Method of Solution of the Quantum-Mechanical Three-Body Problem

Publications, 30 April 2019

The JINR Publishing Department published the preprint «About One Method of Approximate-Analytical Solution of the Quantum-Mechanical Three-Body Problem» P11-2019-9. The authors are Amirkhanov I. V., Sarker N. R., Sarkhadov I.


A method is proposed for investigating the solutions of a single quantum-mechanical three-body problem. The boundary value problem for a system of two particles moving in the potential field of the third particle is studied. The interaction potentials between the particles are chosen to be quadratically growing. In this case, the two-body problem has an exact solution. The solution of the original three-body problem is represented as an expansion in the solutions of the two-body problem. The coefficients of the expansion yield a homogeneous linear matrix equation. Equating the determinant of this equation to zero, we find the eigenvalues of the original problem and the coefficients of the expansion. Since the elements of the matrix of algebraic equations with solutions of the two-body problem are calculated analytically, the eigenvalues of the three-body problem depend explicitly on the parameters of the potentials. Changing these parameters in an arbitrary way, we can obtain the desired spectra, i.e., one can control the eigenvalues of the original problem.