Review of Investigations on the nonrelativistic quantum field theory
1990-1991. Pecar equation for a polaron and the equations for the F-center (describing electron behaviour in polar media) were studied numerically. In the former case we wanted to find solutions to the polaron equation different from spherically-symmetric ones (which are known). In the latter case our aim was to find spherically-symmetric solutions to the F-center equation on rather a large interval (so that we might study the asymptotical behaviour of the solutions at large distances from the coulomb center).
We suggested a procedure for approximation of non-spherically symmetric solutions to the polaron equation (small-dimension approximation – expansion of solutions in terms of spherical functions with distance dependent coefficients). Then the finding of solutions is reduced to the system (theoretically – infinite) of ordinary differential equations. Finite systems were solved with the help of the software package COLCON which makes possible rather an exact approximation of several non-spherically symmetrical solutions. A finite dimensional approximation (less satisfactory) was also considered. As regards the F-center equation, we revealed that the corresponding boundary problem can be solved with the use of the same software package COLCON. In this case both the solutions and their asymptotics can be found rather exactly.
1991-1993. We found several first non-spherical solutions of the polaron equation, known also as Hartri equation. To obtain non-spherical solutions we used the method of small-dimension approximations; the initial nonlinear equation was reduced to a system of nonlinear ordinary differential equations which subsequently were numerically integrated. The results of these investigations are partially presented in /29/ .
As regards the general problems of the polaron theory, we showed that the instability of the excited polaron state is associated with branching of the solutions to the nonlinear polaron equation. Numerical calculations made it clear that the instability leads to a decay of the excited spherically-symmetrical state, giving rise to new symmetrical states. This effect is similar to a pseudo Yan-teller effect and can be observed experimentally.
A theory of bound phonons was developed with due regard for their dispersion ( /28/ ). The results obtained are of importance for the study of dynamical properties of a crystal lattice.
1993-1997. Quantum-field theory of a deuteron. Close ideological relationship between the theory of many particles in a solid body and nuclear physics enables one to transfer the methods successfully employed in one field into the other. The basic problem of nuclear physics, i.e. description of a nucleon in a meson field is similar to the polaron problem, i.e. interaction of an electron with a phonon field. The fundamental difficulty, which arises in the simplest case of the interaction of a nonrelativistic nucleon with a scalar meson field upon the application of the perturbation theory is the ultraviolet divergence.
In the works /34/ , /35/ , /36/ and /44/ we developed a consistent scheme of the strong coupling theory for a nucleon in a meson field which is free of unavoidable divergences. It was shown that in the case of two nucleons in a meson field the problem is reduced to a one-particle one (1.2) with potential U(r) in the form:
| (2.1) |  |
The solution to the nonlinear equation (1.2) with potential (2.1) yields the coupling energy, radius and mass of a deuteron close to experimental values. The results obtained can provide a basis for the development of a nonlinear quantum field theory of nucleons interacting with a meson field.
1994-1997. A quantum-field theory of a deuteron was developed for the case of a pseudoscalar coupling of a nucleon with a meson field. In the limit of strong coupling of the particle with the field we derived an equation for the deuteron coupling energy for scalar and pseudoscalar interactions. The theory yields correct values of the isospin of a deuteron and enables one to develop a consistent theory of a quadrupole momentum of a deuteron.