| Project/Area Number |
13640100
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | University of Tsukuba |
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Principal Investigator |
MINAMI Nariyuki University of Tsukuba, Institute of Mathematics, Associated Professor, 数学系, 助教授 (10183964)
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| Co-Investigator(Kenkyū-buntansha) |
AOSHIMA Makoto University of Tsukuba, Institute of Mathematics, Assosiated Professor, 数学系, 助教授 (90246679)
MORITA Jun University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (20166416)
AKAHIRA Masafumi University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (70017424)
MIKAWA Hiroshi University of Tsukuba, Institute of Mathematics, Assistant Professor, 数学系, 講師 (10219602)
TASAKI Hiroyuki University of Tsukuba, Institute of Mathematics, Associated Professor, 数学系, 助教授 (30179684)
籠谷 恵嗣 筑波大学, 数学系, 助手 (40323258)
土居 伸一 筑波大学, 数学系, 助教授 (00243006)
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| Project Period (FY) |
2001 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2001: ¥2,100,000 (Direct Cost: ¥2,100,000)
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| Keywords | random Schrodinger operators / spectral statistics / branching processes / Galton-Watson trees / ランダムなシュレーディンガー作用素 / エネルギー準位統計 / 量子カオス / ランダム行列 / 分岐過程 |
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| Research Abstract |
1. The definition of one-dimensional Schrodinger operators with singular potentials and its application to random systems : In 1994, H.P. McKean considered the Schrodinger operator with white noise potential on a finite interval, and investigated the probability distribution of its first eigenvalue, but did not mention the fundamental question of the definition of operators with singular potentials like white noise. The formulation of a Schrodinger operator which has as its potential the formal derivative of a continuous function had been already given by Fukushima, Nakao and Minami. But each of their methods had some technical dificulty when applied to the present situation. In our study, we found that the recent concise formulation due to Savchuk and Shkalikov (1999) is effective for our purpose. In particular, their notion of "quasi-derivative "enabled us to prove the Sturm's oscillation theorem needed in filling the gap in McKean's theory. This work was done in collaboration with K. Nagai.
2. The distribution of the number of vertices of a Galton-Watson tree : As is shown e.g. the recent work by A. Khourunzhy (Adv. In Appl. Probab. Vol.33, No.l (2001) 124-140), one needs to count the number of vertices of random trees in order to study the fluctuation of the spectrum of random matrices. On the other hand, random trees are obtained as the trajectory of a Galton- Watson process (a discrete time branching process). We shall call this type of trees the Galton-Watson trees. Continuing the pioneering work of R. Otter (1949), we obtained some new results on the number of vertices of Galton- Watson trees.
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