The infinite product representation of solutions of indefinite problem with a finite number of arbitrary turning points

Abstract

In this paper we consider the Sturm-Liouville equation
\[
y''+(\rho^2\phi^2(x)-q(x))y=0\vspace{-5mm}~~~~~~~~~~~(*) \]\\
on a finite interval $ I $, say
$I=[0,1]$, under the assumption that I contains a finite number of
arbitrary type turning points, which are zeros of $\phi$ in $I $.
According to the four types of turning points, first we obtain the asymptotic forms of the solutions of (*) and then based on Hadamard's factorization theorem we use this asymptotic estimates to study the infinite product representation of solutions of such equations. Infinite product form of the solution has a basic application in studies of inverse spectral problems.