On Eigenvalue Multiplicities of Self-Adjoint Regular Sturm–Liouville Operators
DOI:
https://doi.org/10.15407/mag20.04.04Анотація
Ми надаємо вичерпне дослідження кратності власних значень усіх самоспряжених регулярних задач Штурма--Ліувілля на компактних інтервалах $[a,b] \subset \mathbb{R}$.
Mathematical Subject Classification 2020: 334B09, 34B24, 34L15, 47A75
Ключові слова:
оператори Штурма–Лiувiлля, кратнiсть власних значеньПосилання
P.B. Bailey, W.N. Everitt, and A. Zettl, Regular and Singular Sturm-Liouville problems with coupled boundary conditions, Proc. Roy. Soc. Edinburgh 126A (1996), 505--514. https://doi.org/10.1017/S0308210500022885
B.M. Brown, M.S.P. Eastham, and K.M. Schmidt, Periodic Differential Operators, Operator Th.: Adv. Appls., 230, Birkhäuser, Springer, Basel, 2013. https://doi.org/10.1007/978-3-0348-0528-5
E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Krieger Publ., Malabar, FL, 1985.
M.S.P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh, 1973.
D. Frymark and C. Liaw, Spectral properties of singular Sturm-Liouville operators via boundary triples and perturbation theory, J. Differential Equations 363 (2023), 391--421. https://doi.org/10.1016/j.jde.2023.03.022
F. Gesztesy, R. Nichols, and M. Zinchenko, Sturm-Liouville Operators, Their Spectral Theory, and Some Applications, Colloquium Publications, 67, Amer. Math. Soc., Providence, RI, 2024. https://doi.org/10.1090/coll/067
F. Gesztesy and R. Weikard, Floquet theory revisited, Differential Equations and Mathematical Physics, (Ed. I. Knowles), International Press, Boston, 1995, 67--84.
R. Hryniv, A. Shkalikov, and A. Vladimirov, Spectral analysis of periodic differential operator matrices of mixed order, Trans. Mosc. Math. Soc. 2002, 39--75.
Q. Kong, H. Wu, and A. Zettl, Multiplicity of Sturm-Liouville eigenvalues, J. Comput. Appl. Math. 171 (2004), 291--309. https://doi.org/10.1016/j.cam.2004.01.036
J. Locker, Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators, Math. Surv. Monographs, 73, Amer. Math. Soc., Providence, RI, 2000. https://doi.org/10.1090/surv/073
W. Magnus and S. Winkler, Hill's Equation, Dover, New York, 1979.
M.A. Naimark, Linear Partial Differential Operators Part I. Elementary Theory of Linear Differential Operators, translated by E.R. Dawson, Engl. transl. edited by W.N. Everitt, Ungar Publishing, New York, 1967.
M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adointness, Academic Press, New York, 1975.
B. Simon, Trace Ideals and Their Applications, Mathematical Surveys and Monographs, 120, Amer. Math. Soc., Providence, RI, 2005.
Z. Wang and H. Wu, Equality of multiplicities of a Sturm-Liouville eigenvalue, J. Math. Anal. Appl. 306 (2005), 540--547. https://doi.org/10.1016/j.jmaa.2004.10.041
J. Weidmann, Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics, 68, Springer, New York, 1980. https://doi.org/10.1007/978-1-4612-6027-1
A. Zettl, Sturm-Liouville Theory, Math. Surv. Monographs, 121, Amer. Math. Soc., Providence, RI, 2005.
H. Zhu and Y.Shi, Dependence of eigenvalues on the boundary conditions of Sturm-Liouville problems with one singular endpoint, J. Differential Equations 263 (2017), 5582--5609. https://doi.org/10.1016/j.jde.2017.06.026