A criterion for correct solvability in Lp(ℝ) of a general Sturm-Liouville equation

We consider an equation-(r(x)y′(x))′+q(x)y(x)=f(x),x ∞ℝ, where f ∈ L_p(ℝ) for p ∈ (1, ∞) with the following conditions: r>0,q≥0,1/r ∈ L₁ ^loc(ℝ) q ∈ L₁^loc(ℝ),∫ _-∈⁰dt/r(t)= ∫₀^∈ dt/r(t)= ∞By a solution of the above-mentioned equations, we mean any function y that is absolutely continuous together with ry′ and satisfies it almost everywhere on . Under the above-mentioned conditions, we give a criterion for the correct solvability of the above-mentioned equation in L _p(R) for p ∈ (1, ∞).