A growth condition for Hamiltonian systems related with Krein strings

On 2D newest vertex bisection: Optimality of mesh-closure and H 1-stability of L 2-projection Inverse estimates for elliptic integral operators and application to the adaptive coupling of FEM and BEM 06/2012 Quasi-optimal a priori estimates for fluxes in mixed finite element methods and applications to the Stokes-Darcy coupling 04/2012 M. Langer, H. Woracek Indefinite Hamiltonian systems whose Titchmarsh-Weyl coefficients have no finite generalized poles of non-negativity type 03/2012 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 instanciated by a number ∆(H) ∈ N0 ∪ {∞} and appeared first in connection with a Pontryagin space analogue of the equation (*). It is known that the growth restriction '∆(H) < ∞' has some striking consequences on the spectral theory of the equation; in many respects, the case 'limit point but still ∆(H) < ∞' is similar to the limit circle case. In general, the number ∆(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 ∆(H) for some particular classes of Hamiltonians which occur in connection with Sturm-Liouville equations and Kre˘ ın strings.