITLE = {Equivalence of glancing hypersurfaces},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {37},
YEAR = {1976},
NUMBER = {3},
PAGES = {165--191},
ISSN = {0020-9910},
MRCLASS = {58G15 (58F05)},
MRNUMBER = {0436225},
MRREVIEWER = {Martin Golubitsky},
DOI = {10.1007/BF01390317},
ZBLNUMBER = {0354.53033},
} -
@misc{Mel3,
author={Melrose, R. B.},
TITLE={Polynomial bound on the distribution of poles in scattering by an obstacle},
NOTE={\emph{Journées }``\emph{Équations aux Dérivées partielles\/}'', Saint-Jean de Monts, 1984},
SORTYEAR={1984},
URL = {http://jedp.cedram.org/jedp-bin/fitem?id=JEDP_1984____A3_0},
ZBLNUMBER = {0621.35073},
} -
[MT]
R. B. Melrose and M. E. Taylor, "Near peak scattering and the corrected Kirchhoff approximation for a convex obstacle," Adv. in Math., vol. 55, iss. 3, pp. 242-315, 1985.
@article {MT, MRKEY = {0778964},
AUTHOR = {Melrose, Richard B. and Taylor, Michael E.},
TITLE = {Near peak scattering and the corrected {K}irchhoff approximation for a convex obstacle},
JOURNAL = {Adv. in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {55},
YEAR = {1985},
NUMBER = {3},
PAGES = {242--315},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {35L05 (35J05 35P25 58G15 58G25)},
MRNUMBER = {0778964},
MRREVIEWER = {V. Ivri{\u\i}},
DOI = {10.1016/0001-8708(85)90093-3},
ZBLNUMBER = {0591.58034},
} -
[MSZ] R. B. Melrose, A. Sá Barreto, and M. Zworski, Semilinear Diffraction of Conormal Waves, , 1996, vol. 240.
@book {MSZ, MRKEY = {1636412},
AUTHOR = {Melrose, Richard B. and S{á} Barreto, Ant{ô}nio and Zworski, Maciej},
TITLE = {Semilinear Diffraction of Conormal Waves},
SERIES = {Astérisque},
FJOURNAL = {Astérisque},
YEAR = {1996},
VOLUME = {240},
PAGES = {vi+132 pp. (1997)},
ISSN = {0303-1179},
MRCLASS = {35L70 (58G17)},
MRNUMBER = {1636412},
MRREVIEWER = {Jean-Marc Delort},
ZBLNUMBER = {0902.35004},
} -
@misc{Nau,
author={Naud, F.},
TITLE={Density and localization of resonances for convex co-compact hyperbolic surfaces},
NOTE={\emph{Invent. Math.}; published online 9 March 2013},
DOI = {10.1007/s00222-013-0463-2},
} -
[NSZ1]
S. Nonnenmacher, J. Sjöstrand, and M. Zworski, "From open quantum systems to open quantum maps," Comm. Math. Phys., vol. 304, iss. 1, pp. 1-48, 2011.
@article {NSZ1, MRKEY = {2793928},
AUTHOR = {Nonnenmacher, St{é}phane and Sj{ö}strand, Johannes and Zworski, Maciej},
TITLE = {From open quantum systems to open quantum maps},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {304},
YEAR = {2011},
NUMBER = {1},
PAGES = {1--48},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58J51 (35P25 58J50 81Q50)},
MRNUMBER = {2793928},
MRREVIEWER = {Dieter H. Mayer},
DOI = {10.1007/s00220-011-1214-0},
ZBLNUMBER = {1223.81127},
} -
[NZ1]
S. Nonnenmacher and M. Zworski, "Distribution of resonances for open quantum maps," Comm. Math. Phys., vol. 269, iss. 2, pp. 311-365, 2007.
@article {NZ1, MRKEY = {2274550},
AUTHOR = {Nonnenmacher, St{é}phane and Zworski, Maciej},
TITLE = {Distribution of resonances for open quantum maps},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {269},
YEAR = {2007},
NUMBER = {2},
PAGES = {311--365},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {81Q50 (28A80 81Q20)},
MRNUMBER = {2274550},
DOI = {10.1007/s00220-006-0131-0},
ZBLNUMBER = {1114.81043},
} -
[NZ3]
S. Nonnenmacher and M. Zworski, "Quantum decay rates in chaotic scattering," Acta Math., vol. 203, iss. 2, pp. 149-233, 2009.
@article {NZ3, MRKEY = {2570070},
AUTHOR = {Nonnenmacher, St{é}phane and Zworski, Maciej},
TITLE = {Quantum decay rates in chaotic scattering},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {203},
YEAR = {2009},
NUMBER = {2},
PAGES = {149--233},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {58J51 (35P25 35S05 37D20 81Q20 81Q50)},
MRNUMBER = {2570070},
MRREVIEWER = {Dieter H. Mayer},
DOI = {10.1007/s11511-009-0041-z},
ZBLNUMBER = {1226.35061},
} -
[Ott]
L. Poon, J. Campos, E. Ott, and C. Grebogi, "Wada basin boundaries in chaotic scattering," Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 6, iss. 2, pp. 251-265, 1996.
@article {Ott, MRKEY = {1390157},
AUTHOR = {Poon, Leon and Campos, Jos{é} and Ott, Edward and Grebogi, Celso},
TITLE = {Wada basin boundaries in chaotic scattering},
JOURNAL = {Internat. J. Bifur. Chaos Appl. Sci. Engrg.},
FJOURNAL = {International Journal of Bifurcation and Chaos in Applied Sciences and Engineering},
VOLUME = {6},
YEAR = {1996},
NUMBER = {2},
PAGES = {251--265},
ISSN = {0218-1274},
MRCLASS = {58F13},
MRNUMBER = {1390157},
DOI = {10.1142/S0218127496000035},
ZBLNUMBER = {0870.58069},
} -
[Pedr+09]
J. M. Pedrosa, G. G. Carlo, D. A. Wisniacki, and L. Ermann, "Distribution of resonances in the quantum open baker map," Phys. Rev. E, vol. 79, p. 016215, 2009.
@article{Pedr+09,
author={Pedrosa, J. M. and Carlo, G. G. and Wisniacki, D. A. and Ermann, L.},
TITLE={Distribution of resonances in the quantum open baker map},
JOURNAL={Phys. Rev. E},
VOLUME={79},
YEAR={2009},
PAGES = {016215, 5},
DOI = {10.1103/PhysRevE.79.016215},
} -
[Pest]
V. Petkov and L. Stoyanov, "Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function," Anal. PDE, vol. 3, iss. 4, pp. 427-489, 2010.
@article {Pest, MRKEY = {2718260},
AUTHOR = {Petkov, Vesselin and Stoyanov, Luchezar},
TITLE = {Analytic continuation of the resolvent of the {L}aplacian and the dynamical zeta function},
JOURNAL = {Anal. PDE},
FJOURNAL = {Analysis \& PDE},
VOLUME = {3},
YEAR = {2010},
NUMBER = {4},
PAGES = {427--489},
ISSN = {1948-206X},
MRCLASS = {35P25 (37C30 37D50)},
MRNUMBER = {2718260},
MRREVIEWER = {Leonardo Marazzi},
DOI = {10.2140/apde.2010.3.427},
ZBLNUMBER = {1251.37031},
} -
[Regg]
T. Regge, "Analytic properties of the scattering matrix," Nuovo Cimento, vol. 8, pp. 671-679, 1958.
@article {Regg, MRKEY = {0095702},
AUTHOR = {Regge, T.},
TITLE = {Analytic properties of the scattering matrix},
JOURNAL = {Nuovo Cimento},
VOLUME = {8},
YEAR = {1958},
PAGES = {671--679},
MRCLASS = {81.00 (30.00)},
MRNUMBER = {0095702},
MRREVIEWER = {R. P. Boas, Jr.},
ZBLNUMBER = {0080.41903},
DOI = {10.1007/BF02815247},
} -
[Ramil+09]
J. A. Ramilowski, S. D. Prado, F. Borondo, and D. Farrelly, "Fractal Weyl law behavior in an open Hamiltonian system," Phys. Rev. E, vol. 80, iss. 5, p. 055201, 2009.
@article {Ramil+09, MRKEY = {2593897},
AUTHOR = {Ramilowski, Jordan A. and Prado, S. D. and Borondo, F. and Farrelly, David},
TITLE = {Fractal {W}eyl law behavior in an open {H}amiltonian system},
JOURNAL = {Phys. Rev. E},
FJOURNAL = {Physical Review E. Statistical, Nonlinear, and Soft Matter Physics},
VOLUME = {80},
YEAR = {2009},
NUMBER = {5},
PAGES = {055201, 4},
ISSN = {1539-3755},
CODEN = {PLEEE8},
MRCLASS = {81Q99 (37J05 37N20)},
MRNUMBER = {2593897},
DOI = {10.1103/PhysRevE.80.055201},
} -
[SchoTwor04]
H. Schomerus and J. Tworzydło, "Quantum-to-classical crossover of quasibound states in open quantum systems," Phys. Rev. Lett., vol. 93, p. 154102, 2004.
@article{SchoTwor04,
author={Schomerus, H. and Tworzyd{\l}o, J.},
TITLE={Quantum-to-classical crossover of quasibound states in open quantum systems},
JOURNAL={Phys. Rev. Lett.},
VOLUME={93},
YEAR={2004},
PAGES = {154102, 4},
DOI = {10.1103/PhysRevLett.93.154102},
} -
[shepel]
D. L. Shepelyansky, "Fractal Weyl law for quantum fractal eigenstates," Phys. Rev. E, vol. 77, p. 015202, 2008.
@article{shepel,
author={Shepelyansky, D. L.},
TITLE={Fractal {W}eyl law for quantum fractal eigenstates},
JOURNAL={Phys. Rev. E},
VOLUME={77},
YEAR={2008},
PAGES = {015202, 4},
DOI = {10.1103/PhysRevE.77.015202},
} -
[koppscho]
M. Kopp and H. Schomerus, "Fractal Weyl laws for quantum decay in dynamical systems with a mixed phase space," Phys. Rev. E., vol. 81, p. 026308, 2010.
@article{koppscho,
author={Kopp, M. and Schomerus, H.},
TITLE={Fractal {W}eyl laws for quantum decay in dynamical systems with a mixed phase space},
JOURNAL={Phys. Rev. E.},
VOLUME={81},
YEAR={2010},
PAGES={026308, 6},
DOI = {10.1103/PhysRevE.81.026208},
} -
[SjDuke]
J. Sjöstrand, "Geometric bounds on the density of resonances for semiclassical problems," Duke Math. J., vol. 60, iss. 1, pp. 1-57, 2002.
@article {SjDuke, MRKEY = {1047116},
AUTHOR = {Sj{ö}strand, Johannes},
TITLE = {Geometric bounds on the density of resonances for semiclassical problems},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {60},
YEAR = {2002},
NUMBER = {1},
PAGES = {1--57},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {35P20 (35A27 35P25 47F05 58G15)},
MRNUMBER = {1047116},
MRREVIEWER = {Rainer Picard},
DOI = {10.1215/S0012-7094-90-06001-6},
ZBLNUMBER = {0702.35188},
} -
@misc{SjN,
author={Sjöstrand, Johannes},
TITLE={Lectures on resonances},
SORTYEAR={2014},
URL = {http://sjostrand.perso.math.cnrs.fr/Coursgbg.pdf},
} -
[SjR]
J. Sjöstrand, "Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations," Ann. Fac. Sci. Toulouse Math., vol. 18, iss. 4, pp. 739-795, 2009.
@article {SjR, MRKEY = {2590387},
AUTHOR = {Sj{ö}strand, Johannes},
TITLE = {Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations},
JOURNAL = {Ann. Fac. Sci. Toulouse Math.},
FJOURNAL = {Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série 6},
VOLUME = {18},
YEAR = {2009},
NUMBER = {4},
PAGES = {739--795},
ISSN = {0240-2963},
MRCLASS = {47B80 (35P05 35P20 47B06 47G30)},
MRNUMBER = {2590387},
MRREVIEWER = {Vladimir Tulovsky},
DOI = {10.5802/afst.1223},
ZBLNUMBER = {1194.47058},
} -
[SZ1]
J. Sjöstrand and M. Zworski, "Complex scaling and the distribution of scattering poles," J. Amer. Math. Soc., vol. 4, iss. 4, pp. 729-769, 1991.
@article {SZ1, MRKEY = {1115789},
AUTHOR = {Sj{ö}strand, Johannes and Zworski, Maciej},
TITLE = {Complex scaling and the distribution of scattering poles},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {4},
YEAR = {1991},
NUMBER = {4},
PAGES = {729--769},
ISSN = {0894-0347},
MRCLASS = {35P25 (35B20 58G25)},
MRNUMBER = {1115789},
MRREVIEWER = {George D. Raikov},
DOI = {10.2307/2939287},
ZBLNUMBER = {0752.35046},
} -
[SZ8]
J. Sjöstrand and M. Zworski, "Quantum monodromy and semi-classical trace formulae," J. Math. Pures Appl., vol. 81, iss. 1, pp. 1-33, 2002.
@article {SZ8, MRKEY = {1994881},
AUTHOR = {Sj{ö}strand, Johannes and Zworski, Maciej},
TITLE = {Quantum monodromy and semi-classical trace formulae},
JOURNAL = {J. Math. Pures Appl.},
FJOURNAL = {Journal de Mathématiques Pures et Appliquées. Neuvième Série},
VOLUME = {81},
YEAR = {2002},
NUMBER = {1},
PAGES = {1--33},
ISSN = {0021-7824},
CODEN = {JMPAAM},
MRCLASS = {58J40 (35S30 81Q20)},
MRNUMBER = {1994881},
MRREVIEWER = {Andrew W. Hassell},
DOI = {10.1016/S0021-7824(01)01230-2},
ZBLNUMBER = {1038.58033},
} -
[SZ9]
J. Sjöstrand and M. Zworski, "Elementary linear algebra for advanced spectral problems," Ann. Inst. Fourier $($Grenoble$)$, vol. 57, iss. 7, pp. 2095-2141, 2007.
@article {SZ9, MRKEY = {2394537},
AUTHOR = {Sj{ö}strand, Johannes and Zworski, Maciej},
TITLE = {Elementary linear algebra for advanced spectral problems},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {57},
YEAR = {2007},
NUMBER = {7},
PAGES = {2095--2141},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {35P05 (15A21 35Q40)},
MRNUMBER = {2394537},
MRREVIEWER = {K. Veseli{ć}},
ZBLNUMBER = {1140.15009},
DOI = {10.5802/aif.2328},
} -
[SZ10]
J. Sjöstrand and M. Zworski, "Fractal upper bounds on the density of semiclassical resonances," Duke Math. J., vol. 137, iss. 3, pp. 381-459, 2007.
@article {SZ10, MRKEY = {2309150},
AUTHOR = {Sj{ö}strand, Johannes and Zworski, Maciej},
TITLE = {Fractal upper bounds on the density of semiclassical resonances},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {137},
YEAR = {2007},
NUMBER = {3},
PAGES = {381--459},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {35J10 (35A27 35S05 81Q20)},
MRNUMBER = {2309150},
MRREVIEWER = {Caroline E. Lasser},
DOI = {10.1215/S0012-7094-07-13731-1},
ZBLNUMBER = {1201.35189},
} -
[St1]
P. Stefanov, "Quasimodes and resonances: sharp lower bounds," Duke Math. J., vol. 99, iss. 1, pp. 75-92, 1999.
@article {St1, MRKEY = {1700740},
AUTHOR = {Stefanov, Plamen},
TITLE = {Quasimodes and resonances: sharp lower bounds},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {99},
YEAR = {1999},
NUMBER = {1},
PAGES = {75--92},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {35P25 (47F05)},
MRNUMBER = {1700740},
MRREVIEWER = {Evgeni L. Korotyaev},
DOI = {10.1215/S0012-7094-99-09903-9},
ZBLNUMBER = {0952.47013},
} -
[St2]
P. Stefanov, "Sharp upper bounds on the number of the scattering poles," J. Funct. Anal., vol. 231, iss. 1, pp. 111-142, 2006.
@article {St2, MRKEY = {2190165},
AUTHOR = {Stefanov, Plamen},
TITLE = {Sharp upper bounds on the number of the scattering poles},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {231},
YEAR = {2006},
NUMBER = {1},
PAGES = {111--142},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {35P25 (47A40 81U05)},
MRNUMBER = {2190165},
MRREVIEWER = {Kalyan B. Sinha},
DOI = {10.1016/j.jfa.2005.07.007},
ZBLNUMBER = {1099.35074},
} -
[StVo]
P. Stefanov and G. Vodev, "Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body," Duke Math. J., vol. 78, iss. 3, pp. 677-714, 1995.
@article {StVo, MRKEY = {1334206},
AUTHOR = {Stefanov, Plamen and Vodev, G.},
TITLE = {Distribution of resonances for the {N}eumann problem in linear elasticity outside a strictly convex body},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {78},
YEAR = {1995},
NUMBER = {3},
PAGES = {677--714},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {35J25 (35P05 35Q72 47F05 73C02)},
MRNUMBER = {1334206},
MRREVIEWER = {G{ü}nter Berger},
DOI = {10.1215/S0012-7094-95-07825-9},
ZBLNUMBER = {0846.35139},
} -
[StrainZwo04]
J. Strain and M. Zworski, "Growth of the zeta function for a quadratic map and the dimension of the Julia set," Nonlinearity, vol. 17, iss. 5, pp. 1607-1622, 2004.
@article {StrainZwo04, MRKEY = {2086141},
AUTHOR = {Strain, John and Zworski, Maciej},
TITLE = {Growth of the zeta function for a quadratic map and the dimension of the {J}ulia set},
JOURNAL = {Nonlinearity},
FJOURNAL = {Nonlinearity},
VOLUME = {17},
YEAR = {2004},
NUMBER = {5},
PAGES = {1607--1622},
ISSN = {0951-7715},
CODEN = {NONLE5},
MRCLASS = {37F10 (37C30 37F50 37N20)},
MRNUMBER = {2086141},
MRREVIEWER = {Fr{é}d{é}ric N{ö}el Naud},
DOI = {10.1088/0951-7715/17/5/003},
ZBLNUMBER = {1066.37031},
} -
[TZ] S. Tang and M. Zworski, "Resonance expansions of scattered waves," Comm. Pure Appl. Math., vol. 53, iss. 10, pp. 1305-1334, 2000.
@article {TZ, MRKEY = {1768812},
AUTHOR = {Tang, Siu-Hung and Zworski, Maciej},
TITLE = {Resonance expansions of scattered waves},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {53},
YEAR = {2000},
NUMBER = {10},
PAGES = {1305--1334},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {35P25 (35L05 47F05)},
MRNUMBER = {1768812},
MRREVIEWER = {G{ü}nter Berger},
ZBLNUMBER = {1032.35148},
} -
[Met] M. E. Taylor, Pseudodifferential Operators, Princeton, N.J.: Princeton Univ. Press, 1981, vol. 34.
@book {Met, MRKEY = {0618463},
AUTHOR = {Taylor, Michael E.},
TITLE = {Pseudodifferential Operators},
SERIES = {Princeton Univ. Math. Series},
VOLUME = {34},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, N.J.},
YEAR = {1981},
PAGES = {xi+452},
ISBN = {0-691-08282-0},
MRCLASS = {35Sxx (35-02 47G05 58G15 78A05)},
MRNUMBER = {0618463},
MRREVIEWER = {Vesselin M. Petkov},
ZBLNUMBER = {0453.47026},
} -
[Vai] B. R. Vainberg, "Exterior elliptic problems that depend polynomially on the spectral parameter, and the asymptotic behavior for large values of the time of the solutions of nonstationary problems," Mat. Sb., vol. 134, pp. 224-241, 1973.
@article {Vai, MRKEY = {0346319},
AUTHOR = {Vainberg, B. R.},
TITLE = {Exterior elliptic problems that depend polynomially on the spectral parameter, and the asymptotic behavior for large values of the time of the solutions of nonstationary problems},
JOURNAL = {Mat. Sb.},
VOLUME = {134},
YEAR = {1973},
PAGES = {224--241},
MRCLASS = {35J40 (35L35 47F05)},
MRNUMBER = {0346319},
MRREVIEWER = {Ja. A. Roitberg},
ZBLNUMBER = {0294.35031},
} -
[VasZw00]
A. Vasy and M. Zworski, "Semiclassical estimates in asymptotically Euclidean scattering," Comm. Math. Phys., vol. 212, iss. 1, pp. 205-217, 2000.
@article {VasZw00, MRKEY = {1764368},
AUTHOR = {Vasy, Andr{á}s and Zworski, Maciej},
TITLE = {Semiclassical estimates in asymptotically {E}uclidean scattering},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {212},
YEAR = {2000},
NUMBER = {1},
PAGES = {205--217},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {58J50 (35P25 58J40 81Q20)},
MRNUMBER = {1764368},
MRREVIEWER = {San V{\~u} Ng{\d{o}}c},
DOI = {10.1007/s002200000207},
ZBLNUMBER = {0955.58023},
} -
[Vod]
G. Vodev, "Sharp bounds on the number of scattering poles in even-dimensional spaces," Duke Math. J., vol. 74, iss. 1, pp. 1-17, 1994.
@article {Vod, MRKEY = {1271461},
AUTHOR = {Vodev, Georgi},
TITLE = {Sharp bounds on the number of scattering poles in even-dimensional spaces},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {74},
YEAR = {1994},
NUMBER = {1},
PAGES = {1--17},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {35P25},
MRNUMBER = {1271461},
MRREVIEWER = {Leon S. Farhy},
DOI = {10.1215/S0012-7094-94-07401-2},
ZBLNUMBER = {0813.35075},
} -
[WierMain08]
J. Wiersig and J. Main, "Fractal Weyl law for chaotic microcavities: Fresnel’s laws imply multifractal scattering," Phys. Rev. E, vol. 77, p. 036205, 2008.
@article{WierMain08,
author={Wiersig, J. and Main, J.},
TITLE={Fractal {W}eyl law for chaotic microcavities: {F}resnel's laws imply multifractal scattering},
JOURNAL={Phys. Rev. E},
VOLUME={77},
YEAR={2008},
PAGES = {036205, 8},
DOI = {10.1103/PhysRevE.77.036205},
} -
[schwiema]
H. Schomerus, J. Wiersig, and J. Main, "Lifetime statistics in chaotic dielectric microresonators," Phys. Rev. A, vol. 79, p. 053806, 2009.
@article{schwiema,
author={Schomerus, H. and Wiersig, J. and Main, J.},
TITLE={Lifetime statistics in chaotic dielectric microresonators},
JOURNAL={Phys. Rev. A},
VOLUME={79},
YEAR={2009},
PAGES = {053806, 8},
DOI = {10.1103/PhysRevA.79.053806},
} -
[WuZ]
J. Wunsch and M. Zworski, "Resolvent estimates for normally hyperbolic trapped sets," Ann. Henri Poincaré, vol. 12, iss. 7, pp. 1349-1385, 2011.
@article {WuZ, MRKEY = {2846671},
AUTHOR = {Wunsch, Jared and Zworski, Maciej},
TITLE = {Resolvent estimates for normally hyperbolic trapped sets},
JOURNAL = {Ann. Henri Poincaré},
FJOURNAL = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
VOLUME = {12},
YEAR = {2011},
NUMBER = {7},
PAGES = {1349--1385},
ISSN = {1424-0637},
MRCLASS = {58J40 (35S05 47A40 83C57)},
MRNUMBER = {2846671},
MRREVIEWER = {Sandro Coriasco},
DOI = {10.1007/s00023-011-0108-1},
ZBLNUMBER = {1164.47326},
} -
[EZB] M. Zworski, Semiclassical Analysis, Providence, RI: Amer. Math. Soc., 2012, vol. 138.
@book {EZB, MRKEY = {2952218},
AUTHOR = {Zworski, Maciej},
TITLE = {Semiclassical Analysis},
SERIES = {Grad. Studies in Math.},
VOLUME = {138},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2012},
PAGES = {xii+431},
ISBN = {978-0-8218-8320-4},
MRCLASS = {58J40 (35P05 35S05 81Q20)},
MRNUMBER = {2952218},
MRREVIEWER = {David Borthwick},
ZBLNUMBER = {1252.58001},
}