# 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.

## References

A.S. Gevorkyan and Chin-Kun Hu, On a mathematical approach for the investigation of some statistical phenomena of a diserdored 3D spin system in the external field. Proceedings of the ISAAC Conf. on Analysis, Yerevan, Armenia, Eds. by G.A. Barsegian et al., 165-178, 2004.

A.V. Bogdanov, A.S. Gevorkyan, A.G. Grigoryan, AMS/IP Studies in Advanced Mathematics, 13, 81, 1999.

I.M. Lifshits, S.A. Gredeskul and L.A. Pastur, Introduction to the theory of disordered systems. Moscow, Nauka, (in Russian) 1982.

A.S. Gevorkyan, Exactly solvable models of stochastic quantum mechanics within the framework of Langevin-Schreodinger type equation, Analysis and applications. Proceeding of the NATO Advanced research workshop, Yerevan 2002, Eds. by G.A. Barsegian and H. Begehr, NATO Science publications, 415-442, Kluwer, 2004.

V.I. Klyatskin, Statistical description of dynamical systems with fluctuating parameters. Moscow, Nauka, (in Russian) 1975.

A.N. Vasil'ev, The Quantum-field Renorm group in Theory of Critical Behaviour and of Stochastic Dynamics. Publishing house PINF, St. Petersburg (in Russian) 1998.

M.V. Fedoryuk, Method of Saddle Points, Publisher "Nauka" (in Russian) 1977.

## Downloads

## Published

## How to Cite

*Mathematical Problems of Computer Science*,

*31*, 142–149. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/401

## Issue

## Section

## License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.