Universal superpositions of coherent states and self-similar potentials

A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator A=P(d/dx+x)/ √2 , where P is the parity operator. Such A arises naturally in the q→-1 limit for a symmetry operator of a specific self-similar potential obeying the q-Weyl algebra ${\mathit{AA}}^{\mathrm{°}}$-${\mathit{q}}^2$ ${\mathit{A}}^{\mathrm{°}}$A=1. Coherent states for this and other reflectionless potentials whose discrete spectra consist of N geometric series are analyzed. In the harmonic oscillator limit, the surviving part of these states takes the form of orthonormal superpositions of N canonical coherent states ‖${\mathrm{\epsilon }}^{\mathit{k}}$α〉, k=0,1,...,N-1, where ε is a primitive Nth root of unity, ${\mathrm{\epsilon }}^{\mathit{N}}$=1. A class of q-coherent states related to the bilateral q-hypergeometric series and Ramanujan type integrals is described. It includes an unusual set of coherent states of the free nonrelativistic particle, which is interpreted as a q-algebraic system without a discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural q analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.