Regular Solutions of the Shabat Equation

The Shabat equation whereμandqare parameters, is the simplest self-similar reduction of the so-called dressing chain for constructing and analyzing exactly solvable Schrödinger equations. It is so relevant to the study of theq-oscillator algebra in quantum mechanics. The main objective of this paper is to investigate whether the Shabat equation has a nontrivial global solutionf(t) C1( ) that is normalizable in the sense that the corresponding potentialu(t) f2(t)−f′(t)−μ/(1−q2) is in the function space L1( ) and whether in the caseμ/(1−q2)>0 it has a global solutionf(t) C1( ) that is regular in the sense thatf(t)=± + (t−2) andf′(t)= (t−3) ast→±∞.