A $3\times 3$ singular solution to the Matrix Bochner Problem with $\mathcal{D}(W)$ not of the form $\mathbb{C}[D]$
Ignacio Bono Parisi
TL;DR
The paper addresses the Matrix Bochner Problem by constructing a novel $3\times3$ Hermite-type weight $W$ whose differential-operator algebra $\mathcal{D}(W)$ is not generated by a single operator and is not full, placing it outside the Darboux-transformation classification of CY18. It shows $\mathcal{D}(W)$ is generated as a $\mathbb{C}[D_{1}]$-module by $\{I, D_{2}\}$ with explicit commutation relations $D_{1}D_{2}=D_{2}D_{1}$ and $D_{2}D_{1}=D_{2}^{2}+\frac{2(c^{2}+2)}{c^{2}}(D_{1}-D_{2})$, and demonstrates that $W$ cannot be obtained as a Darboux transformation of a direct sum of classical scalar weights. A key technical move is the factorization $W=T\tilde W T^{*}$, linking the right Fourier algebra $\mathcal{F}_{R}(W)$ to that of the diagonal scalar weight $\tilde W$, which underpins the explicit description of operators in $\mathcal{F}_{R}(W)$. The authors also provide a concrete sequence $Q_n$ of orthogonal polynomials for $W$, derive a three-term recurrence, and interpret the non-fullness geometrically via a two-prime quotient ring $\mathbb{C}[u,v]/(uv)$. Overall, the work augments the Matrix Bochner landscape with a new singular example that resists the existing full/partial Darboux paradigm and expands the known variety of $\mathcal{D}(W)$-algebras in the matrix setting.
Abstract
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.