Transcendence and the Expression of the Spectral Series of a Class of Higher Order Differential Operators

Bing Xie, Jing Li and Jiangang Qi ()
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Bing Xie: School of Mathematics and Statistics, Shandong University, Weihai 264209, China
Jing Li: School of Mathematics and Statistics, Shandong University, Weihai 264209, China
Jiangang Qi: School of Mathematics and Statistics, Shandong University, Weihai 264209, China

Mathematics, 2023, vol. 11, issue 3, 1-19

Abstract: In this paper, a relationship between the spectral zeta series of a class of higher order self-adjoint differential operators on the unit circle and the integral of Green functions is established by Mercer’s Theorem. Furthermore, the explicit expression and the transcendental nature of the spectral series are obtained by the integral representation. Finally, several applications in physics about differential operators’ spectral theory, yellow some further works, and the transcendental nature of some zeta series are listed.

Keywords: self-adjoint differential operators; Mercer’s theorem; spectral series; transcendentality; green functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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