e-mail: svp@thsun1.jinr.ru

**The Factorization Method, Self-Similar Potentials and
Quantum Algebras**

**Abstract**

The factorization method is a convenient operator language formalism for consideration of certain spectral problems. In the simplest differential operators realization it corresponds to the Darboux transformations technique for linear ODE of the second order. In this particular case the method was developed by Schroedinger and became well known to physicists due to the connections with quantum mechanics and supersymmetry. In the theory of orthogonal polynomials its origins go back to Christoffel's theory of kernel polynomials, etc. Special functions are defined in this formalism as the functions associated with similarity reductions of the factorization chains.

We consider in this talk only the original Schroedinger equation case
and review some recent developments in this field. In particular, a class
of self-similar potentials is described whose discrete spectrum consists
of a finite number of geometric progressions. Such spectra are generated
by polynomial quantum algebras which include *q*-analogues of the
harmonic oscillator and *su(1,1)* algebras. Coherent states of these
potentials are described by differential-delay equations of the pantograph
type. Some applications to infinite soliton systems, Ising chains and lattice
Coulomb gases are outlined.

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