Geodesic
The calculation of values of eigenfunctions of the perturbed self-adjoint operators by regularized traces method
Journal of computational and engineering mathematics, Tome 2 (2015) no. 4, pp. 48-60.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: method of regularized traces, perturbed operators, eigenvalues, eigenfunctions, multiple spectrum, «suspended» corrections of perturbation theory. 
@article{JCEM_2015_2_4_a4,
     author = {S. N. Kakushkin and S. I. Kadchenko},
     title = {The calculation of values of eigenfunctions of the perturbed self-adjoint operators by regularized traces method},
     journal = {Journal of computational and engineering mathematics},
     pages = {48--60},
     publisher = {mathdoc},
     volume = {2},
     number = {4},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JCEM_2015_2_4_a4/}
}
TY  - JOUR
AU  - S. N. Kakushkin
AU  - S. I. Kadchenko
TI  - The calculation of values of eigenfunctions of the perturbed self-adjoint operators by regularized traces method
JO  - Journal of computational and engineering mathematics
PY  - 2015
SP  - 48
EP  - 60
VL  - 2
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JCEM_2015_2_4_a4/
LA  - en
ID  - JCEM_2015_2_4_a4
ER  - 
%0 Journal Article
%A S. N. Kakushkin
%A S. I. Kadchenko
%T The calculation of values of eigenfunctions of the perturbed self-adjoint operators by regularized traces method
%J Journal of computational and engineering mathematics
%D 2015
%P 48-60
%V 2
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JCEM_2015_2_4_a4/
%G en
%F JCEM_2015_2_4_a4
S. N. Kakushkin; S. I. Kadchenko. The calculation of values of eigenfunctions of the perturbed self-adjoint operators by regularized traces method. Journal of computational and engineering mathematics, Tome 2 (2015) no. 4, pp. 48-60. http://geodesic.mathdoc.fr/item/JCEM_2015_2_4_a4/

[1] S. I. Kadchenko, S. N. Kakushkin, “The Numerical Methods of Eigenvalues and Eigenfunctions of Perturbed Self-Adjoin Operator Finding”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 13:27 (286) (2012), 45–57 | Zbl

[2] S. I. Kadchenko, S. N. Kakushkin, “Calculation of Values of Eigenfunctions of Discrete Semibounded from Below Operators via the Method of Regularized Traces”, Vestnik Samarskogo Gosudarstvennogo Universitetea. Estestvennonauchnaya Seriya, 2012, no. 6 (97), 13–21 | Zbl

[3] S. N. Kakushkin, “Mathematical Modelling of Finding the Values of Eigenfunctions for the Electrical Oscillations in the Extended Line Problem Using the Method of Regularized Traces”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 6:3 (2013), 125–129 | Zbl

[4] S. I. Kadchenko, S. N. Kakushkin, “Meaning of the First Eigenfunctions of Perturbed Discrete Operators with Simple Spectrum Finding”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 11:5 (264) (2012), 25–32 | Zbl

[5] V. A. Sadovnichii, V. V. Dubrovskii, “Remark on a New Method for Computing of the Eigenvalues and Eigenfunctions of Discrete Operators”, Tr. Seminara imeni I.G. Petrovskogo, 17, MGU, Moscow, 1994, 244–248 | MR

[6] V. A. Sadovnichii, Theory of operators, Drofa, Moscow, 2004, 384 pp. | MR

[7] S. N. Kakushkin, Development of Numerical Methods for Mathematical Modelling of Finding of the Values Eigenfunctions of Perturbed Self-Adjoint Operators, dis.... kand. fiz.-mat. nauk, Magnitogorsk, 2013

[8] S. I. Kadchenko, S. N. Kakushkin, “The Algorithm for Finding of Meanings of Eigenfunctions of Perturbed Self-Adjoint Operators via Method Regularized Traces”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2012, no. 40 (299), 83–88 | Zbl

[9] S. I. Kadchenko, “Method of Regularized Traces”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programmina and Computer Software, 4:37 (170) (2009), 4–23 | Zbl

[10] V. V. Dubrovskii, Perturbation Theory and Traces of Operators, dis.... doc. fiz.-mat. nauk, Moscow, 1992