2015/03/15 by Gregory Natanson, Natanson, Gregory
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1503.04798
112 pages, 1 figure
openalex publication_date 2015/03/15 · arxiv created 2015/12/09 · arxiv updated 2015/12/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.