The Calculation of Values of Eigenfunctions of the Perturbed Self-Adjoint Operators by Regularized Traces Method

Abstract

The authors developed a numerical non-iterative method of finding of the value of eigenfunctions of perturbed self-adjoint operators, which was called the method of regularized traces. It allows to find the value of eigenfunctions of perturbed discrete operators, using the spectral characteristics of the unperturbed operator and the eigenvalues of the perturbed operator. In contrast to the known methods, in the method of regularized traces the value of eigenfunctions are found by the linear equations. It significantly increases the computational efficiency. The difficulty of the method is to find sums of functional series of suspended corrections of perturbation theory, which can be found only numerically. The formulas, which are convenient to find suspended corrections such that one can approximate the amount of these functional series by summing up of them, are presented in the paper. However, if a norm of the perturbing operator is large, then the summation of suspended corrections can be not effective. We obtain analytical formulas, which allow to find the values of sums of functional series of suspended corrections of perturbation theory in the discrete nodes without direct summation of its terms. Computational experiments are performed. These experiments allowed to find the values of the eigenfunctions of the perturbed one-dimensional Laplace operator. The experimental results showed the accuracy and computational efficiency of the developed method.