We consider the spectral function
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025579300007646/resource/name/S0025579300007646_eqnU1.gif?pub-status=live)
associated with the Sturm-Liouville equation
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025579300007646/resource/name/S0025579300007646_eqn1.gif?pub-status=live)
in situations where q(x) →−∞ as x → ∞ and (1.1) is in the Weyl limit-point case at ∞. As usual, q is real-valued and locally integrable in [0, ∞], and our particular concern is where q(x) has the form
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0025579300007646/resource/name/S0025579300007646_eqn2.gif?pub-status=live)
where c (>0) is a parameter, s and p are non-negative on [0, ∞], p(x) → ∞ and p(x) = 0{s(x) } as x → ∞. As the boundary condition at x = 0, we take the Dirichlet condition y(0) = 0 for convenience: we can equally take the Neumann condition y′ (0) = 0 or generally a1y(0) + a2y′ (0) = 0 with real a1 and a2.