In this paper we consider a linear operator on an unbounded interval associated with a matrix linear Hamiltonian system. We characterize its Friedrichs extension in terms of the recessive system of solutions at infinity. This generalizes a similar result obtained by Marletta and Zettl for linear operators defined by even order Sturm-Liouville differential equations.
We investigate the spectral properties of the differential operator −r s∆, s ≥ 0 with the Dirichlet boundary condition in unbounded domains whose boundaries satisfy some geometrical condition. Considering this operator as a self-adjoint operator in the space with the norm ||u|| 2 L2,s(Ω) = ∫ Ω r −s |u| 2 dx, we study the structure of the spectrum with respect to the parameter s. Further we give an estimate of the rate of condensation of discrete spectra when it changes to continuous.