Orthogonality with Respect to the Hermite Product, KP Wave Functions, and the Bispectral Involution

TL;DR

The paper develops a unifying framework connecting KP wave functions in the adelic Grassmannian with Hermite-orthogonality via the Hermite product. By exploiting the bilinear integral form of the KP hierarchy and the bispectral involution on Calogero-Moser matrices, it proves that coefficient sequences from a KP wave function and its bispectral dual are almost bi-orthogonal with respect to the Hermite inner product, with exact orthogonality recovered in special cases such as Exceptional Hermites. It further shows that the same wave functions generate the norms of Exceptional Hermites and, at special parameter values, act as generating functions for those norms, thus explaining a deep link between soliton theory and exceptional orthogonal polynomials. The framework extends to spin Calogero-Moser matrices, producing matrix-valued orthogonality, and suggests broader applicability to higher-rank analogues. Overall, the work provides a structural explanation for observed orthogonality phenomena and opens avenues for generalizations across KP flows and bispectral duals.

Abstract

It is well-known that for any wave function $ψ(x,z)$ of the KP Hierarchy, there is another wave function called its ``adjoint'' such that the path integral of their product with respect to $z$ around any sufficiently large closed path is zero. For the wave functions in the adelic Grassmannian $Gr^{ad}$, the bispectral involution which exchanges the role of $x$ and $z$ implies also the existence of an ``$x$-adjoint wave function'' $ψ^{\star}(x,z)$ so that the product of the wave function, the $x$-adjoint, and the Hermite weight $e^{-x^2/2}$ has no residue. Utilizing this, we show that the sequences of coefficient functions in the power series expansion of any KP wave function in $Gr^{ad}$ and its image under the bispectral involution at $t_2=-1/2$ are always ``almost bi-orthogonal'' with respect to the Hermite product. Whether the sequences have the stronger properties of being (almost) orthogonal can be determined easily in terms of KP flows and the bispectral involution. As a special case, the Exceptional Hermite Orthogonal Polynomials can be recovered in this way. This provides both a generalization of and an explanation of the fact that the generating functions of the Exceptional Hermites are certain special wave functions of the KP Hierarchy. In addition, one new surprise is that the same KP Wave Function which generates the sequences of functions is also a generating function for the norms when evaluated at $t_1=1$ and $t_2=0$. The main results are proved using Calogero-Moser matrices satisfying a rank one condition. The same results also apply in the case of ``spin-generalized'' Calogero-Moser matrices, which produce instances of matrix orthogonality.

Theorems & Definitions (42)

• Remark 2.1
• Definition 2.2
• Remark 2.3
• Remark 3.1
• Theorem 4.1
• proof
• Remark 4.2
• Remark 4.3
• Lemma 5.1
• Definition 5.2