Inverse Nodal Analysis for Vector-Valued Singular Diffusion Operators

Öz

We examine a vector-valued diffusion problem given by the differential equation with boundary conditions where p(x) and q(x) denotes a continuous symmetric matrix-valued function on the interval [0,π] and k and K are real symmetric matrices of size dxd. An eigenfunction y(x) is referred to as being of type (CZ) if all isolated zeros in its components coincide with nodal points of the function itself. We demonstrate that in the two-dimensional case d=2 , the existence of infinitely many (CZ)-type eigenfunctions is both necessary and sufficient for the simultaneous diagonalizability of 〖p(x)〗_k and A,B. Consequently, if all but finitely many eigenfunctions possess the (CZ) property, then the coefficient matrix and the associated boundary data can be uniquely reconstructed up to a constant unitary transformation.

Anahtar Kelimeler

• Inverse nodal problem
• Vector-valued Sturm–Liouville systems
• Singular diffusion operators
• Common zero property

Etik Beyan

Ethics committee approval was not required for this study because of there was no study on animals or humans.

Kaynakça

1. Agranovich, Z. S., and Marchenko, V. A. (1963). The inverse problem of scattering theory. Gordon and Breach.
2. Amirov, R., and Durak, S. (2024). Inverse nodal problems for singular diffusion equation. Mathematical Methods in the Applied Sciences, 47(11), 9067–9083.
3. Avdonin, S., and Ivanov, S. (1995). Families of exponentials: The method of moments in controllability problems for distributed parameter systems. Cambridge University Press.
4. Bondarenko, B. (2015). Inverse spectral problems for matrix Sturm–Liouville operators. Inverse Problems, 31, Article 035002.
5. Brown, M., and Weikard, R. (2000). A Borg–Levinson theorem for matrix-valued potentials. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 130, 1–16.
6. Buterin, S. A. (2015). Inverse spectral problems for matrix Sturm–Liouville operators with transmission conditions. Results in Mathematics, 68, 195–212.
7. Carlson, R. (1998). Inverse spectral problems for Sturm–Liouville equations with discontinuities. Journal of Mathematical Analysis and Applications, 217, 144–167.
8. Carlson, R. (1999). Eigenvalue estimates for matrix Sturm–Liouville operators. Journal of Differential Equations, 151, 1–27.

9. Chen, G. (2002). Inverse nodal problems for Sturm–Liouville operators. Journal of Mathematical Analysis and Applications, 271, 143–156.
10. Durak, S. (2024). Inverse nodal problem for diffusion operator on a star graph with nonhomogeneous edges. Journal of Inverse and Ill-Posed Problems, 32(3), 485–496.
11. Gesztesy, F., and Simon, B. (1997). Inverse spectral analysis with partial information on the potential. I. The case of an a.c. component. Helvetica Physica Acta, 70, 66–71.
12. Gohberg, I. C., and Krein, M. G. (1970). Theory and applications of Volterra operators in Hilbert space (Vol. 24). American Mathematical Society.
13. Hald, J. (1989). Solutions of the inverse nodal problem. Inverse Problems, 5, 307–318.
14. Hochstadt, H., and Lieberman, B. (1978). An inverse Sturm–Liouville problem with mixed given data. SIAM Journal on Applied Mathematics, 34, 676–680.
15. Hryniv, R., and Mykytyuk, Y. (2003). Inverse spectral problems for Sturm–Liouville operators with singular potentials. Inverse Problems, 19, 665–684.
16. Kurasov, P. (1996). Distribution theory for discontinuous test functions and differential operators with generalized coefficients. Journal of Mathematical Analysis and Applications, 201, 297–323.
17. Levitan, B. M., and Sargsjan, I. S. (1991). Sturm–Liouville and Dirac operators. Kluwer Academic Publishers.
18. McLaughlin, J. R. (1988). Inverse spectral theory using nodal points as data. Inverse Problems, 4, 119–128.
19. Pivovarchik, V. (2000). Inverse problem for the Sturm–Liouville equation on a finite interval with transmission conditions. Inverse Problems, 16, 859–870.
20. Pöschel, J., and Trubowitz, E. (1987). Inverse spectral theory. Academic Press.
21. Rundell, W., and Sacks, P. (1992). Reconstruction techniques for classical inverse Sturm–Liouville problems. Mathematics of Computation, 58, 161–183.
22. Serier, F., and Zhou, Y. (2012). Inverse nodal problems for differential operators with eigenparameter dependent boundary conditions. Journal of Mathematical Analysis and Applications, 395, 654–666.
23. Shen, C. L. (1999). Smoothness of the potential function in inverse nodal problems. Inverse Problems, 15, 1265–1274.
24. Shen, C. L., and Shieh, C. T. (1999). On the multiplicity of eigenvalues of vectorial Sturm–Liouville problems. Proceedings of the American Mathematical Society, 127, 1183–1190.
25. Yang, C. F. (1997). A solution of the inverse nodal problem. Inverse Problems, 13, 203–213.
26. Yang, C. F. (1997). Inverse nodal problems of Sturm–Liouville operators. Journal of Differential Equations, 134, 215–234.
27. Yurko, M. (2002). Method of spectral mappings in the inverse problem theory. VSP.
28. Yurko, V. A. (2000). Inverse spectral problems for differential operators and their applications. Gordon and Breach.
29. Zayed, A. (2018). Inverse nodal problems for diffusion-type operators with interior singularities. Boundary Value Problems, 2018, 1–17.
30. Zettl, A. (2005). Sturm–Liouville theory (Vol. 121). American Mathematical Society.