On the limit-point case of singular linear Hamiltonian systems

Abstract

This article is concerned with the limit-point case (l.p.c.) of a Hamiltonian system. We present new proofs for several existing equivalent conditions on the l.p.c. established in terms of the asymptotic behaviour of the square integrable solutions of Hamiltonian systems with different spectral parameters and functions in the domain of the corresponding maximal operator, respectively. Further, we give two equivalent conditions in terms of the asymptotic behaviour of the square integrable solutions of Hamiltonian systems with the same complex and real spectral parameters, respectively. In addition, we establish two limit-point criteria which extend the relevant existing results.

Keywords:

• singular linear Hamiltonian system
• limit-point case
• defect index

AMS Subject Classifications::

• 34B20
• 47B25

Acknowledgements

The author expresses his appreciation to the referees for their useful suggestions. This research was supported by the NNSFs of Shandong Province (Grant Y2008A02).