Principal fundamental system of solutions, The Hartman-Wintner problem and correct solvability of the general Sturm-Liouville equation

We study the problem of correct solvability in the space $L_p(\mathbb{R}),$ $p\in [1,\mathrm{\infty })$ of the equation $$-(r(x)y^{\mathrm{\prime }}(x){)}^{\mathrm{\prime }}+q(x)y(x)=f(x),x\in \mathbb{R}$$ under the conditions $$r>0,q\ge 0,\frac{1}{r}\in L_1(\mathbb{R}),q\in L_1(\mathbb{R})\mathrm{.}$$