Singular solutions of the matrix Bochner problem: the $N$-dimensional cases
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912102287933440 |
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| author | Parisi, Ignacio Bono Pacharoni, Inés |
| author_facet | Parisi, Ignacio Bono Pacharoni, Inés |
| contents | In the theory of matrix-valued orthogonal polynomials, there exists a longstanding problem known as the Matrix Bochner Problem: the classification of all $N \times N$ weight matrices $W(x)$ such that the associated orthogonal polynomials are eigenfunctions of a second-order differential operator. In [4], Casper and Yakimov made an important breakthrough in this area, proving that, under certain hypotheses, every solution to this problem can be obtained as a bispectral Darboux transformation of a direct sum of classical scalar weights.
In the present paper, we construct three families of weight matrices $W(x)$ of size $N \times N$, associated with Hermite, Laguerre, and Jacobi weights, which can be considered 'singular' solutions to the Matrix Bochner Problem because they cannot be obtained as a Darboux transformation of classical scalar weights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00798 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singular solutions of the matrix Bochner problem: the $N$-dimensional cases Parisi, Ignacio Bono Pacharoni, Inés Classical Analysis and ODEs 33C45, 42C05, 34L05, 34L10 In the theory of matrix-valued orthogonal polynomials, there exists a longstanding problem known as the Matrix Bochner Problem: the classification of all $N \times N$ weight matrices $W(x)$ such that the associated orthogonal polynomials are eigenfunctions of a second-order differential operator. In [4], Casper and Yakimov made an important breakthrough in this area, proving that, under certain hypotheses, every solution to this problem can be obtained as a bispectral Darboux transformation of a direct sum of classical scalar weights. In the present paper, we construct three families of weight matrices $W(x)$ of size $N \times N$, associated with Hermite, Laguerre, and Jacobi weights, which can be considered 'singular' solutions to the Matrix Bochner Problem because they cannot be obtained as a Darboux transformation of classical scalar weights. |
| title | Singular solutions of the matrix Bochner problem: the $N$-dimensional cases |
| topic | Classical Analysis and ODEs 33C45, 42C05, 34L05, 34L10 |
| url | https://arxiv.org/abs/2411.00798 |