Connection formulae for spectral functions associated with singular Sturm-Liouville equations

We consider the Sturm-Liouville equation -(py')' + qy = lambda wy on [0, infinity), with the initial condition y(0) cos alpha + p(0)y'(0) sin alpha = 0, alpha is an element of [0, pi), and suppose that Weyl's limit-point case holds at infinity. Let rho(alpha)( mu) be the corresponding spectral function and rho(alpha)'(mu) its symmetri c derivative. We show that for almost all mu is an element of R, if rho(alp ha)'(mu) exists and is positive for some alpha is an element of [0, pi), th en (i) rho(beta)'(mu) exists and is positive for all beta is an element of [0, pi), and (ii) for all alpha(1), alpha(2) is an element of (0, pi)\{1/2 pi}, 2 rho(0)'(mu) {1/sin(2 alpha(1))rho(alpha 1)'(mu) - 1/sin(2 alpha(2))rho(al pha 2)'(mu)} = rho(0)'(mu)/rho(pi/2)'{tan alpha(1) - tan alpha(2)} +cot alp ha(1) - cot alpha(2).