A growth condition for Hamiltonian systems related with Kreĭn strings

Abstract

We study two-dimensional Hamiltonian systems of the form (•) y'(x) = zJH(x)y(x); x ∈ [s-; s+), where the Hamiltonian H is locally integrable on [s-; s+) and nonnegative, and J := (0 -1 | 1 0). The spectral theory of the equation changes depending on the growth of H towards the endpoint s+; the classical distinction into the Weyl alternatives 'limit point' or 'limit circle' case. A refined measure for the growth of a limit point Hamiltonian H can be obtained by comparing with H-polynomials. This growth measure is concretised by a number Δ(H) ∈ N0 ∪ {∞} and appeared first in connection with a Pontryagin space analogue of the equation (*). It is known that the growth restriction 'Δ(H) In general, the number Δ(H) is given in a rather implicit way, difficult to handle and not suitable for concrete calculations. In the present paper we provide a more accessible way to compute Δ(H) for some particular classes of Hamiltonians which occur in connection with Sturm-Liouville equations and Krein strings.