Orthogonal Laurent polynomials in unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

Orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of a Cantero-Moral-Velazquez moment matrix, which is constructed in terms of a complex quasi-definite measure supported in the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials in the unit circle and the corresponding second kind functions. Jacobi operators, 5-term recursion relations and Christoffel-Darboux kernels, projecting to particular spaces of truncated Laurent polynomials, and corresponding Christoffel-Darboux formulae are obtained within this point of view in a completely algebraic way. Cantero-Moral-Velazquez sequence of Laurent monomials is generalized and recursion relations, Christoffel-Darboux kernels, projecting to general spaces of truncated Laurent polynomials and corresponding Christoffel-Darboux formulae are found in this extended context. Continuous deformations of the moment matrix are introduced and is shown how they induce a time dependant orthogonality problem related to a Toda-type integrable system, which is connected with the well known Toeplitz lattice. Using the classical integrability theory tools the Lax and Zakharov-Shabat equations are obtained. The dynamical system associated with the coefficients of the orthogonal Laurent polynomials is explicitly derived and compared with the classical Toeplitz lattice dynamical system for the Verblunsky coefficients of Szeg\H{o} polynomials for a positive measure. Discrete flows are introduced and related to Darboux transformations. Finally, the representation of the orthogonal Laurent polynomials (and its second kind functions), using the formalism of Miwa shifts, in terms of $\tau$-functions is presented and bilinear equations are derived.

💡 Research Summary

This paper establishes a profound and systematic connection between the theory of orthogonal Laurent polynomials on the unit circle (OLPUC) and two-dimensional Toda-type integrable hierarchies, using an algebraic framework centered on the Cantero-Moral-Velazquez (CMV) ordering and Gauss-Borel factorization.

The work begins by constructing a moment matrix g from a complex quasi-definite measure μ supported on the unit circle, using the specific CMV order for the Fourier basis: χ(z) = (1, z⁻¹, z, z⁻², z², ...)^⊤. The quasi-definiteness of μ ensures the existence of a Gauss-Borel (LU) factorization g = S₁⁻¹ S₂, where S₁ is a lower triangular matrix with unit diagonal and S₂ is an upper triangular matrix. This factorization is the primary algebraic engine.

From the factors S₁ and S₂, the authors define two pairs of semi-infinite vectors: Φ₁ = S₁ χ and Φ₂ = (S₂⁻¹)† χ. The components of Φ₁ yield the sequences of orthogonal Laurent polynomials (OLPUC), while the components of Φ₂ are their associated second kind functions. A crucial biorthogonality relation ⟨Φ₁, Φ₂†⟩ = I holds. Within this unified framework, the standard theory is recovered algebraically: the multiplication operator by z is represented by a five-diagonal CMV matrix leading to 5-term recursion relations, Jacobi operators are identified, and Christoffel-Darboux kernels for projection onto specific truncated Laurent polynomial spaces (like `Λ^{