A $3\times 3$ singular solution to the Matrix Bochner Problem with $\mathcal{D}(W)$ not of the form $\mathbb{C}[D]$
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866912692346814464 |
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| author | Parisi, Ignacio Bono |
| author_facet | Parisi, Ignacio Bono |
| contents | The Matrix Bochner Problem aims to classify weight matrices whose sequences of orthogonal polynomials are eigenfunctions of a second-order differential operator. A major breakthrough in this direction was achieved in [7], where it was shown that, under certain natural conditions on the algebra $\mathcal{D}(W)$, all solutions arise from Darboux transformations of direct sums of classical scalar weights. In this paper, we study a new $3 \times 3$ Hermite-type weight matrix and determine its algebra $\mathcal{D}(W)$ as a $\mathbb{C}[D_1]$-module generated by $\{I, D_2\}$, where $D_{1}$ and $D_{2}$ are second-order differential operators. This complete description of the algebra allows us to prove that the weight does not arise from a Darboux transformation of classical scalar weights, showing that it falls outside the classification theorem of [7]. Unlike previous examples in [3,4], which also do not fit within this classification, the algebra $\mathcal{D}(W)$ of this weight matrix is not generated by a single differential operator $D$, making it a fundamentally different case. These results complement the classification theorem of the Matrix Bochner Problem by providing a new type of singular example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A $3\times 3$ singular solution to the Matrix Bochner Problem with $\mathcal{D}(W)$ not of the form $\mathbb{C}[D]$ Parisi, Ignacio Bono Classical Analysis and ODEs 33C45, 42C05, 34L05, 34L10 The Matrix Bochner Problem aims to classify weight matrices whose sequences of orthogonal polynomials are eigenfunctions of a second-order differential operator. A major breakthrough in this direction was achieved in [7], where it was shown that, under certain natural conditions on the algebra $\mathcal{D}(W)$, all solutions arise from Darboux transformations of direct sums of classical scalar weights. In this paper, we study a new $3 \times 3$ Hermite-type weight matrix and determine its algebra $\mathcal{D}(W)$ as a $\mathbb{C}[D_1]$-module generated by $\{I, D_2\}$, where $D_{1}$ and $D_{2}$ are second-order differential operators. This complete description of the algebra allows us to prove that the weight does not arise from a Darboux transformation of classical scalar weights, showing that it falls outside the classification theorem of [7]. Unlike previous examples in [3,4], which also do not fit within this classification, the algebra $\mathcal{D}(W)$ of this weight matrix is not generated by a single differential operator $D$, making it a fundamentally different case. These results complement the classification theorem of the Matrix Bochner Problem by providing a new type of singular example. |
| title | A $3\times 3$ singular solution to the Matrix Bochner Problem with $\mathcal{D}(W)$ not of the form $\mathbb{C}[D]$ |
| topic | Classical Analysis and ODEs 33C45, 42C05, 34L05, 34L10 |
| url | https://arxiv.org/abs/2503.22833 |