The spectral $ζ$-function for quasi-regular Sturm--Liouville operators
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| Format: | Preprint |
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2024
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| author | Fucci, Guglielmo Piorkowski, Mateusz Stanfill, Jonathan |
| author_facet | Fucci, Guglielmo Piorkowski, Mateusz Stanfill, Jonathan |
| contents | In this work we analyze the spectral $ζ$-function associated with the self-adjoint extensions, $T_{A,B}$, 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 $T_{A,B}$. The characteristic function is then employed to construct a contour integral representation for the spectral $ζ$-function of $T_{A,B}$. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $ζ$-function to a larger region of the complex plane. We also present a method for computing the value of the spectral $ζ$-function of $T_{A,B}$ 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 $ζ$-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 $s$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_06922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The spectral $ζ$-function for quasi-regular Sturm--Liouville operators Fucci, Guglielmo Piorkowski, Mateusz Stanfill, Jonathan Mathematical Physics Spectral Theory Primary: 47A10, 47B10, 47G10. Secondary: 34B27, 34L40 In this work we analyze the spectral $ζ$-function associated with the self-adjoint extensions, $T_{A,B}$, 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 $T_{A,B}$. The characteristic function is then employed to construct a contour integral representation for the spectral $ζ$-function of $T_{A,B}$. By assuming a general form for the asymptotic expansion of the characteristic function, we describe the analytic continuation of the $ζ$-function to a larger region of the complex plane. We also present a method for computing the value of the spectral $ζ$-function of $T_{A,B}$ 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 $ζ$-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 $s$. |
| title | The spectral $ζ$-function for quasi-regular Sturm--Liouville operators |
| topic | Mathematical Physics Spectral Theory Primary: 47A10, 47B10, 47G10. Secondary: 34B27, 34L40 |
| url | https://arxiv.org/abs/2409.06922 |