## Mental Enlightenment Scientific-Methodological Journal

#### Abstract

The study of spectral properties, in particular bound states, of multiparticle operators of quantum mechanics and solid state physics is closely related to the problem of solving integral equations with partial integrals (partial integral equations) for functions of three variables. In this article, we study the question of the existence and uniqueness of the solution of a linear partial integral equation for functions of three variables with degenerate kernels and many parameters in a complex Hilbert space. It is proved that under natural conditions, equation (1) has a unique solution, which is expressed through the data and their integrals of the considered equation. The general view of the solution is found. When solving the partial integral equation under study, the Fredholm method was developed for solving a linear integral equation of the second kind with a parameter.

#### First Page

40

#### Last Page

49

#### References

[1]. S.Albeverio, S.N. Lakaev, Zh.I. Abdullaev, The finiteness of the discrete spectrum of a four-particle operator on a lattice. Functional analysis and its applications. 36 (2002), no. 3, 56-60.

[2]. Abdus Salam, Fredholm Solution of Partial Integral Equations.oc. Cambridge Philos.Soc. 49 (1952), 213-217.

[3]. Lakaev SN, Soatov UA On the solution of a certain linear integral equation with partial integrals, Collection of articles. abstracts scientific conf. dedicated 75th anniversary of Tashkent State University, Tashkent, 1995. p. 165.

[4]. Soatov UA, On the solution of a certain linear integral equation with partial integrals. Uzbek Mathematical Journal, Tashkent, 1997, No. 2, pp. 72-79.

[5]. Lakaev S.N., Soatov U., Solition of partial integral equations for three variable functions, ICSTM-96, Samarkand, 1996, p. 62.

[6]. Lakaev SN, Soatov UA On the solution of a partial integral equation for functions of three variables, Dokl. AN RUz, Tashkent, 1997, No. 11, p. 8-10.

[7]. Merkuriev S.P., Faddeev L.D. "Quantum scattering theory for systems several particles ". Moscow," Science ", 1985.

[8]. Albeverio S., Gestesi F., Heeg-Kron Z., Holden H. Solvable models in quantum mechanics. Moscow, Mir. 1991.

[9]. Lakaev S.N. On an infinite number of three-particle bound states systems of three quantum lattice particles. Theoretical and Mathematical Physics, 89 (1991), No. 1, pp. 94-104.

[10]. Lakaev S.N. On the Efimov effect in a system of three identical quantum particles. Functional analysis and its applications, 27 (1993), issue 3, pp. 15-28.

#### Recommended Citation

Soatov, Ulugbek
(2021)
"ON THE SOLUTION OF A LINEAR PARTIALLY INTEGRAL EQUATION IN THE SPACE L2 ([a, b] 3),"
*Mental Enlightenment Scientific-Methodological Journal*: Vol. 2021:
Iss.
1, Article 5.

Available at:
https://uzjournals.edu.uz/tziuj/vol2021/iss1/5