A new construction of solution of Schrodinger equation on the basis of atomic functions

Abstract

Nonlinear Schrodinger equation is used to describe wave processes in many areas of physics such as quantum mechanics, nonlinear optics, theory of solitons, biophysics. There are many different methods and numerical algorithms to approximation solutions of nonlinear Schrodinger equations from simple giving satisfactory results over a sufficiently large number of operations to complex. At present solutions of boundary value problems of mathematical physics are widely used theory of atomic functions (AF). This solution is represent as a sum of shift-scaled AF families. AFs are infinitely differentiable functions and solutions of functional differential equations (FDE). They stay an intermediate position between the splines and trigonometric and algebraic polynomials. They are smoother than splines but less smooth than the polynomial. In this work we construct AFs shift-contraction of which constitutes the solution of Schrodinger equations. On the first stage the operator method solution of FDE is considered. On the second step AF building of N-variables generated by the Schrodinger equation is carried out.