D:\sbornik\...\rev.DVI


Mathematical Problems of Computer Science 31, 142{149, 2008.

N ew M athematical Appr oach for I nvestigation of

Statistical P r oper ties of Random E nvir onment of 1D

Quantum N-P ar ticles System I n E xter nal Field

Ashot S. Gevorkyan y and Arax A. Gevorkyan z

y Institue for Informatics and Automation Problems of NAS of RA
z Yerevan State University

E-mail: g ashot@sci.am, g arax@ipia.sci.am

Abstract

The investigation of 1D quantum N-particles system (PS) with relaxation in the
random environment under the in°uence of external ¯eld is conducted within the limits
of the stochastic di®erential equation (SDE) of Langevin-SchrÄodinger (L-Sch) type.
Using L-Sch equation the 2D second order non-stationary partial di®erential equation
is found, which describes the quantum distribution in the environment, depending on
energy of nonperturbed 1D quantum N-PS and on the external ¯eld's parameters.
It is shown that the average value of interaction potential between 1D disordered
quantum N -PS and on the external ¯eld, 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 ¯eld theory. Critical properties of environment are investigated in detail.
The possibility of ¯rst-order phase transition in environment depending on amplitude
of an external ¯eld is shown.

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