Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia1789310.22363/2312-9735-2018-26-1-49-57Research ArticleOn a Method of Investigation of the Self-Consistent NonlinearBoundary-Value Problem for Eigen-Valueswith Growing PotentialsAmirkhanovI V<p>Amirkhanov I. V. - Senior Researcher, Candidate of Physical and Mathematical Sciences, Head of Sector “Scientiﬁc Division of Computational Physics”. Laboratory of Information Technologies of the Joint Institute for Nuclear Research, Dubna</p>camir@jinr.ruSarkerN R<p>Sarker Nil Ratan - Candidate of Physical and Mathematical Sciences, Senior Researcher “Scientiﬁc Division of Computational Physics”. Laboratory of Information Technologies of the Joint Institute for Nuclear Research, Dubna</p>sarker@jinr.ruJoint Institute for Nuclear Research15122018261495728022018Copyright © 2018, Amirkhanov I.V., Sarker N.R.2018<p>One of the most common methods for investigating multiparticle problems in the frameworkof the variational approach is the transition to a nonlinear one-particle problem by introducinga self-consistent ﬁeld that depends on the states of these particles. The paper considers anonlinear boundary value eigenvalue problem for the Schrodinger equation with a growingpotential including a dependence on the wave function and a power dependence on the coordinate = where = 1,2,3.... For n = 2, the boundary value problem for the Schrodinger equation(linear problem) has an exact solution. For even powers of , it is shown that solutions of sucha problem can be expressed in terms of solutions corresponding to the linear problem, and for= 2 the solution can be obtained in explicit form. The set of solutions obtained for= 2 ischaracterized by equal distances between neighboring eigenvalues. It is shown that the solutionof the nonlinear problem diﬀers from the solution of the linear problem by the shift of theeigenvalues. In the case of a potential higher than the quadratic one, new growing potentialsof a lesser degree appear. For the case of odd values of, the transition is discussed, from theintegro-diﬀerential formulation of the problem to a system of diﬀerential equations which can besolved numerically on the basis of the method of successive approximations, which has provedits eﬀectiveness in the study of the polaron model.</p>self-localizationeigenvaluespolarongrowing potentialsnon-linear boundary value problemавтолокализациясобственные значенияполяронрастущие по-тенциалынелинейная краевая задача[S. I. Pekar, Studies on the Electronic Theory of Crystals, GITGL, Moscow, 1951, in Russian.][N. I. Kashirina, V. D. Lakhno, Mathematical Modeling of Autolocalized States in Condensed Media, Fizmatlit, Moscow, 2013, in Russian.][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, the series of physical 59 (8) (1995) 106–110, in Russian.][I. V. Amirkhanov, E. V. Zemlyanaya, V. D. Lakhno, I. V. Puzynin, T. P. Puzynina, T. A. Strizh, Numerical Investigation of the Quantum Field Model of the Strong- Binding Binucleon, Mathematical Modelling 8 (1997) 51–59, in Russian.][I. V. Amirkhanov, V. D. Lakhno, I. V. Puzynin, T. A. Strizh, V. K. Fedyanin, Numerical Study of a Nonlinear Self-Consistent Eigenvalue Problem in the Generalized Polaron Model, 1988, in Russian.][J. Thompson, Electrons in Liquid Ammonia, Mir, Moscow, 1979, in Russian.][I. V. Amirkhanov, I. V. Puzynin, T. A. Strizh, O. V. Vasilyev, V. D. Lakhno, Numerical Investigation of a Nonlinear Self-Consistent Eigenvalue Problem in the Generalized Model of a Solvated Electron, 1990, in Russian.][V. D. Lakhno, A. V. Volokhova, E. V. Zemlyanaya, I. V. Amirkhanov, I. V. Puzynin, T. P. Puzynina, Polaron Model of the Formation of Hydrated Electron States, Surface. X-ray, synchrotron and neutron studies (1) (2015) 1–6, in Russian.][A. A. Bykov, I. M. Dremin, A. V. Leonidov, Potential models of quarkonium, Successes of Physical Sciences 143 (1984) 3, in Russian.][I. V. Amirkhanov, E. V. Zemlyanaya, I. V. Puzynin, T. P. Puzynina, T. A. Strizh, On Some Problems of Numerical Investigation of the Eigenvalue Problem in the Momentum Representation, Mathematical modeling 9 (10) (1997) 111–119, in Russian.][I. V. Amirkhanov, E. V. Zemlyanaya, I. V. Puzynin, T. P. Puzynina, T. A. Strizh, Numerical Investigation of Relativistic Equations for Bound States with Coulomb and Linear Potentials, Math modeling 12 (12) (2000) 79–96, in Russian.][D. Potter, Computational Methods in Physics, Mir, Moscow, 1975, in Russian.][I. V. Amirkhanov, et al., Numerical Study of the Dynamics of Polaron States, Bulletin of Tver University. Series: Applied Mathematics (17) (2009) 5–14, in Russian.]