A Numerical Method for Solving Inverse Problems Generated by the Perturbed Self-Adjoint Operators

Based on the methods of regularized traces and Bubnov-Galerkin's method a new method for the solution of inverse problems is developed in spectral characteristics perturbed self-adjoint operators. Simple formulas for calculating the eigenvalues of discrete operators without the roots of the corresponding secular equation are found. Computation of eigenvalues of a perturbed self-adjoint operator can be started with any of their numbers, regardless of whether the previous numbers of eigenvalues are known or not. Numerical calculations for eigenvalues of the Sturm-Liouville's operator show that the proposed formulas for large numbers of eigenvalues give more accurate results than the Bubnov-Galerkin's method. In addition, the obtained formulas allow us to calculate the eigenvalues of perturbed self-adjoint operator with very large numbers, where the use of the Bubnov-Galerkin's method becomes difficult. It can be used in problems of hydrodynamic stability theory, if you want to find signs of the real or imaginary parts of the eigenvalues with large numbers.
An integral Fredholm equation of the first kind, restoring the value of the perturbing operator in the nodal points of the sample, is obtained.
The method is tested on inverse problems for the Sturm-Liouville's problem. The results of numerous calculations have shown its computational efficiency.

Keywords

the inverse spectral problem; perturbation theory; discrete and self-adjoint operators; eigenvalues; eigenfunctions; incorrectly formulated problems.

References

1. Dubrovskiy V.V., Kadchenko S.I., Kravchenko V.F., Sadovnichiy V.A. A New Method for the Approximate Calculation of the First Eigenvalues of the Spectral Problem of Hydrodynamic Stability of Poiseuille Flow in a Circular Pipe [Novyy metod priblizhennogo vychisleniya pervyh sobstvennyh chisel spektral'noy zadachi gidrodinamicheskoy ustoychivosti techeniya Puazeylya v krugloy trube]. DAN Russia, 2001, vol. 380, no. 2, pp. 160-163.
2. Dubrovskiy V.V., Kadchenko S.I., Kravchenko V.F., Sadovnichiy V.A. A New Method for the Approximate Calculation of the First Eigenvalues of the Orr-Zomerfeld Spectral Problem [Novyy metod pribizhennogo vychisleniya pervyh sobstvennyh chisel spektral'noy zadachi Orra-Zommerfel'da]. DAN Russia, 2001, vol. 378, no. 4, pp. 443-446.
3. Sadovnichiy V.A., Dubrovskiy V.V., Kadchenko S.I., Kravchenko V.F. Calculation of the First Eigenvalues of the Hydrodynamic Stability of Viscous Flow Between Two Rotating Cylinders [Vychislenie pervyh sobstvennyh znacheniy zadachi gidrodinamicheskoy ustoychivosti tacheniya vyazkoy zhidkosti mezhdu dvumya vrashchayushchimisya tsilindrami]. Differentsial'nye uravneniya [Differential Equations], 2000, vol. 36, no. 6, pp. 742-746.
4. Kadchenko S.I. Computing the Sums of Rayleigh-Schrцdinger Series of Perturbed Self-Adjoint Operators. Computational Mathematics and Mathematical Physics, 2007, vol. 47, no. 9, pp. 1435-1445.
5. Kadchenko S.I. The method of Regularized Traces [Metod regulyarizovannykh sledov]. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2009, no. 37 (170), issue 4, pp. 4-23.
6. Kadchenko S.I., Ryazanova L.S. A Numerical Method for Finding the Eigenvalues of the Discrete Semi-bounded From Below Operators [Chislennyy metod nakhozhdeniya sobstvennykh znacheniy diskretnykh poluogranichennykh snizu operatorov]. Bulletin of the South Ural State University. Series 'Mathematical Modelling, Programming & Computer Software', 2011, no. 17 (234), issue 8, pp. 46-51.
7. Sadovnichiy V.A. Teoriya operatorov [Operator Theory]. Moscow, 1999. 368 p.
8. Mihlin S.G. Variatsionnye metody v matematicheskoy fizike [Variational Methods in Mathematical Physics]. Moscow, 1970. 510 p.
9. Demidovich B.P. Osnovy vychislitel'noy matematiki [Foundations of Computational Mathematics]. Moscow, 1966. 659 p.
10. Vasil'eva A.B. Integral'nye uravneniya [Integral Equations]. Moscow, 1989. 156 p.
11. Verlan' A.F., Sizikov V.S. Integral'nye uravneniya: metody, algoritmy, programmy [Integral Equation Methods, Algorithms, Programs]. Kiev, 1986. 542 p.