Self-similarity in Spectral Problems and q-special Functions

Abstract

Similarity symmetries of the factorization chains for one-dimensional differential and finite-difference Schr\"odinger equations are discussed. Properties of the potentials defined by self-similar reductions of these chains are reviewed. In particular, their algebraic structure, relations to $q$-special functions, infinite soliton systems, supersymmetry, coherent states, orthogonal polynomials, one-dimensional Ising chains and random matrices are outlined.