A Numerical Method for Inverse Spectral Problems

Basing on the Galerkin methods, we develop a new numerical method for solving the inverse spectral problems generated by discrete lower semibounded operators. The restrictions on the perturbing operator are relaxed in comparison with the method based on the theory of regular traces. A Fredholm integral equation of the first kind enables us to recover the values of the perturbing operator at the discretization nodes. We tested the method on spectral problems for the Sturm - Liouville operator, and the results of numerous simulations demonstrate its computational efficiency.
We found simple formulas for the eigenvalues of a discrete lower semibounded operator avoiding the roots of the corresponding secular equations. The calculation of eigenvalues of these operators can start at an arbitrary index independently of the (un)availability of the eigenvalues with smaller indices. For perturbed selfadjoint operators we can calculate eigenvalues with large indices when the Galerkin method becomes difficult to apply.

Keywords

inverse spectral problem; discrete selfadjoint operators; eigenvalues; eigenfunctions; ill-posed problems.

References

1. Ambarzumian V.A. Ueber eine frage der eigengwerttheorie. Zeits. f. phisik, 1929, no. 53, pp. 690-665.
2. Borg G. Eine umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. Acta Math., 1945, vol. 78, no. 3, pp. 1-90.
3. Marchenko V.A. Some Questions in the Theory of a Second Order Differential Operator. Soviet Mathematics, 1950, vol. 72, no. 3, pp. 457-460. (in Russian)
4. Krein M.G. Determination of the Density of an Inhomogeneous Symmetric String from its Frequency Spectrum. Soviet Mathematics, 1951, vol. 76, no. 3, pp. 345-348. (in Russian)
5. Sadovnichii V.A., Dubrovskii V.V. [Some Properties of Operators with Discrete Spectrum]. Differential Equations, 1979. vol. 15, no. 7, pp. 1206-1211. (in Russian)
6. Dubrovskii V.V., Nagorny A.V. [An Inverse Problem for the Laplace Operator with a Continuous Potential]. Differential Equations, 1990, vol. 26, no. 9, pp. 1563-1567. (in Russian)
7. Dubrovskii V.V., Nagorny A.V. [The Inverse Problem for the Degree of the Laplace Operator with Potential in $L^2$]. Differential Equations, 1992, vol. 28, no. 9, pp. 1552-1561. (in Russian)
8. Dubrovskii V.V. Reconstruction of a Potential from the Eigenvalues of Various Problems. Russian Mathematical Surveys, 1996, vol. 51, no. 4, pp. 732-733. DOI:10.1070/RM1996v051n04ABEH002978
9. Dubrovskii V.V. An Existence Theorem in the Inverse Problem of Spectral Analysis. Differential Equations, 1997, vol. 33, no. 12, pp. 1707-1709.
10. Dubrovskii V.V., Dubrovskii V.V. (Jr.) On the Existence Theorem for Solutions to the Inverse Problem of Spectral Analysis. Russian Mathematical Surveys, 2001, vol. 56, no. 1, pp. 154-155. DOI: 10.4213/rm364
11. Zakirova G.A. An Inverse Problem for the Laplace Operator in the Isosceles Rectangular Triangle. Vestnik of Samara State University. Natural Science Series, 2008, no. 2, pp. 34-42. (in Russian)
12. Zakirova G.A. An Inverse Spectral Problem for Laplace Operator and It's Approximate Solution. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2008, no. 27 (127), issue 2, pp. 19-27. (in Russian)
13. Sadovnichii V.A., Dubrovskii V.V., Dubrovskii V.V. (Jr.) The Inverse Problem of Spectral Analysis for the Degree of the Laplace Operator with Potential on the Rectangle. Doklady Mathematics, 2001, vol. 377, no. 3, pp. 310-312. (in Russian)
14. Sadovnichii V.A., Dubrovskii V.V., Dubrovskii V.V. (Jr.), Puzankova E.A. About Restoration the Potential in the Inverse Problem of Spectral Analysis. Doklady Mathematics, 2001, vol. 380, no. 4, pp. 462-464. (in Russian)
15. Sadovnichii V.A., Dubrovskij V.V., Smirnova L.V. [The Uniqueness of the Solution of Inverse Problems of Spectral Analysis]. Doklady Mathematics, 2000, vol. 370, no. 3, pp. 319-321. (in Russian)
16. Sadovnichii V.A., Dubrovskij V.V., Puzankova E.A. [On the Inverse Problem of Spectral Analysis for the Powers of the Laplace Operator with Potential]. Doklady Mathematics, 1999, vol. 367, no. 3, pp. 307-309. (in Russian)
17. Sadovnichii V.A., Dubrovskij V.V., Puzankova E.A. [The Inverse Problem of Spectral Analysis for the Degree of the Laplace Operator on a Rectangle]. Differential Equations, 2000, vol. 36, no. 12, pp. 1693-1698. (in Russian)
18. Sedov A.I. About the Approximate Solution of the Inverse Problem of Spectral Analysis for Laplace Operator. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16(192), issue 5, pp. 73-78.
19. Kadchenko S.I. Numerical Method for the Solution of Inverse Problems Generated by Perturbations of Self-Adjoint Operators by Method of Regularized Traces. Vestnik of Samara State University. Natural Science Series, 2013, no. 6 (107), pp. 23-30. (in Russian)
20. Kadchenko S.I. A Numerical Method for Solving Inverse Problems Generated by the Perturbed Self-Adjoint Operators. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2013. vol. 6, no. 4, pp. 15-25. (in Russian)
21. Kadchenko S.I. [Solution of Inverse Spectral Problems Generated by the Perturbed Self-Adjoint Operators by Method of Regularized Traces]. Bulletin of the Magnitogorsk State University. Mathematics, 2013, issue 15, pp. 34-43. (in Russian)
22. Kadchenko S.I. Numerical Method for Solving Inverse Spectral Problems Generated by Perturbed Self-Adjoint Operators. Vestnik of Samara State University. Natural Science Series, 2013, no. 9 (100), pp. 5-11.(in Russian)
23. Mihlin S.G. Variacionnye metody v matematicheskoj fizike [Variational Methods in Mathematical Physics]. Moscow, Nauka, 1970. 510 p.
24. Demidovich B.P., Maron I.A. Osnovy vychislitel'noj matematiki [Fundamentals of Computational Mathematics]. Moscow, Nauka, 1966. 659 p.
25. Vasileva A.B., Tihonov N.A. Integralnye uravneniya [Integral Equations]. Мoscow, MGU, 1989. 156 p.