Spectral analysis and eigenvalue problems for relativistic operators

2017 Fiscal Year Final Research Report

• Project/Area Number: 26400175
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: University of Hyogo
• Principal Investigator: Umeda Tomio 兵庫県立大学, 物質理学研究科, 教授 (20160319)
• Co-Investigator(Kenkyū-buntansha): 山岸 弘幸 東京都立産業技術高等専門学校, ものづくり工学科, 准教授 (10448053)
• Research Collaborator: Schmidt Karl M. ; Richard Serge ; UCHIYAMA Jun
• Project Period (FY): 2014-04-01 – 2018-03-31
• Keywords: スペクトル解析 / ディラック作用素 / シュレディンガー作用素 / 相対論的シュレディンガー作用素 / 固有値問題

Outline of Final Research Achievements

This project is about spectral analysis of relativistic operators, which plays important roles in theoretical investigation of quantum mechanics, the essence of modern sciences and engineerings. In particular, focused are low energy states of quantum particles under relativistic effect.
Dirac and relativistic Schroedinger operators, each of which is a relativistic modification of the Schroedinger operator, describing energy states of particles, are examined to explore their energy spectra. It was shown that the spectrum of massless Dirac operators with "any" potentials has no gap. As for relativistic Schroedinger operators, it was obtained that whether the zero energy is an eigenvalue (the energy state of the particle bound to the nucleus) or not is dominant in the states of the particle with low energy.

Free Research Field

スペクトル理論