IJPAM: Volume 61, No. 4 (2010)

THE STURM-LIOUVILLE PROBLEM AND
THE POLAR REPRESENTATION THEOREM

Jorge RezendeGrupo de Física-Matemática, Universidade de Lisboa, Professor Gama Pinto Av. 2, Lisboa, 1649-003, PORTUGAL
Grupo de Física-Matemática
Universidade de Lisboa
Professor Gama Pinto Av. 2, Lisboa, 1649-003, PORTUGAL
and
Departamento de Matemática
Faculdade de Ciências
Universidade de Lisboa, PORTUGAL
e-mail: [email protected]


Abstract.The polar representation theorem for the $n$-dimensional time-dependent linear Hamiltonian system \begin{equation*}
\dot{Q}=BQ+CP\text{, \ }\dot{P}=-AQ-B^{*}P\text{,}
\end{equation*} with continuous coefficients, states that, given two isotropic solutions $%
\left( Q_1,P_1\right) $ and $\left( Q_2,P_2\right) $, with the identity matrix as Wronskian, the formula \begin{equation*}
Q_2=r\cos \varphi \text{, \ }Q_1=r\sin \varphi \text{,}
\end{equation*} holds, where $r$ and $\varphi $ are continuous matrices, $\det r\neq 0$ and $%
\varphi $ is symmetric.

In this article we use the monotonicity properties of the matrix $\varphi $ eigenvalues in order to obtain results on the Sturm-Liouville problem.

Dedicated to the memory of Professor Ruy Luís Gomes.


Received: May 10, 2010

AMS Subject Classification: 34B24, 34C10, 34A30

Key Words and Phrases: Sturm-Liouville theory, Hamiltonian systems, polar representation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 61
Issue: 4