Spectral and scattering theory of Schroedinger equations

2013 Fiscal Year Final Research Report

• Project/Area Number: 21244008
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: The University of Tokyo
• Principal Investigator: NAKAMURA Shu 東京大学, 数理(科)学研究科(研究院), 教授 (50183520)
• Co-Investigator(Kenkyū-buntansha): ADACHI Tadayoshi 神戸大学, 理学研究科, 教授 (30281158) ; NAKANO Fumihiko 学習院大学, 理学部, 教授 (10291246) ; FUJIIE Setsuro 立命館大学, 理工学部, 教授 (00238536)
• Co-Investigator(Renkei-kenkyūsha): TAMURA Hideo 岡山大学, 理学部, 名誉教授 (30022734) ; YAJIMA Kenji 学習院大学, 理学部, 教授 (80011758) ; ISOZAKI Hiroshi 筑波大学, 数理物質科学研究科, 教授 (90111913) ; IWATSUKA Akira 京都工芸繊維大学, 工芸科学研究科, 教授 (40184890) ; MINAMI Nariyuki 慶應義塾大学, 医学部, 教授 (10183964) ; UEKI Naomasa 京都大学, 人間・環境学研究科(研究院), 教授 (80211069) ; NOMURA Yuji 愛媛大学, 大学院・理工学研究科, 准教授 (40282818) ; DOI Shin-ichi 大阪大学, 理学(系)研究科(研究院), 教授 (00243006) ; TATE Tatsuya 名古屋大学, 大学院・多元数理科学研究科, 准教授 (00317299) ; MINE Takuya 京都工芸繊維大学, 工芸科学研究科, 准教授 (90378597) ; ITO Kenichi 神戸大学, 理学研究科, 准教授 (90512509)
• Project Period (FY): 2009-04-01 – 2014-03-31
• Keywords: シュレディンガー方程式 / スペクトル理論 / 散乱理論 / 半古典極限 / ランダム作用素 / 解の超局所的特異性

Research Abstract

We have investigated the spectral theory, scattering theory and the structure of solutions for Schrodinger equations, using the functional analysis, microlocal analysis, real analysis and probability theory methods. The research results include results on: 1. Microlocal singularities for solutions of Schrodinger equations, 2. Scattering theory for Schrodinger operators on manifolds, 3. Scattering theory for multi-particles in electric/magnetic fields, 4. Semiclassical analysis for Schrodinger equations, 5. Random Schrodinger equations and charge transport. These results are published as 39 publications (including in press, at present), and several projects are still in progress.

Research Products

• [Journal Article] The spectrum of the scattering matrix near resonant energies in the semiclassical limit (2014)
  • Author(s): Nakamura, S., Pushnitski, A
  • Journal Title: Trans. American Math. Soc Volume: 366 Pages: 1725-1747
  • DOI: 10.1090/S0002-9947-2013-06077-1
• [Journal Article] A generalization of carries processes and Eulerian numbers (2014)
  • Author(s): Nakano, F., Sadahiro, T
  • Journal Title: Adv. in Appl. Math Volume: 53 Pages: 28-43
  • DOI: 10.1016/j.aam.2013.09.005
• [Journal Article] Width of resonances created by homoclinic orbits - isotropic fixed point case (2014)
  • Author(s): Bony, J.F., Fujiie, S., Ramond, T., Zerzeri, M
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B45 Pages: 31-43
  • URL: http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/B45-contents.pdf
• [Journal Article] The spectral shift function and the Friedel sum rule (2013)
  • Author(s): Kohmoto, M., Koma, T., Nakamura, S
  • Journal Title: Ann. H. Poincare Volume: 14 Pages: 1413-1424
  • DOI: 10.1007/s00023-012-0219-3
• [Journal Article] Microlocal properties of scattering matrices for Schrodinger equations on scattering manifolds (2013)
  • Author(s): Ito, K., Nakamura, S
  • Journal Title: Analysis and PDE Volume: 6(No. 2) Pages: 257-286
  • DOI: 10.2140/apde.2013.6.257
• [Journal Article] Hitting matrix and domino tiling with diagonal impurities (2013)
  • Author(s): Nakano, F., Sadahiro, T
  • Journal Title: J. Stat. Phys Volume: 151(no. 6) Pages: 1035-1055
  • DOI: 10.1007/s10955-013-0744-6
• [Journal Article] On multidimensional inverse scattering in time-dependent electric fields (2013)
  • Author(s): Adachi, T., Fujiwara, Y., Ishida, A
  • Journal Title: Inverse Problems Volume: 29(no. 8, 085012) Pages: 24
  • DOI: 10.1088/0266-5611/29/8/085012
• [Journal Article] Remarks on the fundamental solution to Schrodinger equation with variable coefficients (2012)
  • Author(s): Ito, K., Nakamura, S
  • Journal Title: Ann. Inst. Fourier Volume: 62 Pages: 1091-1121
  • URL: http://aif.cedram.org/item?id=AIF_2012__62_3_1091_0
• [Journal Article] Localization for the random displacement model (2012)
  • Author(s): Klopp, F., Loss, M., Nakamura, S., Stolz, G
  • Journal Title: Duke Math. J Volume: 161(No.4) Pages: 587-621
  • DOI: 10.1215/00127094-1548353
• [Journal Article] A note on the analyticity of density of states (2012)
  • Author(s): Kaminaga, M., Krishna, M., Nakamura, S
  • Journal Title: J. Stat. Phys Volume: 149 Pages: 496-504
  • DOI: 10.1007/s10955-012-0603-x
• [Journal Article] Understanding the random displacement model : From ground-state properties to localization. Operator Theory (2012)
  • Author(s): Klopp, F., Loss, M., Nakamura, S., Stolz, G
  • Journal Title: Advances and Applications Volume: 224 Pages: 183-219
  • DOI: 10.1007/978-3-0348-0414-1_10
• [Journal Article] Domino tilings with one diagonal impurity (2012)
  • Author(s): Nakano, F., Sadahiro, T
  • Journal Title: Fund. Inform Volume: 117(no. 1-4) Pages: 249-264
  • DOI: 10.3233/FI-2012-699
• [Journal Article] A (-β)-expansion associated to Sturmian sequences (2012)
  • Author(s): Nakano, F., Sadahiro, T
  • Journal Title: Integers Volume: 12(no. 4) Pages: 571-599
  • DOI: 10.1515/integers-2011-0123
• [Journal Article] Avron-Herbst type formula in crossed constant magnetic and time-dependent electric fields (2012)
  • Author(s): Adachi, T., Kawamoto, M
  • Journal Title: Lett. Math. Phys Volume: 102(no. 1) Pages: 65-90
  • DOI: 10.1007/s11005-012-0555-8
• [Journal Article] Notes on vertex atlas of Danzer tiling (2011)
  • Author(s): Hayashi, H., Kawachi, Y., Komatsu, K., Konda, A., Kurozoe, M., Nakano, F., Odawara, N., Onda, R., Sugio, A., Yamauchi, M
  • Journal Title: Nihonkai Math. J Volume: 22(no. 1) Pages: 49-58
  • URL: http://projecteuclid.org/euclid.nihmj/1339694050
• [Journal Article] On multidimensional inverse scattering in an external electric field asymptotically zero in time (2011)
  • Author(s): Adachi, T., Kamada, T., Kazuno, M., Toratani, K
  • Journal Title: Inverse Problems Volume: 27(no. 6, 065006) Pages: 17
  • DOI: 10.1088/0266-5611/27/6/065006
• [Journal Article] Scattering in an external electric field asymptotically constant in time (2011)
  • Author(s): Adachi, T., Ishida, A
  • Journal Title: J. Math. Phys Volume: 52(no. 6, 062101) Pages: 15
  • DOI: 10.1063/1.3592600
• [Journal Article] Width of shape resonances for non globally analytic potentials (2011)
  • Author(s): Fujiie, S., Lahmar-Benbernou, A., Martinez, A
  • Journal Title: J. Math. Soc Volume: 63(no. 1) Pages: 1-78
  • DOI: 10.2969/jmsj/06310001
• [Journal Article] Spectral projection, residue of the scattering amplitude and Schrodinger group expansion for barrier-top resonances (2011)
  • Author(s): Bony, J.F., Fujiie, S., Ramond, T., Zerzeri, M
  • Journal Title: Ann. Inst. Fourier (Grenoble) Volume: 61(no. 4) Pages: 1351-1406
  • DOI: 10.5802/aif.2643
• [Journal Article] Time-dependent scattering theory for Schrodinger operators on scattering manifolds (2010)
  • Author(s): Ito, K., Nakamura, S
  • Journal Title: J.London Math. Soc Volume: 81 Pages: 774-792
  • DOI: 10.1112/jlms/jdq018
• [Journal Article] Lifshitz tails for generalized alloy type random Schrodinger operators (2010)
  • Author(s): Klopp, F., Nakamura, S
  • Journal Title: Analysis and PDE Volume: 3-4 Pages: 409-426
  • DOI: 10.2140/apde.2010.3.409
• [Journal Article] Analytic wave front set for solutions to Schrodinger equations II- Long range perturbations (2010)
  • Author(s): Martinez, A., Nakamura, S., Sordoni, V
  • Journal Title: Comm. Partial Differential Equations Volume: 35 Pages: 2279-2309
  • DOI: 10.1080/03605302.2010.523918
• [Journal Article] A bijection theorem for domino tilings with diagonal impurities (2010)
  • Author(s): Nakano, F., Sadahiro, T
  • Journal Title: J. Stat. Phys Volume: 139(no. 4) Pages: 565-597
  • DOI: 10.1007/s10955-010-9954-3
• [Journal Article] Local move connectedness of domino tilings with diagonal impurities (2010)
  • Author(s): Nakano, F., Ono, H., Sadahiro, T
  • Journal Title: Discrete Math Volume: 310(no. 13-14) Pages: 1918-1931
  • DOI: 10.1016/j.disc.2010.02.015
• [Journal Article] Scattering theory for two-body quantum systems with singular potentials in a time-periodic electric field (2010)
  • Author(s): Adachi, T., Kimura, T., Shimizu, Y
  • Journal Title: J. Math. Phys Volume: 51(no. 3, 032103) Pages: 23
  • DOI: 10.1063/1.3317902
• [Journal Article] Wave front set for solutions to Schrodinger equations. J (2009)
  • Author(s): Nakamura, S
  • Journal Title: Functional Analysis 256 Pages: 1299-1309
  • DOI: 10.1016/j.jfa.2008.06.007
• [Journal Article] Semiclassical singularity propagation property for Schrodinger equations (2009)
  • Author(s): Nakamura, S
  • Journal Title: J. Math. Soc Volume: 61(1) Pages: 177-211
  • DOI: 10.2969/jmsj/06110177
• [Journal Article] Spectral extrema and Lifshitz tails for non monotonous alloy type models (2009)
  • Author(s): Klopp, F., Nakamura, S
  • Journal Title: Commun. Math. Phys Volume: 287 Pages: 1133-1143
  • DOI: 10.1007/s00220-008-0666-3
• [Journal Article] Wave front set for solutions to perturbed harmonic oscillators (2009)
  • Author(s): Mao, S., Nakamura, S
  • Journal Title: Comm. Partial Differential Equations Volume: 34(5) Pages: 506-519
  • DOI: 10.1080/03605300902768891
• [Journal Article] Analytic wave front for solutions to Schrodinger equation (2009)
  • Author(s): Martinez, A., Nakamura, S., Sordoni, V
  • Journal Title: Advances in Math Volume: 222 Pages: 1277-1307
  • DOI: 10.1016/j.aim.2009.06.002
• [Journal Article] Singularities of solutions to Schrodinger equation on scattering manifold (2009)
  • Author(s): Ito, K., Nakamura, S
  • Journal Title: American J. Math Volume: 131(6) Pages: 1835-1865
  • URL: http://muse.jhu.edu/journals/american_journal_of_mathematics/v131/131.6.ito.html
• [Journal Article] Infinite divisibility of random measures associated to some random Schrodinger operators (2009)
  • Author(s): Nakano, F
  • Journal Title: Osaka J. Math Volume: 46(no. 3) Pages: 845-862
  • URL: http://projecteuclid.org/euclid.ojm/1256564209
• [Journal Article] Semiclassical resonances for a two-level Schrodinger operator with a conical intersection (2009)
  • Author(s): Fujiie, S., Lasser, C., Nedelec, L
  • Journal Title: Asymptot. Anal Volume: 65(no. 1-2) Pages: 17-58
  • DOI: 10.3233/ASY-2009-0946
• [Journal Article] Propagation of microlocal solutions through a hyperbolic fixed point. Differential equations and exact WKB analysis (2008)
  • Author(s): Bony, J.F., Fujiie, S., Ramond, T., Zerzeri, M
  • Journal Title: Inst. Math. Sci. (RIMS) Volume: B10 Pages: 1-32
  • URL: http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B10/pdf/B10-01.pdf
• [Journal Article] Propagation of singularities for Schrodinger equations with modestly long range type potentials
  • Author(s): Horie, K., Nakamura, S
  • Journal Title: Publ. RIMS Volume: (印刷中)
  • URL: http://arxiv.org/abs/1202.4702
• [Journal Article] A Remark on the Mourre theory for two body Schrodinger operators
  • Author(s): Nakamura, S
  • Journal Title: J. Spectral Theory Volume: (印刷中)
  • URL: http://arxiv.org/abs/1305.4716
• [Journal Article] Level statistics for the one-dimensional Schrodinger operators with random decaying potential, Festschrift Masatoshi Fukushima, in honor of Masatoshi Fukushima's Sanju
  • Author(s): S. Kotani and F. Nakano
  • Journal Title: World Scientific Volume: (印刷中)
  • URL: http://jp.arxiv.org/abs/1210.4224
• [Journal Article] Level statistics for one-dimensional Schrodinger operators and Gaussian beta ensemble
  • Author(s): F. Nakano
  • Journal Title: J. Stat. Phys Volume: (印刷中)
  • URL: http://jp.arxiv.org/abs/1312.6901
• [Journal Article] "WKB solutions near an unstable equillibrium" Nonlinear Physical Systems : Spectral Analysis
  • Author(s): Bony, J.F., Fujiie, S., Ramond, T., Zerzeri, M
  • Journal Title: Stability and Bifurcations Pages: 15-39
  • DOI: 10.1002/9781118577608.ch2
• [Presentation] Applications of phase space analysis to scattering theory for discrete Schrodinger operators (2013)
  • Author(s): Nakamura, S
  • Organizer: "Microlocal Analysis and Spectral Theory" in the honor of Johannes Sjostrand
  • Place of Presentation: CIRM, Marseille, France
  • Year and Date: 20130923-27
• [Presentation] Microlocal singularities and scattering theory for Schrodinger equations on manifold (2012)
  • Author(s): Nakamura, S
  • Organizer: International Congress on Mathematical Physics 2012
  • Place of Presentation: Aalborg, Denmark
  • Year and Date: 20120803, 20120806-11
• [Presentation] Semi-classical resonances associated with unstable equilibria (2012)
  • Author(s): Fujiie, S
  • Organizer: Spectral Analysis, Stability and Bifurcation in Modern Nonlinear Physical Systems
  • Place of Presentation: Banff
  • Year and Date: 2012-11-05
• [Presentation] Microlocal properties of scattering matrices for Schrodinger operators on manifolds (2012)
  • Author(s): Nakamura, S
  • Organizer: The fourth Birman Conference
  • Place of Presentation: Euler International Mathematical Institute, St. Petersburg, Russia
  • Year and Date: 2012-07-06
• [Presentation] シュレディンガー方程式の超局所特異性の解析と散乱理論 (2011)
  • Author(s): 中村周
  • Organizer: 日本数学会・秋季総合分科会・関数方程式論分科会
  • Year and Date: 2011-09-29
• [Presentation] Distribution of eigenfunctions in the Anderson model (2010)
  • Author(s): Nakano, F
  • Organizer: Journees Mathematiques
  • Place of Presentation: Kairouan, Tunigia
  • Year and Date: 20101104-06
• [Presentation] A remark on the discrete Schrodinger operators with random vector potential (2010)
  • Author(s): Nakano, F
  • Organizer: Random Schrodinger Operators
  • Place of Presentation: Centre Interfacultaire Bernoulli at the EPFL in Lausanne, Switzerland
  • Year and Date: 20100531-0604
• [Presentation] 時間周期的に変動する電場中の量子力学系に対する散乱理論 (2009)
  • Author(s): 足立匡義
  • Organizer: 日本数学会函数方程式論分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-25
• [Book] 量子力学のスペクトル理論 (2012)
  • Author(s): 中村周
  • Publisher: 共立出版