New Mathematical Approach for Investigation of Statistical Properties of Random Environment of 1D Quantum N-Particles System In External Field

Abstract

The investigation of 1D quantum N-particles system (PS) with relaxation in the random environment under the influence of external field is conducted within the limits of the stochastic differential equation (SDE) of Langevin-Schrödinger (L-Sch) type. Using L-Sch equation the 2D second order non-stationary partial differential equation is found, which describes the quantum distribution in the environment, depending on energy of nonperturbed 1D quantum N-PS and on the external field's parameters. It is shown that the average value of interaction potential between 1D disordered quantum N-PS and on the external field, has the ultraviolet divergence. This problem is solved by renormalization of equation for the function of quantum distribution. It is shown that it has a sense of dimensional renormalization which is characteristic for the quantum field theory. Critical properties of environment are investigated in detail. The possibility of first-order phase transition in environment depending on amplitude of an external field is shown.