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

TL;DR

This paper investigates the correct solvability of a Sturm-Liouville differential equation in the space L_p(R) under specific conditions on the coefficients, focusing on the fundamental system of solutions and the Hartman-Wintner problem.

Contribution

It provides new criteria for the correct solvability of Sturm-Liouville equations with particular coefficient conditions, extending existing theory.

Findings

Established conditions for correct solvability in L_p(R)

Analyzed the principal fundamental system of solutions

Connected the Hartman-Wintner problem to solvability criteria

Abstract

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