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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE015322136  
Author(s):  
Loutsenko, Igor; Spiridonov, Vyacheslav:  
Shorttitle:  
Selfsimilarity in spectral problems and qspecial functions  
Source:  
In: Levi, Decio (ed.) et al, Symmetries and integrability of difference equations, CRM Proc. Lecture Notes 25, 273293 (2000).  
Classification:  
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
Schroedinger factorization, chains of Hamiltonians, seldsimilar reductions, qspecial functions, infinite soliton systems, supersymmetry, coherent states, orthgonal polynomials, onedimensional Ising chains, random matrices  
Review:  
A review of several subjects related to the finitedifference equations, unified by the idea of contemplating Schroedinger equation in one dimension and of transforming its solutions and spectra in the way which could mimic a selfsimilar pattern. The first connection is found in the inverse scattering (or Darboux or factorization or supersymmetric) context where the selfsimilarity of the (chains of) partner potentials is interpreted as related to the so called quantum algebra symmetries. Their realization using coherent states reveals nontrivial connections to Ramanujan qbeta integral. In a discrete analogue of this case (using recurrences with orthogonal polynomials) one gets close to the qanalogues of some Painleve transcendents. In solitonic context one studies the onedimensional Ising chains and gets basic hypergeometric series and their relationship to KP solitons and random matrix models. All these exciting ideas are very fresh and present just certain first steps towards a formation of a Galois theory for operator equations.  
Remarks to the editors:  