On symmetry of the product of two higher-order quasi-differential operators

Yanyu Xiang; Aiping Wang; Yanyu Xiang; Aiping Wang

doi:10.3934/math.2023478

AIMS Mathematics

2023, Volume 8, Issue 4: 9483-9505. doi: 10.3934/math.2023478

Previous Article Next Article

Research article

On symmetry of the product of two higher-order quasi-differential operators

Yanyu Xiang ,
Aiping Wang ^,

School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China

Received: 29 December 2022 Accepted: 13 February 2023 Published: 20 February 2023
MSC : 34B20, 34B24, 47B25

The symmetric realizations of the product of two higher-order quasi-differential expressions in Hilbert space are investigated. By means of the construction theory of symmetric operators, we characterize symmetric domains determined by two-point boundary conditions for product of two symmetric differential expressions with regular or limit-circle singular endpoints. The presented result contains the characterization of self-adjoint domains as a special case. Several examples of singular symmetric product operators are given.
- product of differential operators,
- symmetric boundary conditions,
- singular operators
Citation: Yanyu Xiang, Aiping Wang. On symmetry of the product of two higher-order quasi-differential operators[J]. AIMS Mathematics, 2023, 8(4): 9483-9505. doi: 10.3934/math.2023478

Related Papers:

Abstract

The symmetric realizations of the product of two higher-order quasi-differential expressions in Hilbert space are investigated. By means of the construction theory of symmetric operators, we characterize symmetric domains determined by two-point boundary conditions for product of two symmetric differential expressions with regular or limit-circle singular endpoints. The presented result contains the characterization of self-adjoint domains as a special case. Several examples of singular symmetric product operators are given.

References

[1]	A. Wang, A. Zettl, Ordinary Differential Operators, American Mathematical Soc., 2019.
[2]	A. Zettl, Powers of real symmetric differential expressions without smoothness assumptions, Quaest. Math., 1 (1976), 83–94. https://doi.org/10.1080/16073606.1976.9632515 doi: 10.1080/16073606.1976.9632515
[3]	A. Wang, A. Zettl, Characterization of domains of symmetric and self-adjoint ordinary differential operators, Electron. J. Differ. Eq., 15 (2018), 1–18.
[4]	A. Wang, A. Zettl, A symmetric GKN-type theorem and symmetric differential operators, J. Differ. Equations, 265 (2018), 5156–5176. https://doi.org/10.1016/j.jde.2018.06.028 doi: 10.1016/j.jde.2018.06.028
[5]	A. Wang, A. Zettl, A GKN theorem for symmetric operators, J. Math. Anal. Appl., 452 (2017), 780–791. https://doi.org/10.1016/j.jmaa.2017.03.004 doi: 10.1016/j.jmaa.2017.03.004
[6]	A. Wang, J. Sun, A. Zettl, Characterization of domains of self-adjoint ordinary differential operators, J. Differ. Equations, 246 (2009), 1600–1622. https://doi.org/10.1016/j.jde.2008.11.001 doi: 10.1016/j.jde.2008.11.001
[7]	C. Yang, Self-adjointness of products of the limit-point Sturm-Liouville operators, Journal of Systems Science and Mathematical Sciences (in Chinese), 26 (2006), 368–374.
[8]	J. An, J. Sun, On self-adjointness of the product of two n-order differential operators on $[a, b]$, Annals Of Differential Equations, 18 (1998), 50–57.
[9]	J. An, J. Sun, the self-adjointness of the product operators of two mth-order differential operators on $[0, +\infty)$, Acta Math. Sin. (English Seris), 20 (2004), 793–802. https://doi.org/10.1007/s10114-004-0386-3 doi: 10.1007/s10114-004-0386-3
[10]	J. Sun, On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices, Acta Math. Sin., 2 (1986), 152–167. https://doi.org/10.1007/BF02564877 doi: 10.1007/BF02564877
[11]	J. Zheng, On the self-adjointness of product of two odd-order differential operators, Inner Mongolia University (in Chinese), 2009.
[12]	M. M$\ddot{o}$ller, A. Zettl, Symmetrical differential operators and their friedrichs extension, J. Differ. Equations, 115 (1995), 50–69. https://doi.org/10.1006/jdeq.1995.1003 doi: 10.1006/jdeq.1995.1003
[13]	Q, Lin, Self-adjointness of class 4th-order and 6th-order differential operator products, Journal of Inner Mongolia Normal University (Natural Science Edition) (in Chinese), 32 (2019), 74–79.
[14]	Q. Qing, G. Wang, H. Kong, Self-adjointness of the products of high-order differential operators, Journal of Inner Mongolia Normal University(Natural Science Edition)(in Chinese), 41 (2012), 227–230.
[15]	R. M. Kauffman, T. T. Read, A. Zettl, The Deficiency Index Problem for Powers of Ordinary Differential Expressions, Lecture Notes in Mathematics, 1977.
[16]	W. D. Evans, A. Zettl, On the deficiency indices of powers of real 2nth-order symmetric differential expressions, J. Lond. Math. Soc., 13 (1976), 543–556. https://doi.org/10.1112/jlms/s2-13.3.543 doi: 10.1112/jlms/s2-13.3.543
[17]	X. Guo, The self-adjointness ofproduct of four second-orderdifferential operators, Inner Mongolia University, 2012.
[18]	X. Hao, J. Sun, A.Wang, A. Zettl, Characterization of domains of self-adjoint ordinary differential operators Ⅱ, Results Math., 61 (2012), 255–281. https://doi.org/10.1007/s00025-011-0096-y doi: 10.1007/s00025-011-0096-y
[19]	X. Zhang, W. Wang, Q. Yang, On self-adjointness of the product of three second-order differential operators, Journal of Inner Mongolia Normal University (Natural Science Edition) (in Chinese), 36 (2007), 35–42.
[20]	Z, Cao, J. Sun, D. E. Edmunds, On self-adjointness of the product of two second-order differential operators, Acta Math. Sin. (English Series), 15 (1999), 375–386. https://doi.org/10.1007/BF02650732 doi: 10.1007/BF02650732

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)