Green-Liouville approximation and correct solvability in L_p(ℝ) of the general Sturm-Liouville equation

We consider the equation

\(-(r(x) y^{\prime}(x))^{\prime}+q(x)y(x)=f(x),\quad x\in\mathbb R,\) $$-(r(x)y^{\mathrm{\prime }}(x){)}^{\mathrm{\prime }}+q(x)y(x)=f(x),x\in \mathbb{R},$$

where f ∈ L_p(ℝ), p ∈ (1, ∞) and

\(r>0,\quad{{1}\over{{r}}}\in L_1^{\text{loc}}(\mathbb R),\quad q\in L_1^{\text{loc}}(\mathbb R).\) $$r>0,\frac{1}{r}\in L_1^{\text{loc}}(\mathbb{R}),q\in L_1^{\text{loc}}(\mathbb{R}).$$

For particular equations of this form, we suggest some methods for the study of the question on requirements to the functions r and q under which the above equation is correctly solvable in the space L_p(ℝ), p ∈ (1, ∞).