Doctor of science

Yes　Doctor

Analysis

0000-0002-3435-9957

02years06months

I am studying the spectral analysis of operators on an infinite dimensional space. Especially, from the mathematical standpoint, we investigate the quantum field theory on pseudo-Riemannian manifolds by using operator theory, micro-local analysis, theory of one-parameter semigroup, stochastic analysis, functional integral.

The Hamiltonian that appears in quantum field theory can be mathematically regarded as an unbounded self-adjoint operator on Hilbert space. The spectrum of the self-adjoint operator is analyzed nonperturbatively and abstractly. The Hamiltonian of quantum field theory has an eigenvalue embedded in a continuous spectrum when the coupling constant is zero. It has been found that unlike discrete eigenvalues, the analysis of embedded eigenvalues ​​is subtle, and is different from the behavior of discrete eigenvalues. Kato's regular theory exists in the perturbation theory of discrete eigenvalues, and very many things have been established. However, there is no general theory about our analysis of the perturbations of the eigenvalues.

The goal is to analyze the existence / absence of the Hamiltonian ground state and the degree of degeneracy, analysis of infrared divergence / ultraviolet divergence, spectral scattering theory, resonance, renormalization theory, Gibbs measure, etc. In recent years, we have also been studying CCR representation theory and lattice Schrodinger-operators.

Keywords:

Feynmann-Kac formula, Gibbs measure, renormalization, Boson-Fock space, Euclidean field, infinite-dimensional Ornstein-Uhlenbeck process, Levy process, subordinator, exponential decay, Gaussian decay, ultraviolet cut, infrared cut, ground state, self-adjoint operator, noncommutative Perron-Frobenius theorem, local convergence of measure, Kato potential, Bernstein function, Pauli-Fierz model, Nelson model, spin-boson model, non-commutative harmonic oscillator (NCHO), Rabi model, spectrum zeta function, time operator, CCR, weak Weyl relation, Laplacian on lattice, semi-classical analysis

Research Interests

• semiclassical analysis
  keyword : Wigner measure, Pauli-Fierz model
  2019.05～2022.12.
• Spectral analysis of QFT
  keyword : quantum field theory, Fock space, embedded eigenvalues, ground states, scattering theory, pseudo Riemann manifold, resonances, renormalization group, functional integrations, Gibbs measures
  1993.04.
• Feynman-Kac formula and related topics
  keyword : Schroedinger operator, relativistic Schroedinger operator, Schroedinger operator with spin, relativistic Schroedinger operator with spin, PF model, Nelson model, spin-boson model
  2006.04.
• Gibbs measure　and localizations
  keyword : Gibbs measure, ground states, UV renormalization, exponential decay
  2000.04.
• Spectral zeta functions
  keyword : Rabi model, non-commutative harmonic oscillator
  2013.04.
• Time operators and CCR representations
  keyword : time operator, CCR
  2009.04.
• Schrodinger operator on lattice
  keyword : lattice, spectrum
  2012.04.

Academic Activities Awards

• MSJ 2019 Analysis prize

Educational Activities