Quadratic Functionals with General Boundary Conditions

The purpose of this paper is to give the Reid ``Roundabout Theorem' for quadratic functionals with general boundary conditions. In particular, we describe the so-called coupled point and regularity condition introduced in [16] in terms of Riccati equation solutions. Accepted 27 February 1996     

Spectral Measures and Jacobi Matrices Related to Laguerre-Type Systems of Orthogonal Polynomials

This paper considers systems of Laguerre-type orthogonal polynomials for which the corresponding Jacobi matrices represent unbounded self-adjoint operators which are bounded above or below. Under appropriate assumptions on the coefficient sequences in the recursion formula, results are obtained on the uniform boundedness of the polynomials on bounded intervals, the absence of eigenvalues for the corresponding operator, and the absolute continuity of the measure of orthogonality. Date received: September 7, 1995. Date revised: April 17, 1996.     

Singular Sturmian separation theorems on unbounded intervals for linear Hamiltonian systems

In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case.