Geometric analysis for unitary transition operators

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03267
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Tohoku University
• Principal Investigator: Tate Tatsuya 東北大学, 理学研究科, 教授 (00317299)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: ユニタリ推移作用素 / 量子ウォーク / 結晶格子 / 局在 / 一般固有関数展開

Outline of Final Research Achievements

Unitary transition operators are unitary operators defined on graphs with certain finite propagation property. Quantum walks are examples of unitary transition operators. But recently the word "quantum walk" means unitary transition operators in the above sense. In the first stage of the research, the aim was to study problems on geometrical deduction for weak limits of quantum walks on integer lattices, on semiclassical analysis and on quantum walks with singular continuous spectrum. Although the situation on these problems is still far from resolved, the researches in this program have made a certain contribution, in particular to the last problem, because the generalized eigenfunction expansion formula has been obtained. Many of previous researches on quantum walks were made for each individual models in case-by-case. But it will be expected that the generalized eigenfunction expansion formula will give a unified method to handle 1-dimensional quantum walks systematically.

Free Research Field

離散幾何解析

Academic Significance and Societal Importance of the Research Achievements

量子ウォークは量子論的な事象のコンピュータ・シミュレーションにしばしば応用される。したがって量子ウォークを定義するコイン行列と量子ウォークの挙動との理論上の関連を調べることは，応用に対する理論的な裏付けを与える重要な研究である。本研究においては，ある程度一般な１次元量子ウォークに対して，量子ウォークを簡単な作用素に変換するフーリエ変換の類似物，つまり一般固有関数展開定理，をコイン行列の言葉で書き下すことに成功した。さらにその計算に必要なレゾルベントに関する性質も導くことができた。今後は，これを用いてコイン行列と力学的挙動との関連を理論的かつ定量的に調べることが可能になるものと期待している。