Compactness conditions for the green operator in L^p(double-struck R sign) corresponding to a general Sturm-Liouville operator

We consider the equation (0.1) -(r(x)y′(x))′ + q(x)y(x) = f(x), x ∈ double-struck R sign, where r(x) > 0, q(x) ≥ 0 for x ∈ double-struck R sign, 1/r(x) ∈ L^loc₁(double-struck R sign), q(x) ∈ L^loc₁ (double-struck R sign), f(x) ∈ L_p(double-struck R sign), p ∈ [1, ∞] (L_∞(double-struck R sign) := C(double-struck R sign)). We give necessary and sufficient conditions under which, regardless of p ∈ [1, ∞], the following statements hold simultaneously: I) For any f(x) ∈ L_p(double-struck R sign) Equation (0.1) has a unique solution y(x) ∈ L_p(double-struck R sign) where y(x) = (Gf)(x) ^def= ∫^∞_-∞ G(x,t)f(t)dt, x ∈ double-struck R sign. II) The operator G : L_p (double-struck R sign) → L_p(double-struck R sign) is compact. Here G(x,t) is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm-Liouville operator.