Asymptotic form and infinite product representation of solution of a second order initial value problem with a complex parameter and a finite number of turning points

Abstract

The paper studies the differential equation $$y'' + (\rho ^2 \varphi ^2 (x) - q(x))y = 0$$ (*) on the interval I = [0, 1], containing a finite number of zeros 0 < x1 < x2 < ... < xm < 1 of ϕ2, i.e. so-called turning points. Using asymptotic estimates from [6] for appropriate fundamental systems of solutions of (*) as |ρ| → ∞, it is proved that, if there is an asymptotic solution of the initial value problem generated by (*) in the interval [0, x1), then the asymptotic solutions in the remaining intervals can be obtained recursively. Furthermore, an infinite product representation of solutions of (*) is studied. The representations are useful in the study of inverse spectral problems for such equations.