A criterion for correct solvability of the Sturm-Liouville equation in the space $L_{p}(R)$

Abstract:

We consider an equation \begin{equation}\tag {1} -y''(x) + q(x)\ y(x) = f(x),\qquad x\in R, \end{equation} where $f(x) \in L_{p}(R),\ p\in [1,\infty ]\ \left (\| f \|_{\infty } := C (R) \right )$, and $0 \le q(x)\in L_{1}^{\operatorname {loc}} (R).$ By a solution of equation (1), we mean any function $y(x)$ such that $y(x), y’(x) \in AC^{\operatorname {loc}} (R),$ and equality (1) holds almost everywhere on $R.$ In this paper, we obtain a criterion for the correct solvability of (1) in $L_{p} (R)$, $p \in [1,\infty ].$