Criteria for correct solvability of a general Sturm–Liouville equation in the space $L_1({\mathbb {R}})$

Abstract

We consider the equation

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

(1)

where $f\in L_1({\mathbb {R}})$ and

$$\begin{aligned}&r >0,\quad q\ge 0,\quad 1/r\in L_1^{\mathrm{loc}}({\mathbb {R}}),\quad q\in L_1^{\mathrm{loc}}({\mathbb {R}}),\end{aligned}$$

(2)

$$\begin{aligned}&\lim _{|d|\rightarrow \infty }\int _{x-d}^x\frac{dt}{r(t)}\cdot \int _{x-d}^x q(t)dt=\infty . \end{aligned}$$

(3)

By a solution of (1), we mean any function y,  absolutely continuous in ${\mathbb {R}}$ together with $ry'$, which satisfies (1) almost everywhere in ${\mathbb {R}}.$ Under conditions (2) and (3), we give a criterion for correct solvability of (1) in the space $L_1({\mathbb {R}})$.