Asymptotics of Eigenvalues of the Matrix Diffusion Operators

Abstract

In this paper, matrix diffusion equations with boundary conditions and jump conditions on $\left[0,\pi \right]\backslash \left\{a\right\}$ are considered. Under these conditions, the asymptotic of the eigenvalues of the matrix diffusion operator is obtained, while the Rouche theorem and the Gaussian elimination method are used.

Keywords

• Matrix diffusion operator,
• Eigenvalue asymptotics,
• Spectral data.

References

1. [1] Titchmarsh E.C., The Theory of Functions, Oxford University Press, 1932.
2. [2] Papanicolaou V.G., Trace formulas and the behaviour of large eigenvalues, SIAM Journal on Mathematical Analysis, 26(1), 218-237, 1995.
3. [3] Levitan B.M., Inverse Sturm-Liouville Problems, De Gruyter, 1987.
4. [4] Carlson R., Large eigenvalues and trace formulas for matrix Sturm-Liouville problems, SIAM Journal on Mathematical Analysis, 30(5), 949-962, 1999.
5. [5] Shen C.L., Shieh C., On the multiplicity of eigenvalues of a vectorial Sturm-Liouville differential equations and some related spectral problems, Proceedings of the American Mathematical Society, 127, 2943-2952, 1999.
6. [6] Beals R., Henkin G.M., Novikova N.N., The inverse boundary problem for the Rayleigh system, Journal of Mathematical Physics, 36(12), 6688-6708, 1995.
7. [7] Boutet de Monvel A., Shepelsky D., Inverse scattering problem for anisotropic media, Journal of Mathematical Physics, 36(7), 3443-3453, 1995.
8. [8] Chabanov V.M., Recovering the M-channel Sturm-Liouville operator from M + 1 spectra, Journal of Mathematical Physics, 45(11), 4255-4260, 2004.

9. [9] Harmer M., Inverse scattering on matrices with boundary conditions, Journal of Physics A: Mathematical and Theoretical, 38(22), 4875-4885, 2005.
10. [10] Yurko V.A., Inverse spectral problems for differential operators on spatial networks, Russian Mathematical Surveys, 71(3), 539-584, 2016.
11. [11] Amirov R.K., Nabiev A.A., Inverse problems for the quadratic pencil of the Sturm-Liouville equations with impulse, Abstract and Applied Analysis, Article ID 361989, 2013.