New proof of Weyl's theorem

TL;DR

This paper presents a new proof of Weyl's theorem, which states that for certain differential operators, solutions exist in L^2 space for complex spectral parameters, using an innovative argument.

Contribution

The paper introduces a novel proof technique for Weyl's theorem, providing an alternative to classical methods.

Findings

Validated Weyl's theorem with a new proof approach

Confirmed existence of L^2 solutions for complex spectral parameters

Enhanced understanding of spectral theory for differential operators

Abstract

Let $lu = -u^{\prime \prime} + q(x)u$, where $q(x)$ is a real-valued $L^2_{loc}(0, \infty)$ function. H. Weyl has proved in 1910 that for any $z$, $Imz \neq 0$, 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.