New proof of Weyl's theorem

Abstract

Let $lu=-u^{\prime \prime }+q(x)u$, where $q(x)$ is a real-valued $L_{loc}^2(0,\infty )$ function. H. Weyl has proved in 1910 that for any $z$, $Imz\ne 0$, the equation $(l-z)w=0$, $x>0$, has a solution $w\in L^2(0,\infty )$. We prove this classical result using a new argument.