Correct solvability, embedding theorems and separability for the Sturm–Liouville equation

Abstract

For $p\in [1,\infty ),$ $f\in L_p(\mathbb {R})$ and $0\le q\in L_1^{{\text {loc}}}(\mathbb {R})$, we show that the weighted function space

$$\begin{aligned} S_p^{(2)}(R,q)=\left\{ y\in AC_{{\text {loc}}}^{(1)}(\mathbb {R}):\Vert y''-qy\Vert _p+\Vert q^{1/p}y\Vert _p<\infty \right\} \end{aligned}$$

is embedded into $L_p(\mathbb {R})$ if and only if the equation

$$\begin{aligned} - y''(x)+q(x)y(x)=f(x),\quad x\in \mathbb {R}, \end{aligned}$$

is correctly solvable in $L_p(\mathbb {R})$.