[Paper Review] Orthogonal Polynomials for Potentials of two Variables with External Sources

M. C. Bergère|ArXiv.org|Nov 24, 2003

Advanced Combinatorial Mathematics1 references3 citations

TL;DR

This paper extends Christoffel's one-variable orthogonal polynomial construction to two complex variables (z, z*) with external sources, introducing biorthogonal polynomials for potentials V(z, z*) via a determinant formula involving generalized Schur polynomials. The key contribution is a closed-form expression for monic orthogonal polynomials qn(z; ξi, ξi*) in terms of auxiliary polynomials Qn(z; ξi*) satisfying a biorthogonality condition, with the norm kn(ξi, ξi*) proven to be a ratio of determinants of Cauchy kernels, confirming a conjecture by Akemann and Vernizzi.

ABSTRACT

This publication is an exercise which extends to two variables the Christoffel's construction of orthogonal polynomials for potentials of one variable with external sources. We generalize the construction to biorthogonal polynomials. We also introduce generalized Schur polynomials as a set of orthogonal, symmetric, non homogeneous polynomials of several variables, attached to Young tableaux.

Motivation & Objective

• To generalize Christoffel's one-variable orthogonal polynomial construction with external sources to two complex variables (z, z*).
• To define and construct monic biorthogonal polynomials qn(z; ξi, ξi*) for potentials V(z, z*) with L complex external sources.
• To prove the orthogonality of the constructed polynomials using a determinant-based formula involving auxiliary polynomials Qn(z; ξi*).
• To derive a closed-form expression for the norm kn(ξi, ξi*) of the biorthogonal polynomials, confirming a conjecture by Akemann and Vernizzi.
• To introduce generalized Schur polynomials as symmetric, non-homogeneous orthogonal polynomials associated with Young tableaux in the two-variable setting.

Proposed method

• Define the two-variable potential V(z, z*) and the associated inner product ∫∫ d²z p∗m(z) pn(z) e^{-V(z,z*)} = hn δnm.
• Introduce L complex external sources (ξ1, ..., ξL) and define the weighted inner product with weight ∏|z − ξi|² e^{-V(z,z*)}.
• Construct the monic biorthogonal polynomials qn(z; ξi, ξi*) as a determinant ratio: qn = det[Qn+k(z; ξi*)] / det[Qn+k(ξi; ξi*)] × ∏(z − ξi)^{-1} × [⟨n,L⟩]^{-1}, where Qn are auxiliary polynomials.
• Define auxiliary monic polynomials Qn(z; ξi*) via the biorthogonality condition: ∫∫ d²z p∗m(z) Qn(z; ξi*) ∏(z* − ξi*) e^{-V(z,z*)} = 0 for m < n.
• Prove the orthogonality of qn by showing that the determinant structure ensures vanishing inner products for m < n, relying on the properties of Qn.
• Derive the norm kn(ξi, ξi*) as kn = hn+L × det[KN+L(ξi, ξj*)] / det[KN+L−1(ξi, ξj*)], where KN(z, ξ*) is the Cauchy kernel defined as ∑(1/hk) p∗k(ξ) pk(z).

Experimental results

Research questions

• RQ1 How can Christoffel's one-variable orthogonal polynomial construction with external sources be generalized to two complex variables?
• RQ2 What is the explicit form of the monic biorthogonal polynomials qn(z; ξi, ξi*) for a two-variable potential V(z, z*) with L complex external sources?
• RQ3 What is the norm kn(ξi, ξi*) of the biorthogonal polynomials, and can it be expressed in terms of determinants of Cauchy kernels?
• RQ4 How are the auxiliary polynomials Qn(z; ξi*) constructed, and what biorthogonality condition do they satisfy?
• RQ5 Can the conjectured formula for kn(ξi, ξi*) by Akemann and Vernizzi be rigorously proven in the two-variable case?

Key findings

• The monic biorthogonal polynomials qn(z; ξi, ξi*) are constructed as a determinant ratio involving Qn+k(z; ξi*), ensuring orthogonality under the weighted inner product with ∏|z − ξi|².
• The auxiliary polynomials Qn(z; ξi*) are uniquely defined by the biorthogonality condition ∫∫ d²z p∗m(z) Qn(z; ξi*) ∏(z* − ξi*) e^{-V(z,z*)} = 0 for m < n.
• For one external source, Qn(z; ξ*) = hn p∗n(ξ) Kn(z, ξ*), where Kn(z, ξ*) = ∑_{i=0}^n (1/hi) p∗i(ξ) pi(z) is the Cauchy kernel.
• For L sources, Qn(z; ξi*) is expressed as a determinant of p∗k(ξi) and Kn(z, ξi*), scaled by the inverse of the determinant of p∗k(ξi).
• The norm kn(ξi, ξi*) is rigorously proven to be kn = hn+L × det[KN+L(ξi, ξj*)] / det[KN+L−1(ξi, ξj*)], confirming the conjecture by Akemann and Vernizzi.
• The final expression for qn(z; ξi, ξi*) is given by a determinant of pn+L(z), pn+L(ξi), and Kn+L−1(z, ξi*), divided by the determinant of Kn+L−1(ξi, ξj*), with an additional ∏(z − ξi)^{-1} factor.