University of Cambridge on a Functional Riccati Equation Regular Solutions of the Shabat Equation

The Shabat equation f 0 (t) + q 2 f 0 (qt) + f 2 (t) ? q 2 f 2 (qt) = ; t 2 R; where and q are parameters, is the simplest self-similar reduction of the so called dressing chain for constructing and analysing exactly solvable Schrr odinger equations. It is also relevant to the study of the q-oscillator algebra in quantum mechanics. The main objective of this paper is to investigate whether the Shabat equation has a nontrivial global solution f(t) 2 C 1 (R) that is normalisable in the sense that the corresponding potential u(t) := f 2 (t) ? f 0 (t) ? =(1 ? q 2) is in the function space L 1 (R) and whether in the case =(1 ? q 2) > 0 it has a global solution f(t) 2 C 1 (R) that is regular in the sense that f(t) = p =(1 ? q 2) + O ? t ?2 and f 0 (t) = O ? t ?3 as t ! 1.