Search

The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behaviour of the survival and transition amplitudes for the process of decay of the excited states of the charged particle, and the emission of photons in the decay process. In particular, for the survival amplitude the exact time behaviour is found. This is completely determined by the resonances of the systems plus a tail term prevailing in the asymptotic, long time regime. Moreover, the survival amplitude exhibites in a fairly clear way the Lamb shift correction to the unperturbed frequencies of the oscillator., Shortened version. To appear in J. Math. Phys

A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equations with their symmetries by using symplectic operators. It leads to a simple and sufficiently general model of integrable Hamiltonian equation, of which the Korteweg–de Vries equation, the modified Korteweg–de Vries equation, the nonlinear Schrodinger equation and the so‐called Harry Dym equation turn out to be particular examples.

A systematic approach to the study of nonlinear evolution equations based on the theory of the equivalence transformations is suggested. In this paper it is applied to the Burgers and to the Korteweg-de Vries equations. The main result is that the Hopf-Cole transformation for the Burgers equation and the Miura, Bäcklund, and Hirota transformations for the Korteweg-de Vries equation (together with the linear equations of the inverse scattering theory) are all deduced from a single general equivalence condition. Copyright © 1977 American Institute of Physics.