2002 Fiscal Year Annual Research Report

量子ホワイトノイズと非線形無限次元解析

Research Project

Project/Area Number	12440036
Research Institution	Tohoku University
Principal Investigator	尾畑伸明東北大学, 大学院・情報科学研究科, 教授 (10169360)
Co-Investigator(Kenkyū-buntansha)	齊藤公明名城大学, 理工学部, 教授 (90195983) 浦川肇東北大学, 大学院・情報科学研究科, 教授 (50022679) 日合文雄東北大学, 大学院・情報科学研究科, 教授 (30092571) 田崎秀一早稲田大学, 理工学部, 助教授 (10260150) 有光敏彦筑波大学, 物理学系, 教授 (50134200)
Keywords	ホワイトノイズ理論 / 量子確率論 / 量子ホワイトノイズ / 量子分解 / 量子中心極限定理 / 複素スペクトル / レヴィラプラシアン / 散逸量子系
Research Abstract	無限次元解析・量子確率論における最近の発展を取り入れながら、未解決問題へのアプローチを図り、応用面における課題を探索した。また、仙台ワークショップ「量子確率と量子情報」を主催するとともに、内外から研究者を招聘しセミナーと共同研究を実施した。 1.正規積ホワイトノイズ方程式。従来の伊藤型方程式を著しく拡張し、その解の存在・一意性・滑らかさを詳細に調べるために、シンボルの理論を2つの方向で深化させた。第一は、核型性に基づく無限次元正則関数の空間に帰着させる特徴づけ定理であり、第二は複素ホワイトノイズを用いる新しい方法である。 2.ホワイトノイズの高次冪とレヴィラプラシアン。ホワイトノイズの2乗とレヴィラプラシアンの関連を明らかにし、レヴィラプラシアンに付随する確率過程を直積分によって構成した。 3.複素ホワイトノイズ。シーガル・バーグマン同型を再考し、核型性に依存しない作用素シンボルの特徴づけ定理を証明した。これによって、作用素のユニタリ性の判定条件が得られ、正規積ホワイトノイズ方程式の解がユニタリになるための必要十分条件を導いた。 4.物理への応用。散逸過程を記述する種々の方程式の相互関係を組織的に研究したが、物理特有の形式的議論を超える数理的諸問題が残された。量子ホワイトノイズをフォック空間系列に基づいて議論し、特異なボゴリューボフ変換が現れるのを確認した。 5.代数的確率論からの視点。新しい概念である「量子分解」によるグラフの隣接作用素の漸近的スペクトル解析を確立させた。新しい例を構成し、極限分布のみたす必要条件を導いた。この制限を乗り越えて、q-変形ガウス分布を議論するための予備的考察を行うとともに、別のタイプの極限定理も証明した。

Research Products

(37 results)

All Other

All Publications (37 results)

[Publications] D.M.Chung, U.C.Ji, N.Obata: "Quantum stochastic analysis via white noise operators in weighted Fock space"Rev. Math. Phys.. 14. 241-272 (2002)
[Publications] N.Obata: "Inverse S-transform, Wick product and overcompleteness of exponential vectors"Quantum Information. 4. 147-176 (2002)
[Publications] H.-H.Kuo, N.Obata, K.Saito: "Diagonalization of the Levy Laplacian and related stable processes"Infin. Dimen. Anal. Quantum Probab. Rel. Top.. 5. 317-331 (2002)
[Publications] U.C.Ji, N.Obata, H.Ouerdiane: "Analytic characterization of generalized Fock space operators as two-variable entire functions with growth condition"Infin. Dimen. Anal. Quantum Probab. Rel. Top.. 5. 395-407 (2002)
[Publications] U.C.Ji, N.Obata: "Quantum white noise calculus"Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads. 143-191 (2002)
[Publications] N.Obata, K.Saito: "Cauchy processes and the Levy Laplacian"Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads. 360-373 (2002)
[Publications] U.C.Ji, N.Obata: "Segal-Bargmann transform of white noise operators and white noise differential equations"京都大学数理解析研究所講究録. 1266. 59-81 (2002)
[Publications] 尾畑伸明: "量子分解法によるスペクトル解析"物性研究. 77. 784-805 (2002)
[Publications] N.Obata: "Quantum white noise calculus based on nuclear algebras of entire functions"京都大学数理解析研究所講究録. 1278. 130-157 (2002)
[Publications] 尾畑伸明, 洞彰人: "量子分解法によるグラフ上のスペクトル解析I:個数作用素が現れない場合"京都大学数理解析研究所講究録. 1291. 11-44 (2002)
[Publications] Y.Hashimoto, A.Hora, N.Obata: "Central limit theorems for large graphs : Method of quantum decomposition"J. Math. Phys.. 44. 71-88 (2003)
[Publications] U.C.Ji, N.Obata: "A unified characterization theorem in white noise theory"Infin. Dimen. Anal. Quantum Probab. Rel. Top. (印刷中). (2003)
[Publications] U.C.Ji, N.Obata: "A role of Bargmann--Segal spaces in characterization and expansion of operators on Fock space"J. Math. Soc. Japan. (掲載決定). (2003)
[Publications] F.Hiai, X.Zhan: "Inequalities involving unitarily invariant norms and operator monotone functions"Linear Algebra Appl.. 341. 151-169 (2002)
[Publications] F.Hiai, M.Mizuo, D.Petz: "Free relative entropy for measures and a corresponding perturbation theory"J. Math. Soc. Japan. 54. 679-718 (2002)
[Publications] F.Hiai, Y.Ueda: "Automorphisms of free product-type and their crossed-products"J. Operator Theory. (掲載決定). (2003)
[Publications] H.Urakawa: "Graph theory"Selected Topics in Geometry and Mathematical Physics. 1. 189-237 (2002)
[Publications] H.Urakawa: "The Cheeger constant, the heat kernel, and the Green kernel of an infinite graph"Monatsh. Math.. (印刷中). (2002)
[Publications] 浦川肇: "スペクトル幾何学とグラフ理論"応用数理. 12. 29-45 (2002)
[Publications] N.Matsuura, H.Urakawa: "Discrete improper affine sphere"J. Geom. Phys.. 45. 164-183 (2003)
[Publications] S.Dragomir, T.Ichiyama, H.Urakawa: "Yang--Mills theory and conjugate connections"Diff. Geom. Appl.. 18. 229-238 (2003)
[Publications] A.Ishikawa, K.Saito, A.H.Tsoi: "Poisson noise analysis based on the Levy Laplacian"Quantum Information. 4. 103-114 (2002)
[Publications] H.-H.Kuo, K.Saito, A.Stan: "A Hausdorff-Young inequality for white noise analysis"Quantum Information. 4. 115-126 (2002)
[Publications] K.Saito: "Infinite dimensional Cauchy process associated with the Levy Laplacian"京都大学数理解析研究所講究録. 1278. 172-182 (2002)
[Publications] T.Arimitsu, N.Arimitsu: "Multifractal analysis of turbulence by using statistics based on non-extensive Tsallis' or extensive Renyi's entropy"J.Korean Phys. Soc.. 40. 1032-1036 (2002)
[Publications] T.Arimitsu, N.Arimitsu: "PDF of velocity fluctuation in turbulence by a statistics based on generalized entropy"Physica A. 305. 218-226 (2002)
[Publications] T.Arimitsu, N.Arimitsu: "Analysis of velocity fluctuation in turbulence based on generalized statistics"J. Phys. : Condens. Matter. 14. 2237-2246 (2002)
[Publications] T.Arimitsu, N.Arimitsu: "Analysis of velocity derivatives in turbulence based on generalized statistics"Europhys. Lett.. 60. 60-65 (2002)
[Publications] S.Tasaki: "Thermodynamic behaviors of large dynamical systems -Quantum 1d conductor and classical multibaker map"Lecture Notes in Physics. 597. 395-412 (2002)
[Publications] S.Tasaki, P.Gaspard: "Spectral properties of a piecewise linear intermittent map"J. Stat. Phys.. 109(印刷中). (2002)
[Publications] S.Tasaki: "Irreversibility in reversible multibaker maps"Adv. Chem. Phys.. 122. 70-107 (2002)
[Publications] H.Honda, S.Tasaki, A.Chiba, H.Ogura: "Relaxation phenomenon measured as dynamical specific heat in first-order phase transition of molecular crystal"Phys. Rev. B. 65(Art.No.104112). (2002)
[Publications] H.Honda, H.Ogura, S.Tasaki, A.Chiba: "Two-phase coexisting state of n-hexatriacontane in the first-order phase transition"Thermochimica Acta. (印刷中). (2003)
[Publications] N.Obata, T.Matsui, A.Hora(編集): "Non-Commutativity, Infinite-Dimensionality and probability at the Crossroads"World Scientific. 438 (2002)
[Publications] T.Nakamura, N.Obata(編集): "Exploring the Human Brain, Mind and Information Technology"Tohoku University. 220 (2002)
[Publications] L.Accardi, S.Tasaki(編集): "Fundamental Problems in Quantum Mechanics"World Scientific(印刷中). (2003)
[Publications] T.Hida, K.Saito(編集): "Quantum Information IV"World Scientific. 200 (2002)

2002 Fiscal Year Annual Research Report

量子ホワイトノイズと非線形無限次元解析

Principal Investigator

尾畑 伸明 東北大学, 大学院・情報科学研究科, 教授 (10169360)

Research Products

[Publications] D.M.Chung, U.C.Ji, N.Obata: "Quantum stochastic analysis via white noise operators in weighted Fock space"Rev. Math. Phys.. 14. 241-272 (2002)

[Publications] N.Obata: "Inverse S-transform, Wick product and overcompleteness of exponential vectors"Quantum Information. 4. 147-176 (2002)

[Publications] H.-H.Kuo, N.Obata, K.Saito: "Diagonalization of the Levy Laplacian and related stable processes"Infin. Dimen. Anal. Quantum Probab. Rel. Top.. 5. 317-331 (2002)

[Publications] U.C.Ji, N.Obata, H.Ouerdiane: "Analytic characterization of generalized Fock space operators as two-variable entire functions with growth condition"Infin. Dimen. Anal. Quantum Probab. Rel. Top.. 5. 395-407 (2002)

[Publications] U.C.Ji, N.Obata: "Quantum white noise calculus"Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads. 143-191 (2002)

[Publications] N.Obata, K.Saito: "Cauchy processes and the Levy Laplacian"Non-Commutativity, Infinite-Dimensionality and Probability at the Crossroads. 360-373 (2002)

[Publications] U.C.Ji, N.Obata: "Segal-Bargmann transform of white noise operators and white noise differential equations"京都大学数理解析研究所講究録. 1266. 59-81 (2002)

[Publications] 尾畑伸明: "量子分解法によるスペクトル解析"物性研究. 77. 784-805 (2002)

[Publications] N.Obata: "Quantum white noise calculus based on nuclear algebras of entire functions"京都大学数理解析研究所講究録. 1278. 130-157 (2002)

[Publications] 尾畑伸明, 洞彰人: "量子分解法によるグラフ上のスペクトル解析I:個数作用素が現れない場合"京都大学数理解析研究所講究録. 1291. 11-44 (2002)

[Publications] Y.Hashimoto, A.Hora, N.Obata: "Central limit theorems for large graphs : Method of quantum decomposition"J. Math. Phys.. 44. 71-88 (2003)

[Publications] U.C.Ji, N.Obata: "A unified characterization theorem in white noise theory"Infin. Dimen. Anal. Quantum Probab. Rel. Top. (印刷中). (2003)

[Publications] U.C.Ji, N.Obata: "A role of Bargmann--Segal spaces in characterization and expansion of operators on Fock space"J. Math. Soc. Japan. (掲載決定). (2003)

[Publications] F.Hiai, X.Zhan: "Inequalities involving unitarily invariant norms and operator monotone functions"Linear Algebra Appl.. 341. 151-169 (2002)

[Publications] F.Hiai, M.Mizuo, D.Petz: "Free relative entropy for measures and a corresponding perturbation theory"J. Math. Soc. Japan. 54. 679-718 (2002)

[Publications] F.Hiai, Y.Ueda: "Automorphisms of free product-type and their crossed-products"J. Operator Theory. (掲載決定). (2003)

[Publications] H.Urakawa: "Graph theory"Selected Topics in Geometry and Mathematical Physics. 1. 189-237 (2002)

[Publications] H.Urakawa: "The Cheeger constant, the heat kernel, and the Green kernel of an infinite graph"Monatsh. Math.. (印刷中). (2002)

[Publications] 浦川 肇: "スペクトル幾何学とグラフ理論"応用数理. 12. 29-45 (2002)

[Publications] N.Matsuura, H.Urakawa: "Discrete improper affine sphere"J. Geom. Phys.. 45. 164-183 (2003)

[Publications] S.Dragomir, T.Ichiyama, H.Urakawa: "Yang--Mills theory and conjugate connections"Diff. Geom. Appl.. 18. 229-238 (2003)

[Publications] A.Ishikawa, K.Saito, A.H.Tsoi: "Poisson noise analysis based on the Levy Laplacian"Quantum Information. 4. 103-114 (2002)

[Publications] H.-H.Kuo, K.Saito, A.Stan: "A Hausdorff-Young inequality for white noise analysis"Quantum Information. 4. 115-126 (2002)

[Publications] K.Saito: "Infinite dimensional Cauchy process associated with the Levy Laplacian"京都大学数理解析研究所講究録. 1278. 172-182 (2002)

[Publications] T.Arimitsu, N.Arimitsu: "Multifractal analysis of turbulence by using statistics based on non-extensive Tsallis' or extensive Renyi's entropy"J.Korean Phys. Soc.. 40. 1032-1036 (2002)

[Publications] T.Arimitsu, N.Arimitsu: "PDF of velocity fluctuation in turbulence by a statistics based on generalized entropy"Physica A. 305. 218-226 (2002)

[Publications] T.Arimitsu, N.Arimitsu: "Analysis of velocity fluctuation in turbulence based on generalized statistics"J. Phys. : Condens. Matter. 14. 2237-2246 (2002)

[Publications] T.Arimitsu, N.Arimitsu: "Analysis of velocity derivatives in turbulence based on generalized statistics"Europhys. Lett.. 60. 60-65 (2002)

[Publications] S.Tasaki: "Thermodynamic behaviors of large dynamical systems -Quantum 1d conductor and classical multibaker map"Lecture Notes in Physics. 597. 395-412 (2002)

[Publications] S.Tasaki, P.Gaspard: "Spectral properties of a piecewise linear intermittent map"J. Stat. Phys.. 109(印刷中). (2002)

[Publications] S.Tasaki: "Irreversibility in reversible multibaker maps"Adv. Chem. Phys.. 122. 70-107 (2002)

[Publications] H.Honda, S.Tasaki, A.Chiba, H.Ogura: "Relaxation phenomenon measured as dynamical specific heat in first-order phase transition of molecular crystal"Phys. Rev. B. 65(Art.No.104112). (2002)

[Publications] H.Honda, H.Ogura, S.Tasaki, A.Chiba: "Two-phase coexisting state of n-hexatriacontane in the first-order phase transition"Thermochimica Acta. (印刷中). (2003)

[Publications] N.Obata, T.Matsui, A.Hora(編集): "Non-Commutativity, Infinite-Dimensionality and probability at the Crossroads"World Scientific. 438 (2002)

[Publications] T.Nakamura, N.Obata(編集): "Exploring the Human Brain, Mind and Information Technology"Tohoku University. 220 (2002)

[Publications] L.Accardi, S.Tasaki(編集): "Fundamental Problems in Quantum Mechanics"World Scientific(印刷中). (2003)

[Publications] T.Hida, K.Saito(編集): "Quantum Information IV"World Scientific. 200 (2002)

尾畑伸明東北大学, 大学院・情報科学研究科, 教授 (10169360)

[Publications] 浦川肇: "スペクトル幾何学とグラフ理論"応用数理. 12. 29-45 (2002)