[Paper Review] Single-Source Nets of Algebraically-Quantized Reflective Liouville Potentials on the Line I. Almost-Everywhere Holomorphic Solutions of Rational Canonical Sturm-Liouville Equations with Second-Order Poles

Gregory Natanson|arXiv (Cornell University)|Mar 15, 2015

Spectral Theory in Mathematical Physics102 references5 citations

TL;DR

This paper introduces a unified method to construct SUSY ladders of rational Liouville potentials (RLPs) via canonical Liouville-Darboux transformations (CLDTs), using almost-everywhere holomorphic (AEH) solutions of rational canonical Sturm-Liouville equations (RCSLEs) with second-order poles. The key contribution is the systematic generation of networks of polynomial solutions—r-, c-, or i-Gauss-seed Heine polynomials—by varying free terms in equations with energy-dependent polynomial coefficients, rooted in Jacobi, Laguerre, or Routh polynomials.

ABSTRACT

The paper presents the unified technique for constructing SUSY ladders of rational Liouville potentials (RLPs) starting from the so-called "Gauss-reference" (GRef) potentials exactly quantized on the line via classical Jacobi, classical (generalized) Laguerre, or Romanovski-Routh polynomials with energy-dependent indexes. Each RLP is obtained by means of the Liouville transformation (LT) of the appropriate rational canonical Sturm-Liouville equation (RCSLE) with second-order poles. The presented analysis takes advantage of the generic factorization of canonical Sturm-Liouville equations (CSLEs) in terms of intertwining "generalized" Darboux operators. We refer to the latter operators as the canonical Liouville-Darboux transformations (CLDTs) to stress that they are equivalent to three-step operations: i) the LT from the CSLE to the Schrodinger equation; ii) the Darboux transformation (DT) of the appropriate LP; and iii) the inverse LT from the Schrodinger equation to the new CSLE. It is proven that the CLDT preserves the rational form of the RCSLE if its factorization function (FF) is an almost-everywhere holomorphic (AEH) solution of the RCSLE (or, in other words, a solution with a rational logarithmic derivative). As explained in the paper there are up to four gauge transformations which convert each RCSLE of our interest into the second-order differential equations with energy-dependent polynomial coefficients. The most important result of the paper is that polynomial solutions of these equations belong to sequences of Heine polynomials obtained by varying free terms at fixed values of singular points and the appropriate characteristic exponents. This allows us to construct networks of polynomial solutions -- the so-called "r-, c-, or i-Gauss-seed" (r-, c-, or i-GS) Heine polynomials -- starting from Jacobi, (generalized) Laguerre or Routh polynomials, respectively.

Motivation & Objective

• To develop a unified framework for generating SUSY ladders of rational Liouville potentials (RLPs) on the line.
• To identify conditions under which canonical Liouville-Darboux transformations (CLDTs) preserve the rational form of rational canonical Sturm-Liouville equations (RCSLEs) with second-order poles.
• To establish a connection between AEH solutions of RCSLEs and sequences of Heine polynomials with energy-dependent coefficients.
• To classify and construct networks of polynomial solutions—designated as r-, c-, or i-Gauss-seed (r-, c-, i-GS) Heine polynomials—starting from classical orthogonal polynomials.
• To demonstrate that polynomial solutions of transformed equations arise from varying free terms while fixing singular points and characteristic exponents.

Proposed method

• Employing the generic factorization of canonical Sturm-Liouville equations (CSLEs) via intertwining 'generalized' Darboux operators, termed canonical Liouville-Darboux transformations (CLDTs).
• Applying a three-step CLDT process: (i) Liouville transformation (LT) from CSLE to Schrödinger equation, (ii) Darboux transformation (DT) of the resulting Liouville potential, and (iii) inverse LT back to a new CSLE.
• Requiring the factorization function (FF) to be an almost-everywhere holomorphic (AEH) solution, i.e., a solution with a rational logarithmic derivative, to preserve the rational form of the RCSLE.
• Utilizing up to four gauge transformations to convert each RCSLE into second-order differential equations with energy-dependent polynomial coefficients.
• Analyzing the resulting equations as Heine-type equations whose polynomial solutions form sequences under variation of free terms at fixed singular points and characteristic exponents.
• Constructing single-source nets of solutions—r-, c-, or i-Gauss-seed (r-, c-, i-GS) Heine polynomials—by starting from Jacobi, generalized Laguerre, or Routh polynomials, respectively.

Experimental results

Research questions

• RQ1 How can SUSY ladders of rational Liouville potentials be systematically generated from classical orthogonal polynomials?
• RQ2 Under what conditions does the canonical Liouville-Darboux transformation (CLDT) preserve the rational form of a rational canonical Sturm-Liouville equation (RCSLE) with second-order poles?
• RQ3 What is the role of almost-everywhere holomorphic (AEH) solutions with rational logarithmic derivatives in maintaining the rational structure of RCSLEs?
• RQ4 How do gauge transformations convert RCSLEs into second-order equations with energy-dependent polynomial coefficients, and what polynomial families emerge from such transformations?
• RQ5 Can networks of polynomial solutions—designated as r-, c-, or i-Gauss-seed Heine polynomials—be constructed by varying free terms while fixing singular points and characteristic exponents?

Key findings

• The canonical Liouville-Darboux transformation (CLDT) preserves the rational form of the RCSLE if and only if the factorization function (FF) is an almost-everywhere holomorphic (AEH) solution with a rational logarithmic derivative.
• Each RCSLE of interest can be transformed into a second-order differential equation with energy-dependent polynomial coefficients via up to four gauge transformations.
• Polynomial solutions of the transformed equations are identified as sequences of Heine polynomials formed by varying the free term while keeping singular points and characteristic exponents fixed.
• The resulting solution networks—r-, c-, or i-Gauss-seed (r-, c-, i-GS) Heine polynomials—originate from Jacobi, generalized Laguerre, or Routh polynomials, respectively.
• The construction establishes a unified algebraic framework for generating SUSY ladders of rational Liouville potentials (RLPs) from classical orthogonal polynomials via AEH solutions and CLDTs.
• The method provides a systematic algebraic-quantization procedure for reflective Liouville potentials on the line, rooted in the structure of Heine polynomials and energy-dependent coefficients.