Research activity information

• 2015 日本数学会 函数方程式論分科会, 第七回 福原賞, 多様体上のシュレーディンガー方程式の数学解析

  ITO Kenichi

  Japan society

• Low-energy resolvent estimates for slowly decaying attractive potentials

  Kenichi Ito, Tomoya Tagawa

  Springer Science and Business Media LLC, Jul. 2026, Letters in Mathematical Physics, 116(4) (4)

  [Refereed]

  Scientific journal


• Scattering theory for $C^2$ long-range potentials

  Kenichi Ito, Erik Skibsted

  We develop a complete stationary scattering theory for Schrödinger operators on \R^{d} , d\ge 2 , with C^{2} long-range potentials. This extends former results in the literature, in particular Isozaki (1980) and (1982), Ikebe and Isozaki (1982), and Gâtal and Yafaev (1999), which all require a higher degree of smoothness. In this sense, the spirit of our paper is similar to Hörmander [The Analysis of Linear Partial Differential Operators IV (1985), Chapter XXX] and J. Dereziński and C. Gérard [Scattering Theory of Classical and Quantum N -Particle Systems (1997), Section 4.7], which also develop a scattering theory under the C^{2} condition, however being very different from ours. While the Agmon–Hörmander theory is based on the Fourier transform and a momentum-space representation, our theory is entirely position-space based and may be seen as more related to our previous approach to scattering theory on manifolds, Ito and Skibsted (2013), (2019), and (2021). The C^{2} regularity is natural in the Agmon–Hörmander theory as well as in our theory, in fact probably being “optimal” in the Euclidean setting. We prove equivalence of the stationary scattering theory and a developed position-space based time-dependent scattering theory. Furthermore, we develop a related stationary scattering theory at fixed energy in terms of asymptotics of generalized eigenfunctions of minimal growth. A basic ingredient of our approach is a solution to the eikonal equation constructed from the geometric variational scheme of Cruz-Sampedro and Skibsted (2013). Another key ingredient is strong radiation condition bounds for the limiting resolvents originating in Herbst and Skibsted (1991). They improve formerly known ones by Isozaki (1980) and Saitō (1979) and considerably simplify the stationary approach. We obtain the bounds by a new commutator scheme whose elementary form allows a small degree of smoothness.

  European Mathematical Society - EMS - Publishing House GmbH, Feb. 2025, Journal of Spectral Theory, 15(1) (1), 353 - 439, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Stationary scattering theory for one-body Stark operators, I

  Tadayoshi Adachi, Kyohei Itakura, Kenichi Ito, Erik Skibsted

  2022, Pure and Applied Functional Analysis, 7(3) (3), 825 - 861, English, Co-authored internationally

  [Refereed][Invited]

  Scientific journal


• Stationary Scattering Theory for One-body Stark Operators, II

  Kenichi Ito, Erik Skibsted

  Springer Science and Business Media LLC, Sep. 2021, Annales Henri Poincaré, 23(2) (2), 513 - 548, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Stationary scattering theory on manifolds

  Kenichi Ito, Erik Skibsted

  Cellule MathDoc/CEDRAM, Aug. 2021, Annales de l'Institut Fourier, 71(3) (3), 1065 - 1119, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions

  Kenichi Ito, Arne Jensen

  Springer Science and Business Media LLC, Jun. 2021, Integral Equations and Operator Theory, 93(3) (3), English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• New methods in spectral theory of N-body Schrödinger operators

  Tadayoshi Adachi, Kyohei Itakura, Kenichi. Ito, Erik Skibsted

  We develop a new scheme of proofs for spectral theory of the [Formula: see text]-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem which is unified with exponential decay estimates studied previously only for [Formula: see text]-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with ‘zeroth order’ operators. In particular, they do not rely on Mourre’s differential inequality technique.

  World Scientific Pub Co Pte Lt, Jan. 2021, Reviews in Mathematical Physics, 33(05) (05), 2150015, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Spectral theory on manifolds

  Kenichi Ito, Erik Skibsted

  Mathematical Society of Japan, Dec. 2020, Advanced Studies in Pure Mathematics, 85, 193 - 203, English, Co-authored internationally

  [Refereed][Invited]

  International conference proceedings


• Commutator Methods for N-Body Schrödinger Operators

  Tadayoshi Adachi, Kyohei Itakura, Kenichi Ito, Erik Skibsted

  Springer International Publishing, Nov. 2020, Spectral Theory and Mathematical Physics, 1 - 15, English, International magazine, Co-authored internationally

  [Refereed][Invited]

  International conference proceedings


• Radiation condition bounds on manifolds with ends

  Kenichi Ito, Erik Skibsted

  Elsevier BV, May 2020, Journal of Functional Analysis, 278(9) (9), 108449, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Spectral theory for 1-body Stark operators

  Tadayoshi Adachi, Kyohei Itakura, Kenichi Ito, Erik Skibsted

  Elsevier BV, Apr. 2020, Journal of Differential Equations, 268(9) (9), 5179 - 5206, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Time-dependent scattering theory on manifolds

  Kenichi Ito, Erik Skibsted

  Elsevier BV, Sep. 2019, Journal of Functional Analysis, 277(5) (5), 1423 - 1468, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacians

  Kenichi Ito, Arne Jensen

  Elsevier BV, Aug. 2019, Journal of Functional Analysis, 277(4) (4), 965 - 993, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Resolvent expansion for the Schrödinger operator on a graph with infinite rays

  Kenichi Ito, Arne Jensen

  Academic Press Inc., Aug. 2018, Journal of Mathematical Analysis and Applications, 464(1) (1), 616 - 661, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Resolvent expansions for the Schrödinger operator on the discrete half-line

  Kenichi Ito, Arne Jensen

  May 2017, Journal of Mathematical Physics, 58(5) (5), English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal

  Link to Kobe Univ. Repository (Kernel)


• Rellich's theorem and N-body Schrödinger operators

  Kenichi Ito, Erik Skibsted

  Jun. 2016, Reviews in Mathematica Physics, 28(5) (5), English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal

  Link to Kobe Univ. Repository (Kernel)


• A complete classification of threshold properties for one-dimensional discrete Schrödinger operators

  Kenichi Ito, Arne Jensen

  Feb. 2015, Reviews in Mathematical Physics, 27(1) (1), English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Absence of positive eigenvalues for hard-core N-body systems

  Kenichi Ito, Erik Skibsted

  Dec. 2014, Annales Henri Poincaré, 15(12) (12), 2379 - 2408, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Resolvent expansion for the discrete one-dimensional Schrödinger operator

  Kenichi Ito, Arne Jensen

  WORLD SCIENTIFIC, Nov. 2014, Mathematical Results in Quantum Mechanics, 253 - 257, English, International magazine, Co-authored internationally

  [Refereed][Invited]

  International conference proceedings


• Absence of embedded eigenvalues for the Schrödinger operator on manifold

  Kenichi Ito, Erik Skibsted

  Apr. 2014, RIMS Kôkyûroku Bessatsu, B45, 69 - 75, English

  [Refereed][Invited]

  International conference proceedings


• Absence of embedded eigenvalues for Riemannian Laplacians

  Kenichi Ito, Erik Skibsted

  Nov. 2013, Advances in Mathematics, 248, 945 - 962, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Absence of positive eigenvalues for hard-core N-body systems

  Kenichi Ito, Erik Skibsted

  Oct. 2013, XVIIth International Congress on Mathematical Physics, 520 - 527, English, International magazine, Co-authored internationally

  [Refereed][Invited]

  International conference proceedings


• Scattering theory for Riemannian Laplacians

  Kenichi Ito, Erik Skibsted

  Apr. 2013, Journal of Functional Analysis, 264(8) (8), 1929 - 1974, English, International magazine, Co-authored internationally

  [Refereed]

  Scientific journal


• Microlocal properties of scattering matrices for Schrödinger equations on scattering manifolds

  Kenichi Ito, Shu Nakamura

  2013, Analysis & PDE, 6(2) (2), 257 - 286, English, International magazine

  [Refereed]

  Scientific journal


• Remarks on the fundamental solution to Schrödinger equation with variable coefficients

  Kenichi Ito, Shu Nakamura

  2012, Annales de l’Institut Fourier, 62(3) (3), 1091 - 1121, English, International magazine

  [Refereed]

  Scientific journal


• Time-dependent scattering theory for Schrödinger operators on scattering manifolds

  Kenichi Ito, Shu Nakamura

  Jun. 2010, Journal of the London Mathematical Society - Second Series, 81(3) (3), 774 - 792, English, International magazine

  [Refereed]

  Scientific journal


• Schrödinger equations on scattering manifolds and microlocal singularities

  Kenichi Ito, Shu Nakamura

  Apr. 2010, RIMS Kôkyûroku Bessatsu, B16, 91 - 100, English

  [Refereed][Invited]

  International conference proceedings


• Singularities of solutions to the Schrödinger equation on scattering manifold

  Kenichi Ito, Shu Nakamura

  Dec. 2009, American Journal of Mathematics, 131(6) (6), 1835 - 1865, English, International magazine, Co-authored internationally

  [Refereed]


• Multilinear eigenfunction estimates for the harmonic oscillator and the nonlinear Schrödinger equation with the harmonic potential

  Takafumi Akahori, Kenichi Ito

  Jul. 2009, Annales Henri Poincaré, 10(4) (4), 673 - 709, English, International magazine

  [Refereed]

  Scientific journal


• Propagation of singularities for Schrödinger equations on the Euclidean space with a scattering metric

  Kenichi Ito

  Lead, Dec. 2006, Communications in Partial Differential Equations, 31(12) (12), 1735 - 1777, English, International magazine

  [Refereed]

  Scientific journal

• $C^2$級長距離型ポテンシャルに対する散乱理論 (Spectral and Scattering Theory and Related Topics)

  Kenichi Ito, Erik Skibsted

  Apr. 2026, 2341, 80 - 97


• $N$体Schrödinger作用素に対する交換子法 (Mathematical aspects of quantum fields and related topics)

  Tadayoshi Adachi, Kyohei Itakura, Kenichi Ito, Erik Skibsted

  Jun. 2021, RIMS Kokyuroku, 2187, 98 - 114


• Resolvent expansion for the Schrödinger operator on a graph with infinite rays (Tosio Kato Centennial Conference)

  Kenichi Ito, Arne Jensen

  Jul. 2018, RIMS Kokyuroku, 2074, 47 - 54


• 楳田登美男: 数理物理における固有値問題

  ITO Kenichi

  Apr. 2016, サイエンス社「数理科学」, 54(4) (4), p. 60, Japanese

  Book review


• Rellich's theorem, limiting absorption principle and radiation condition on manifold with ends (Spectral and Scattering Theory and Related Topics)

  Kenichi Ito, Erik Skibsted

  Nov. 2015, RIMS Kokyuroku, 1975, 117 - 126


• N. Lerner: Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators (Pseudo Diff. Oper., 3)

  ITO Kenichi

  日本数学会, Oct. 2014, 日本数学会雑誌「数学」, 66(4) (4), p. 421. - 426, Japanese

  [Refereed][Invited]

  Book review


• Scattering theory from a geometric view point (Regularity and Singularity for Geometric Partial Differential Equations and Conservation Laws)

  ITO Kenichi

  Kyoto University, Jul. 2013, RIMS Kokyuroku, 1845, 13 - 19, English


• Scattering theory from a geometric view point (Spectral and Scattering Theory and Related Topics)

  ITO Kenichi

  Kyoto University, Sep. 2011, RIMS Kokyuroku, 1760, 44 - 58, English


• Remarks on Fundamental Solutions to Schrödinger Equation with Variable Coefficients (Spectral and Scattering Theory and Related Topics)

  Ito Kenichi, Nakamura Shu

  Kyoto University, Jul. 2010, RIMS Kokyuroku, 1696, 1 - 8, English


• Remarks on Scattering on Scattering Manifolds (Spectral and Scattering Theory and Related Topics)

  NAKAMURA Shu, ITO Kenichi

  Kyoto University, Jul. 2008, RIMS Kokyuroku, 1607, 85 - 92, English


• On the propagation of the homogeneous wavefront set for Schrödinger equations and on the equivalence of the homogeneous and the qsc wavefront sets (Spectral and Scattering Theory and Related Topics)

  ITO Kenichi

  Kyoto University, Jul. 2006, RIMS Kokyuroku, 1510, 182 - 198, English

• Characterization of space-time singularities for the Schrödinger equation by initial state

  Kenichi Ito

  Himeji Conference on Partial Differential Equations, Mar. 2026, English

  [Invited]


• Stationary scattering theory for C^2 long-range potentials

  Kenichi Ito

  作用素論セミナー, Jun. 2025

  [Invited]


• Stationary scattering theory for C^2 long-range potentials

  Kenichi Ito

  神戸大学解析セミ ナー, May 2025

  [Invited]


• Schrödinger 方程式の解の時空間特異性

  伊藤健一

  神戸大学理学研究科数学教室談話会, Apr. 2025

  [Invited]


• Stationary scattering theory on manifold with ends (4回連続講演)

  ITO Kenichi

  Tokyo-Berkeley Mathematics Workshop "Partial Differential Equations and Mathematical Physics", Jan. 2017, English, Tokyo University, Tokyo University, International conference

  [Invited]

  Invited oral presentation


• Resolvent expansions for the Schrödinger operator on the discrete half-line

  ITO Kenichi

  HMAセミナー・冬の研究会 2017, Jan. 2017, Japanese, Hiroshima University, Hiroshima University, Domestic conference

  [Invited]

  Invited oral presentation


• Branching form of the resolvent at threshold for discrete Laplacians

  ITO Kenichi

  研究集会「第27回 数理物理と微分方程式」, Nov. 2016, Japanese, かんぽの宿 富山, Domestic conference

  Oral presentation


• Branching form of the resolvent at threshold for discrete Laplacians

  ITO Kenichi

  QMath13: Mathematical Results in Quantum Physics, Oct. 2016, English, Georgia Institute of Technology, International conference

  Oral presentation


• Branching form of the resolvent at threshold for discrete Laplacians

  ITO Kenichi

  Math/Phys Seminar, Aug. 2016, English, Aarhus University, International conference

  [Invited]

  Invited oral presentation


• Interpretation of complex resonances (and a remark on Rellich's theorem)

  ITO Kenichi

  RIMS共同研究「線形及び非線形分散型方程式に関する最近の進展」, May 2016, Japanese, Kyoto University, Kyoto University, Domestic conference

  [Invited]

  Invited oral presentation


• Asymptotic behavior of eigenfunctions on manifold with ends

  ITO Kenichi

  微分トポロジーセミナー, Nov. 2015, Japanese, 京都大学, Domestic conference

  [Invited]

  Invited oral presentation


• A remark on Rellich's theorem

  ITO Kenichi

  波動セミナー, Nov. 2015, Japanese, 北海道大学, Domestic conference

  [Invited]

  Invited oral presentation


• A remark on Rellich's theorem

  ITO Kenichi

  研究集会「第26回 数理物理と微分方程式」, Nov. 2015, Japanese, ニューサンピア姫路ゆめさき兵庫県姫路市, Domestic conference

  Oral presentation


• A complete classification of threshold properties for one-dimensional discrete Schrödinger operators

  ITO Kenichi

  Young Researchers Symposium, Jul. 2015, English, Santiago, Chile, International conference

  [Invited]

  Oral presentation


• A complete classification of threshold propertiesfor one-dimensional discrete Schrodinger operators

  ITO Kenichi

  Young Researchers Symposium, Jul. 2015, English, Santiago, Chile, International conference

  [Invited]

  Oral presentation


• Complete classification of threshold properties for one-dimensional discrete Schrödinger operators

  ITO Kenichi

  Topics in Analysis and Mathematical Physics,A conference in honor of Arne Jensen's 65th birthday, May 2015, English, Aalborg University, International conference

  [Invited]

  Invited oral presentation


• Complete classification of threshold propertiesfor one-dimensional discrete Schrodinger operators

  ITO Kenichi

  Topics in Analysis and Mathematical Physics,A conference in honor of Arne Jensen's 65th birthday, May 2015, English, Aalborg University, International conference

  [Invited]

  Invited oral presentation


• Stationary scattering theory on manifold with ends (LAP and RC)

  ITO Kenichi

  偏微分方程式姫路研究集会, Mar. 2015, English, イーグレ姫路, International conference

  [Invited]

  Invited oral presentation


• Stationary scattering theory on manifold with ends

  ITO Kenichi

  第7回名古屋微分方程式研究集会(The 7th Nagoya Workshop on Differential Equations), Mar. 2015, English, 名古屋大学, International conference

  [Invited]

  Invited oral presentation


• Stationary scattering theory on manifold with ends

  ITO Kenichi

  第3回 信州関数解析シンポジウム, Dec. 2014, Japanese, 信州大学, Domestic conference

  [Invited]

  Invited oral presentation


• Stationary scattering theory on manifold with ends

  ITO Kenichi

  熊本大学応用解析セミナー, Dec. 2014, Japanese, 熊本大学, Domestic conference

  [Invited]

  Invited oral presentation


• Stationary scattering theory on manifold with ends

  ITO Kenichi

  東京大学火曜解析セミナー, Nov. 2014, Japanese, 東京大学, Domestic conference

  [Invited]

  Invited oral presentation


• Stationary scattering theory on manifold with ends

  ITO Kenichi

  研究集会「第25回 数理物理と微分方程式」, Nov. 2014, Japanese, 神奈川県足柄下郡箱根町強羅 四季の湯強羅静雲荘, Domestic conference

  Oral presentation


• Stationary scattering theory on manifold with ends

  ITO Kenichi

  第20回南大阪応用数学セミナー, Oct. 2014, Japanese, 大阪市立大学, Domestic conference

  [Invited]

  Invited oral presentation


• Stationary scattering theory on manifold with ends

  ITO Kenichi

  NLPDEセミナー, Oct. 2014, Japanese, 京都大学, Domestic conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends

  ITO Kenichi

  RIMS 研究集会「スペクトル理論と その周辺(Spectral and Scattering Theory and Related Topics)」, Oct. 2014, Japanese, 京都大学, Domestic conference

  [Invited]

  Invited oral presentation


• Stationary scattering theory on manifold with ends（4回連続講演）

  ITO Kenichi

  RIMS共同研究「多様体・格子・グラフ上のシュレディンガー方程式のスペクトル・散乱理論」, Sep. 2014, Japanese, 京都大学, Domestic conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends

  ITO Kenichi

  RIMS研究集会「偏微分方程式に付随する確率論的問題」, Sep. 2014, Japanese, 京都大学, Domestic conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends

  ITO Kenichi

  Analysis Seminar, Aug. 2014, English, Aarhus University, International conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends

  ITO Kenichi

  偏微分方程式セミナー, Jul. 2014, Japanese, 北海道大学, Domestic conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends

  ITO Kenichi

  愛媛大学解析セミナー, Jun. 2014, Japanese, 愛媛大学, Domestic conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends

  ITO Kenichi

  大阪大学数学教室 微分方程式セミナー, May 2014, Japanese, 大阪大学, Domestic conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions and LAP on manifold with ends

  ITO Kenichi

  RIMS共同研究「線形および非線形分散型方程式に関する最近の進展」, May 2014, Japanese, 京都大学, Domestic conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions for the Schröodinger operator on manifold with ends

  ITO Kenichi

  神戸大学理学研究科数学教室談話会, Apr. 2014, Japanese, 神戸大学, Domestic conference

  [Invited]

  Invited oral presentation


• Absence of $B_0^*$-eigenfunctions for the Schr"odinger operator on manifold with ends

  ITO Kenichi

  神戸大学理学研究科数学教室談話会, Apr. 2014, Japanese, 神戸大学, Domestic conference

  [Invited]

  Invited oral presentation

• Mathematical Society of Japan

  - Present

• シュレーディンガー方程式の数学解析

  伊藤 健一

  日本学術振興会, 科学研究費助成事業, 基盤研究(C), 東京大学・神戸大学, Apr. 2023 - Mar. 2027, Principal investigator

  当該年度には主に，無限遠方でゆっくりと減衰する引力ポテンシャルを持つSchrodinger作用素を研究した．対応する古典力学系の性質から，0以上のすべてのエネルギーにおいて量子力学的粒子は散乱することが期待され，それを実証する形で，Rellichの定理，極限吸収原理，放射条件評価，Sommerfeldの一意性定理を得た．一般にエネルギー0周りでのSchrodinger作用素の解析は難しく，これまでの先行研究では超局所解析学の手法を改良をして適用するのが定石であった．本研究では，超局所解析学的手法の代わりに，研究代表者自身の考案した交換子法を適用する点が新しい．この交換子法は長い計算を必要とするが，その代わり初等的式変形のみで閉じており，これにより特に，ポテンシャルに対する滑らかさの仮定を自然な形にまで緩めることができた．また，得られた評価はエネルギー0周りで一様である．このような一様評価を得るには，対応する古典力学系の特徴を反映した適切なエスケープ関数を，エネルギーに依存する形で構成し，それに応じて，エネルギーに依存する関数空間を導入することが重要な鍵となる．この関数空間はエネルギー0において重みの次数が不連続に変化する特徴を持つ．以上の結果は，それら自身が興味の対象であるだけでなく，定常散乱理論を論じるための重要なステップにもなっている．散乱行列のエネルギー0における挙動には重要な情報が含まれているため，今後の応用が期待される．


• Geometric methods for the Schrödinger equation

  Kenichi Ito

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Kobe University / The University of Tokyo, Apr. 2017 - Mar. 2023, Principal investigator

  In this research project I applied a commutator method that I had invented to the Schrodinger operators with N-body, Stark or long-range potentials, and obtained a series of results including the stationary scattering theory. In addition, I completely classified threshold resonances of the Schrodinger operator on a graph with rays. Moreover, I obtained asymptotic expansions of the resolvent around thresholds for the Laplacian on the 2-dimensional square lattice.


• Spectral and scattering theory on geometric objects

  Kenichi Ito

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Young Scientists (B), University of Tuskuba / Kobe University, Apr. 2013 - Mar. 2017, Principal investigator

  I constructed a new general framework for the stationary scattering theory for the Schroedinger operator on a manifold with asymptotically Euclidean and/or hyperbolic funnel ends. I also succeeded in formulating the threshold resonances in a natural manner for the discrete Schroedinger operators on the discrete line and discrete half-line. Moreover, I obtained explicit expressions for the branching parts around thresholds for the resolvent of the discrete Laplacian on the square lattice.


• Singularities of the Schrödinger equation

  Kenichi Ito

  Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Young Scientists (B), University of Tsukuba, Apr. 2009 - Mar. 2013, Principal investigator

  I discussed the propagation of singularities and the spectral and scattering theory for the Schrodinger operator on a noncompact manifold with ends. Iextended results on the Euclidean space to those manifolds and studied what geometric structure is essentially needed for such analysis. I found that the existence of an endand its volume growth rate can be rephrased in terms of the existence of an unbounded convex function and its strength of convexity, respectively, and formulated geometrically, without coordinates, one model manifold on which methods of the Schrodinger operator theory applies.


• シュレーディンガー方程式による特異性の伝播について

  伊藤 健一

  日本学術振興会, 科学研究費助成事業, 特別研究員奨励費, 東京大学, Apr. 2006 - Mar. 2008, Principal investigator

  1.多様体上でのシュレーディンガー方程式 シュレーディンガー時間推進作用素による伝播速度は無限大である.それゆえ超局所的特異性を考察するうえでは,通常の波面集合だけでは不十分であり,無限遠方での増大度も合わせた形で特異性を考えなければならない.中村周教授(東京大学)は,遠方での増大度を自由系の時間推進作用素で再び元の波面集合型特異性に戻すというアイデアを用いて漸近的ユークリッド空間において時間推進作用素の特異性について研究した(2003年).しかしこのアイデアは多様体上ではある意味であまり有効ではない.この問題を,動径方向と方位角方向を完全に分離する新しい自由系を導入することにより回避し,散乱多様体上において特異性をとらえることに成功した.またさらに有限時間の波動作用素が,フーリエ積分作用素になることも得られた.これに続き,散乱多様体上でも適当なポテンシャル条件のもとで波動作用素の存在および完全性を証明することに成功した.すると上と同様の方法で(無限時間の)波動作用素がフーリエ積分作用素であることが分かる.さらに散乱作用素のフーリエ変換が,運動量の同位角成分が0となる部分の近くを除いて,フーリエ積分作用素になることも分かった. 2.エルミート固有関数の多重積積分評価 調和振動子ポテンシャルを持つシュレーディンガー作用素の固有関数の多重積積分を評価し,非線形問題に応用した.時間推進作用素をフーリエ積分作用素型の作用素で近似することで多重積積分を評価するのであるが,その際,コンパクト多様体上のラプラシアンの固有関数の場合と異なり,相関数を定義できない領域が相空間上に現れる.ここでは空間方向と運動量方向を等価にみることで,パラメトリックスを二つの部分に分けて議論し,結果を得ることに成功した.