Study of spectra of Laplacians on infinite graphs and BEC

2017 Fiscal Year Final Research Report

• Project/Area Number: 26800054
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Shinshu University
• Principal Investigator: SUZUKI Akito 信州大学, 学術研究院工学系, 准教授 (70585611)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: 無限グラフ / スペクトル / BEC / 離散ラプラシアン / 離散シュレーディンガー / 量子ウォーク / 結晶格子

Outline of Final Research Achievements

The spectrum of the discrete Lalacian on a graph describes the energy of a free particle moving in a substance and the Bose-Einstein condensation is a phenomenon that a macroscopic number of particles occupy the lowest energy state. In this case, the graph represents the structure of the substance. In this study, we have clarified the relation between the spectra of the Laplacians on infinite graphs and the shapes of the graphs. We have also given examples of graphs where BEC occurs. As by-products, we have revealed the structure of the discrete Schroedinger operator consisting of the Laplacian and an interaction term and that of a quantum walk, which is a quantum mechanical counterpart of a random walk. Moreover, we have proved the weak limit theorem, which gives the long-time limit distribution of the quantum walk.

Free Research Field

数理物理