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The spectral ζ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta $$\end{document}-function for quasi-regular Sturm–Liouville operatorsThe spectral ζ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta $$\end{document}-function for quasi-regular...G. Fucci et al.
被引：0
作者：
Guglielmo Fucci
[1
]
Mateusz Piorkowski
[2
]
Jonathan Stanfill
[3
]
机构：
[1] East Carolina University,Department of Mathematics
[2] Royal Institute of Technology (KTH),Department of Mathematics
[3] The Ohio State University,Department of Mathematics
关键词：
-Function;
(Quasi-regular) Sturm–Liouville operators;
Traces;
Zeta regularized functional determinants;
(Generalized) Bessel operators;
Legendre differential equation;
Primary 47A10, 47B10, 47G10;
Secondary 34B27, 34L40;
D O I：
10.1007/s11005-024-01893-x
中图分类号：
学科分类号：
摘要：
In this work, we analyze the spectral ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}-function associated with the self-adjoint extensions, TA,B\documentclass[12pt]{minimal}
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\begin{document}$$T_{A,B}$$\end{document}, of quasi-regular Sturm–Liouville operators that are bounded from below. By utilizing the Green’s function formalism, we find the characteristic function, which implicitly provides the eigenvalues associated with a given self-adjoint extension TA,B\documentclass[12pt]{minimal}
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\begin{document}$$T_{A,B}$$\end{document}. The characteristic function is then employed to construct a contour integral representation for the spectral ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}-function of TA,B\documentclass[12pt]{minimal}
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\begin{document}$$T_{A,B}$$\end{document}. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}-function to a larger region of the complex plane. We also present a method for computing the value of the spectral ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}-function of TA,B\documentclass[12pt]{minimal}
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\begin{document}$$T_{A,B}$$\end{document} at all positive integers. We provide two examples to illustrate the methods developed in the paper: the generalized Bessel and Legendre operators. We show that in the case of the generalized Bessel operator, the spectral ζ\documentclass[12pt]{minimal}
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\begin{document}$$\zeta $$\end{document}-function develops a branch point at the origin, while in the case of the Legendre operator it presents, more remarkably, branch points at every nonpositive integer value of s.
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