Correct Solvability, Embedding Theorems and Separability for the Sturm-Liouville Equation

We consider the equation - y"(x)+q(x)y(x)=f(x), x\in R and the weighted function space S_p^{(2)}(R,q)={y\in AC_{\loc}^{(1)}®:|y"-qy|_p+|q^{1/p}y|_p<\infty}; p\in[1,\infty), f\in L_p®$and$0\le q\in L_1^{\loc}®$. We show that there exists an embedding S_p^{(2)}(R,q)\hookrightarrow L_p® if and only if equation above is correctly solvable in $L_p®.