### Speaker

Dr
Nil Ratan Sarker
(JINR)

### Description

Success in solving multiparticle problems is in many cases connected with the choice of an adequate model. As a simple example, we can cite the concept of a polaron as a problem of an autolocalized electron in an ionic crystal. At present, there are a large number of physical examples [1-3], the theory of polarons, bipolarons, a strong-coupling binucleon model, a generalized polaron model, etc. The effect of autolocalization in liquids leads to the formation of solvated electrons in them, which play an important role in many chemical processes [4, 5]. Similar problems arise in the nonrelativistic potential model at the description of the spectrum of heavy quarkonia [6]. To study such problems, one can involve methods of self-consistent description of multiparticle systems.
A method is proposed for investigating the properties of solutions of a nonlinear self-consistent boundary value problem with increasing potentials of even and odd powers. A comparative analysis of the solutions of the linear boundary value problem for a quadratic growing potential with a nonlinear self-consistent boundary value problem for this potential is carried out. Formulas are obtained which allow us to calculate the shift of the eigenvalues. If the distances between the levels of a linear problem are equidistant, then in a self-consistent problem this property is also satisfied. In addition, when investigating problems with potentials above the quadratic one, new growing potentials appear to a lesser degree in the self-consistent problem than the original potential.
References
1. N.I. Kashirina, V.D. Lakhno, Mathematical modeling of autolocalized states in condensed media. Moscow: Fizmatlit. -2013. -292 p.
2. I.V. Amirkhanov, V.D. Lakhno, I.V. Puzynin, T.A. Strizh, V.K. Fedyayin. Numerical investigation of a nonlinear self-consistent eigenvalue problem in the generalized polaron model. Preprint, Biological Studies of the Academy of Sciences of the USSR, Pushchino. -1988. -23 p.
3. I.V. Amirkhanov, I.V. Puzynin, T.A. Strizh, V.D. Lakhno. Solution of LLP equations in bipolaron theory. Bulletin of the Academy of Sciences, series of physical. -V.59. -N.8. -1995. -P.106-110.
4. Thompson J. Electrons in liquid ammonia. Moscow: Mir. - 1979.-138 p.
5. I.V. Amirkhanov, I..V. Puzynin, T.A. Strizh, O.V. Vasilyev, V.D. Lakhno. Numerical study of a nonlinear self-consistent eigenvalue problem in the generalized model of a solvated electron. Preprint, Biological Studies of the USSR, Pushchino, -1990, -24 p.
6. A.A. Bykov, I.M. Dremin, A.V. Leonidov. -UFM. -V.143. -P.3. -1984.
The work was carried out at the financial support of the RFBR grant, No. 17-01-00661a.

### Primary author

Dr
Nil Ratan Sarker
(JINR)

### Co-author

Dr
Ilkizar Amirkhanov
(JINR)