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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE015322136
Author(s):
Loutsenko, Igor; Spiridonov, Vyacheslav:
Shorttitle:
Self-similarity in spectral problems and q-special functions
Source:
In: Levi, Decio (ed.) et al, Symmetries and integrability of difference equations, CRM Proc. Lecture Notes 25, 273-293 (2000).
Classification:
Primary Classification:
33D80Connections with quantum groups, Chevalley groups, p-adic groups, Hecke algebras, and related topics
Secondary Classification:
33DxxBasic hypergeometric functions
34M50Inverse problems Riemann-Hilbert, inverse differential Galois, etc.
34C28Complex behavior, chaotic systems
44A30Multiple transforms
81Q60Supersymmetric quantum mechanics
05A30q-calculus and related topics
81R30Coherent states
82B20Lattice systems Ising, dimer, Potts, etc. and systems on graphs
Keywords:
Schroedinger factorization, chains of Hamiltonians, seld-similar reductions, q-special functions, infinite soliton systems, supersymmetry, coherent states, orthgonal polynomials, one-dimensional Ising chains, random matrices
Review:


A review of several subjects related to the finite-difference
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 self-similar pattern. The
first connection is found in the inverse scattering (or Darboux or
factorization or supersymmetric) context where the self-similarity
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 q-beta
integral. In a discrete analogue of this case (using recurrences
with orthogonal polynomials) one gets close to the q-analogues of
some Painleve transcendents. In solitonic context one studies the
one-dimensional 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: