The Sturm-Liouville problem and the Polar Representation Theorem

TL;DR

This paper explores the polar representation theorem for Hamiltonian systems and applies it to derive new results on the Sturm-Liouville problem using eigenvalue monotonicity properties.

Contribution

It introduces a novel application of the polar representation theorem to analyze the Sturm-Liouville problem through eigenvalue monotonicity.

Findings

Derived new results on Sturm-Liouville problems

Utilized eigenvalue monotonicity properties

Extended understanding of Hamiltonian systems

Abstract

The polar representation theorem for the n-dimensional time-dependent linear Hamiltonian system with continuous coefficients, states that, given two isotropic solutions (Q1, P1) and (Q2, P2), with the identity matrix as Wronskian,the formula Q2 = rcos(f), Q1 = rsin(f), holds, where r and f are continuous matrices, r is non-singular and f is symmetric. In this article we use the monotonicity properties of the matrix f eigenvalues in order to obtain results on the Sturm-Liouville problem.