Lectures on exceptional orthogonal polynomials and rational solutions to Painleve equations

These are the lecture notes for a course on exceptional polynomials taught at the \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications} that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten years. In addition, some new results are presented on the construction of rational solutions to Painlev'e equation PIV and its higher order generalizations that belong to the $A_{2n}^{(1)}$-Painlev'e hierarchy. The construction is based on dressing chains of Schr"odinger operators with potentials that are rational extensions of the harmonic oscillator. Some of the material presented here (Sturm-Liouville operators, classical orthogonal polynomials, Darboux-Crum transformations, etc.) are classical and can be found in many textbooks, while some results (genus, interlacing and cyclic Maya diagrams) are new and presented for the first time in this set of lecture notes.

💡 Research Summary

The lecture notes by Gómez‑Ullate and Milson provide a comprehensive synthesis of two rapidly developing areas of mathematical physics: exceptional orthogonal polynomials (EOPs) and rational solutions of Painlevé equations, especially those belonging to the A₂ₙ⁽¹⁾‑Painlevé hierarchy. The authors begin by recalling the classical Sturm‑Liouville framework in which Hermite, Laguerre, and Jacobi polynomials arise as eigenfunctions of exactly solvable Schrödinger operators (the harmonic, isotonic, and Darboux‑Pöschl‑Teller potentials). They then introduce Darboux transformations in a formal setting, emphasizing the intertwining relations that connect a given Schrödinger operator L with its Darboux partner \tilde L. A key definition is “exact solvability by polynomials”: an operator L is exactly solvable if its eigenfunctions can be written as a fixed prefactor μ(x) multiplied by a polynomial y_k(z(x)) of degree k, for all but finitely many k. This captures the classical families and sets the stage for generating new solvable operators via rational Darboux transformations.

The authors identify the seed functions that preserve exact solvability. For the harmonic oscillator, they combine the physical bound states ϕ_n = e^{−x²/2} H_n(x) (n ≥ 0) with the non‑square‑integrable “virtual” states ϕ̃_n = e^{x²/2} \tilde H_n(x) (n ≥ 1), unifying them into a single integer‑indexed family ϕ_n, n∈ℤ. This unified family provides all possible rational Darboux transformations of the oscillator. By iterating Darboux steps, they obtain the Darboux‑Crum formula: after n successive transformations with seed functions ϕ₁,…,ϕ_n, the new potential is U_n = U – 2 (log W