On the relation between Darboux transformations and polynomial mappings

2012/12/05 by Maxim Derevyagin, Derevyagin, Maxim
Mathematics · Physics and Astronomy · #30B70 #30E20 (Secondary) #42C05 #47B36 (Primary) 15A23 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.CA #math.MP #math.SP #msc:15A23 #msc:30B70 #msc:30E20 #msc:42C05 #msc:47B36

paper · pdf · doi:10.48550/arxiv.1212.1134

18 pages

arxiv created 2012/12/05 · arxiv updated 2012/12/06

Abstract

Let dμ(t) be a probability measure on [0,+∞) such that its moments are finite. Then the Cauchy-Stieltjes transform S of dμ(t) is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. In the present paper, we consider a matrix interpretation of the unwrapping transformation from S(λ) to λS(λ²), which is intimately related to the simplest case of polynomial mappings. More precisely, it is shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to the Carlitz problem appears as a shifted Darboux transformation.

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