Öz
This paper is concerned with a finite-dimensional example of a linear pencil which leads to a class of non-self-adjoint matrices. We consider the linear pencil $H_c-\lambda L$, where $H_c$ is a tri-diagonal matrix with a constant parameter $c$ on the main diagonal and off-diagonal entries equal to one, and $L$ is a diagonal matrix whose elements decrease linearly from one to minus one. In general, the spectra of operator polynomials may contain non-real eigenvalues as well as real eigenvalues. Nevertheless, they exhibit certain patterns. Our aim in this research is to carry out a variety of numerical investigation on the eigenvalues so as to understand the eigenvalue behaviour of such pencils from different points of view. In accordance with our numerical findings, a series of conjectures are offered and various heuristics has been discussed.
Anahtar Kelimeler
Linear operator pencils non-self-adjoint matrices eigenvalues spectral theory
Kaynakça
- Bagarello, F., Gazeau, J.P., Szafraniec, F.H., Znojil, M., Non-selfadjoint Operators in Quantum Physics, John Wiley & Sons, Inc., Hoboken, NJ, 2015.
- Bai, Z., Day, D., Demmel, J., Dongarra, J., A test matrix collection for non-Hermitian eigenvalue problems, Technical Report CS-97-355, (1996).
- Bora, S., Mehrmann, V., Linear perturbation theory for structured matrix pencils arising in control theory, SIAM J. Matrix Anal. Appl. 28(2006), 148–169.
- Cullum, J., Kerner, W., Willoughby, R., A generalized nonsymmetric Lanczos procedure, Comput. Phys. Commun., 53(1989), 19–48.
- Davies, E.B., Levitin, M., Spectra of a class of non-self-adjoint matrices, Linear Algebra Appl., 448(2014), 55–84.
- Elton, D.M., Levitin, M., Polterovich, I., Eigenvalues of a one-dimensional Dirac operator pencil, Ann. Henri Poincar´e, 15(2014), 2321–2377.
- Jeribi, A., Moalla, N., Yengui, S., S -essential spectra and application to an example of transport operators, Math. Methods Appl. Sci., 37(2014), 2341–2353.
- Levitin, M., Öztürk, H.M., A two-parameter eigenvalue problem for a class of block-operator matrices, Oper. Theory Adv. Appl., 268(2018), 367–380.
- Levitin, M., Seri, M., Accumulation of complex eigenvalues of an indefinite Sturm-Liouville operator with a shifted Coulomb potential, Oper. Matrices, 10(2016), 223–245.
- Öztürk, H.M., On a conjecture of Davies and Levitin, Math. Methods Appl. Sci., 46(2023), 4391–4412.
- Markus, A.S., Introduction to the Spectral Theory of Polynomial Operator Pencils. Transl. from the Russian by H.H. McFaden, American Mathematical Society, 1988.
- Möller, M., Pivovarchik, V., Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and Their Applications, Birkh¨auser/Springer, Cham, 2015.
- Tisseur, F., Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev., 43(2001), 235–286.
- Tretter, C., Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2008.
Öz
Kaynakça
- Bagarello, F., Gazeau, J.P., Szafraniec, F.H., Znojil, M., Non-selfadjoint Operators in Quantum Physics, John Wiley & Sons, Inc., Hoboken, NJ, 2015.
- Bai, Z., Day, D., Demmel, J., Dongarra, J., A test matrix collection for non-Hermitian eigenvalue problems, Technical Report CS-97-355, (1996).
- Bora, S., Mehrmann, V., Linear perturbation theory for structured matrix pencils arising in control theory, SIAM J. Matrix Anal. Appl. 28(2006), 148–169.
- Cullum, J., Kerner, W., Willoughby, R., A generalized nonsymmetric Lanczos procedure, Comput. Phys. Commun., 53(1989), 19–48.
- Davies, E.B., Levitin, M., Spectra of a class of non-self-adjoint matrices, Linear Algebra Appl., 448(2014), 55–84.
- Elton, D.M., Levitin, M., Polterovich, I., Eigenvalues of a one-dimensional Dirac operator pencil, Ann. Henri Poincar´e, 15(2014), 2321–2377.
- Jeribi, A., Moalla, N., Yengui, S., S -essential spectra and application to an example of transport operators, Math. Methods Appl. Sci., 37(2014), 2341–2353.
- Levitin, M., Öztürk, H.M., A two-parameter eigenvalue problem for a class of block-operator matrices, Oper. Theory Adv. Appl., 268(2018), 367–380.
- Levitin, M., Seri, M., Accumulation of complex eigenvalues of an indefinite Sturm-Liouville operator with a shifted Coulomb potential, Oper. Matrices, 10(2016), 223–245.
- Öztürk, H.M., On a conjecture of Davies and Levitin, Math. Methods Appl. Sci., 46(2023), 4391–4412.
- Markus, A.S., Introduction to the Spectral Theory of Polynomial Operator Pencils. Transl. from the Russian by H.H. McFaden, American Mathematical Society, 1988.
- Möller, M., Pivovarchik, V., Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and Their Applications, Birkh¨auser/Springer, Cham, 2015.
- Tisseur, F., Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev., 43(2001), 235–286.
- Tretter, C., Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London, 2008.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Deneysel Analiz, Sayısal ve Hesaplamalı Matematik (Diğer), Operatör Cebirleri ve Fonksiyonel Analiz |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 31 Aralık 2024 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 16 Sayı: 2 |