I am conducting a mathematical study on a class of systems comprising quantum mechanical particles interacting with a quantum field. A quantum system consisting of electromagnetic fields interacting with electrons is an example of such a system. In quantum mechanical models of such systems, electrons are treated as quantum mechanical particles obeying the Schrödinger operator or the Dirac operator, and electromagnetic fields are described in terms of the quantum field theory. By analyzing these models, one can observe several phenomena: Bare electrons are unstable in vacuum. Under favorable conditions, the stable electron exists in which a cloud of photons surrounds the bare electron. Under the influence of the Coulomb potential, excited electrons are unstable, and they fall into a lower orbit.

Other models that I have studied include, for example, a system of nucleons interacting with a scalar field or a model that describes the dynamics of electrons moving within a conductor. Many interesting phenomena have been observed in the case of each model.

The common settings of my research for these models are the following: First, I define the Hilbert space of the system. Then, I define a self-adjoint operator (called Hamiltonian) corresponding to the energy of the system. All the phenomena stated above are analyzed on the basis of the properties of the spectrum of the Hamiltonian and are proved mathematically. One of the important characteristic states is the ground state. Information regarding the existence or nonexistence of the ground state in models is important many times. In general, however, the eigenvalue equation for the quantum field models cannot be solved explicitly. How to prove the existence of a solution to the eigenvalue equation or how to extract the properties of the eigenvector (if they exist) is the main subject of my research.