Український математичний конгрес - 2009

Елена Мищенко (Одесский государственный экологический университет, Одесса, Украина)

Quantization of quasistationary states for two-centre 3D-Schrodinger equation: New Numerical Approach

We develop a new method to calculation of the eigen-values and eigen functions for the two-centre 3D-Schrodinger equation and elaborate a new procedure for quantization of the quasistationary states for this equation. For the last purpose we use the operator perturbation theory formalism [1,2]. For numerical solution of the corresponding Schrödinger equation in the cylindrical coordinates we use the model potential method and finite differences numeral approach. Under the differences solution of the corresponding Schrödinger equation, an infinite region is exchanged by a grid (r; z). For z < 0 it is used an condition of the smallness of wave function on the boundary. For z > 0 the boundary condition has the form of plane divergent wave. It has sufficiently large size; inside it a rectangular uniform grid with steps h( r), h(z) is constructed. The eigen values of Hamiltonian are calculated by means of the inverse iterations method. The corresponding system of inhomogeneous equations is solved by the Thomas method. To calculate the width or energy the quasistationary state we use a formalism of operator perturbation theory [1]. We present some numerical examples, which demonstrate an efficiency of the proposed approach [3].

[1] Glushkov A.V., Relativistic Quantum Theory (Odessa, Astroprint, 2008).-900P.
[2] Glushkov A.V, in: New projects and new lines of research in nuclear physics. Eds. G. Fazio and F.Hanappe (Singapore, World Sci.) 126-142 (2003);
[3] Glushkov A.V., Mischenko E.V., etal, Int. J.Quant. Chem. 109,N13 (2009).