Erratum: 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, AA†− q2A†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 |ǫkα〉, k = 0, 1, . . . , N−1, where ǫ is a primitive Nth root of unity, ǫN = 1. A class of q-coherent states related to the bilateral q-hypergeometric series and Ramanujan type integrals is described. It includes a curious set of coherent states of the free nonrelativistic particle which is interpreted as a q-algebraic system without 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. PACS numbers: 03.65.-w, 42.50.-p, 02.30.-f Published in Phys. Rev. A52, no. 3 (1995) 1909-1935. On leave of absence from the Institute for Nuclear Research, Russian Academy of Sciences, Moscow, Russia