Complete systems of solutions, transmutations and Darboux transform for Sturm-Liouville equations in impedance form
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908414704091136 |
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| author | Vicente-Benítez, Víctor A. |
| author_facet | Vicente-Benítez, Víctor A. |
| contents | We present the construction of a complete system of functions associated with the Sturm-Liouville equation in impedance form on a finite interval $I$, given an impedance function $a\in L^2(I)$. The system, known as the formal powers, is generated through recursive integration of the impedance function $a$ and its reciprocal. We establish the completeness of this system in the space $L^p$ with the weight function $a^2$. Under additional conditions on $a$, we extend the completeness of this completeness to Sobolev spaces $W^{1,p}(I)$, along with a generalized Taylor formula. We show that the completeness of the formal powers implies key analytic properties for a transmutation operator associated with the Sturm-Liouville equation in impedance form, including the existence of a continuous inverse. Finally, we introduce a formulation of the Darboux-transformed equation and establish a relation between the transmutation operators to the original and transformed equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_16604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete systems of solutions, transmutations and Darboux transform for Sturm-Liouville equations in impedance form Vicente-Benítez, Víctor A. Classical Analysis and ODEs 34B24, 34A25, 34L40, 41A30, 42A65, 47G20 We present the construction of a complete system of functions associated with the Sturm-Liouville equation in impedance form on a finite interval $I$, given an impedance function $a\in L^2(I)$. The system, known as the formal powers, is generated through recursive integration of the impedance function $a$ and its reciprocal. We establish the completeness of this system in the space $L^p$ with the weight function $a^2$. Under additional conditions on $a$, we extend the completeness of this completeness to Sobolev spaces $W^{1,p}(I)$, along with a generalized Taylor formula. We show that the completeness of the formal powers implies key analytic properties for a transmutation operator associated with the Sturm-Liouville equation in impedance form, including the existence of a continuous inverse. Finally, we introduce a formulation of the Darboux-transformed equation and establish a relation between the transmutation operators to the original and transformed equations. |
| title | Complete systems of solutions, transmutations and Darboux transform for Sturm-Liouville equations in impedance form |
| topic | Classical Analysis and ODEs 34B24, 34A25, 34L40, 41A30, 42A65, 47G20 |
| url | https://arxiv.org/abs/2506.16604 |