On an Invertible Extension of a Nonself-Adjoint Singular Differential Operator on the Half-Line

Abstract

We consider the differential operator generated in the class of compactly supported functions on the positive half-line by the nonself-adjoint differential expression ${-a_{2}(x)}f^{\prime {}\prime }+a_{1}(x)f^{\prime }+a_{0}(x)f$ with locally Lebesgue integrable coefficients. Cases are not excluded where $a_{j}(x)$ ($j = 0,1$) are sign-alternating or can have infinite limits as $x\to \infty$ (with any sign). Descriptions of the internal connections between the coefficients $a_{j}$ ($j = 0,1,2$) are given under which the operator in question admits a closed invertible extension in the space $L_{2}(0,\infty )$.