Neumann series of Bessel functions for the solutions of the Sturm-Liouville equation in impedance form and related boundary value problems
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911360114229248 |
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| author | Márquez-Hernández, Abigail G. Vicente-Benítez, Víctor A. |
| author_facet | Márquez-Hernández, Abigail G. Vicente-Benítez, Víctor A. |
| contents | We present a Neumann series of spherical Bessel functions representation for solutions of the Sturm--Liouville equation in impedance form \[ (κ(x)u')' + λκ(x)u = 0,\quad 0 < x < L, \] in the case where $κ\in W^{1,2}(0,L)$ and has no zeros on the interval of interest. The $x$-dependent coefficients of this representation can be constructed explicitly by means of a simple recursive integration procedure. Moreover, we derive bounds for the truncation error, which are uniform whenever the spectral parameter $ρ=\sqrtλ$ satisfies a condition of the form $|\operatorname{Im}ρ|\leq C$. Based on these representations, we develop a numerical method for solving spectral problems that enables the computation of eigenvalues with non-deteriorating accuracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04513 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Neumann series of Bessel functions for the solutions of the Sturm-Liouville equation in impedance form and related boundary value problems Márquez-Hernández, Abigail G. Vicente-Benítez, Víctor A. Classical Analysis and ODEs Numerical Analysis Mathematical Physics 34A25, 34B09, 34B24, 34L16, 41A30, 47G20 We present a Neumann series of spherical Bessel functions representation for solutions of the Sturm--Liouville equation in impedance form \[ (κ(x)u')' + λκ(x)u = 0,\quad 0 < x < L, \] in the case where $κ\in W^{1,2}(0,L)$ and has no zeros on the interval of interest. The $x$-dependent coefficients of this representation can be constructed explicitly by means of a simple recursive integration procedure. Moreover, we derive bounds for the truncation error, which are uniform whenever the spectral parameter $ρ=\sqrtλ$ satisfies a condition of the form $|\operatorname{Im}ρ|\leq C$. Based on these representations, we develop a numerical method for solving spectral problems that enables the computation of eigenvalues with non-deteriorating accuracy. |
| title | Neumann series of Bessel functions for the solutions of the Sturm-Liouville equation in impedance form and related boundary value problems |
| topic | Classical Analysis and ODEs Numerical Analysis Mathematical Physics 34A25, 34B09, 34B24, 34L16, 41A30, 47G20 |
| url | https://arxiv.org/abs/2601.04513 |