Finding of Values for Sums of Functional Rayleigh - Schredinger Series for Perturbed Self-Adjoint Operators

Authors of the article developed non-iteration method for calculating the values of eigenfunctions for perturbed self-adjoint operators, namely the method of regularized traces (RT). It allows to find the values of eigenfunctions of perturbed operators aware the spectral characteristics of unperturbed operator and the eigenvalues of the perturbed operator. In contrast to the known methods of finding the eigenfunctions, the RT method does not use the matrix, and the values of eigenfunctions are searched by linear formulas. This greatly increases its computational efficiency compared with classical methods. For application of the RT method in practice one should be able to summarize the functional Rayleigh - Schrodinger series of perturbed discrete operators. Previously authors obtained formulas for finding the "weighted" corrections of the perturbation theory, that allowed to approximate the sum of functional Rayleigh - Schrodinger series, by partial sums consisting of these corrections. In the article formulas for finding the values of sums of functional Rayleigh - Schrodinger series of perturbed discrete operators in the the nodal points were obtained. Computational experiments for finding the values of the eigenfunctions of the perturbed one-dimensional Laplace operator were conducted. The results of the experiment showed the high computational efficiency of this method of summation of the Rayleigh - Schrodinger series.

Keywords

perturbed operators; eigenvalues, eigenfunctions; multiple spectrum; the sum of functional Rayleigh - Schrodinger series; "weighted".

References

1. Sadovnichiy V.A., Dubrovskiy V.V. Remark on a New Method of Calculation of Eigenvalues and Eigenfunction For Discrete Operators. Journal of Mathematical Sciences, 1995, vol. 75, no. 3, pp. 1770-1772. DOI: 10.1007/BF02368675
2. Dubrovskiy V.V., Sedov A.I. An Estimate for the Difference of Spectral Functions of the Selfadjoint Operator. Elektromagnitnye volny i elektronnye sistemy [Electromagnetic Waves and Electronic Systems], 2000, vol. 5, no. 5, pp. 10-13.
3. Dubrovskiy V.V. Teoriya vozmushcheniy i sledy operatorov [Perturbation Theory and Traces of Operators. Dissertation of the Doctor of Physical and Mathematical Sciences]. Moscow, 1992.
4. Kadchenko S.I., Kakushkin S.N. Chislennye metody regulyarizovannyh sledov spektral'nogo analiza [Numerical Methods of Regularized Traces of Spectral Analysis]. Chelyabinsk, Publishing Center of the South Ural State University, 2015. 206 p.