Darboux transformability of the symmetric block-nilpotent Hermite-type family

Determine whether the Hermite-type weight matrices W(x) = e^{-x^2} e^{B x^2} e^{B^* x^2}, where B is the block-nilpotent matrix defined by equation (ex-B) in Section 6, can be obtained as Darboux transformations of classical diagonal weights, specifically the scalar Hermite weight w(x) = e^{-x^2} I, in the sense of Definition 2.3 (existence of degree-preserving differential operators intertwining the corresponding orthogonal polynomial sequences).

Background

The paper establishes a matrix-valued Hermite–Laguerre correspondence under the quadratic change of variables y = x^2, requiring the symmetry W(x) = W(-x) to preserve differential operator structures and Darboux transformations. In Sections 3–5, the authors show that symmetric Hermite-type weights yield Laguerre-type weights with parameters α = −1/2 and α = 1/2 and that differential and Darboux structures are preserved across this transformation.

Section 6 introduces an explicit, arbitrary-dimension family of symmetric Hermite-type weight matrices of the form W(x) = e^{-x^2} e^{B x^2} e^{B^* x^2}, where B is a block-nilpotent matrix (equation (ex-B)). This family admits a second-order differential operator in its algebra and naturally produces Laguerre-type weights with α = ±1/2 under y = x^2. The authors remark that, to their knowledge, this is the only explicit family satisfying W(x) = W(-x) with a second-order operator known, and they pose the open question of whether this family itself can be realized via Darboux transformations from classical diagonal weights, conjecturing that the answer is affirmative.