Some qualitative properties of the spectral density function for Hamiltonian systems

For Hamiltonian systems with one singular endpoint of limit point type, the problem is considered of determining when the continuous spectrum is absolutely continuous or continuously differentiable.

Following a method of Atkinson for (-py')' + qy = λwy , a sequence of regular nonself-adjoint boundary value problems is examined and their Titchmarsh-Weyl M(λ) - functions derived. The Pick-Nevanlinna representation of each M(λ) - function produces a density function that is C¹(R) , and whose derivative is related to an energy-like functional for the Hamiltonian system. A sequence of densities {ρ_n(λ)} associated with these regular nonself-adjoint operators is shown to converge to a density ρ(μ) associated with the M(λ)-function for the singular self-adjoint operator. The energy-like functionals are used to establish the absolute continuity and continuous differentiability of ρ(μ).

Examples are considered of a general second order system, and of a Dirac system, in which the potentials are of long range, short range, and oscillatory type. Conditions are stated in terms of these potentials such that the continuous spectrum is C¹. Examples are also considered of a coupled second order system and of a Dirac system in which the potentials include terms which are smooth and unbounded. Conditions are stated such that the continuous spectrum is C¹ , and is all of R.

In the last chapter, following a technique used by Hinton and Shaw in their consideration of the equation y⁽⁴⁾ + q(x)y = &lamdba;y , first order asymptotics are developed for the M(&lamdba;)-function associated with a second order system with coupling terms present and with minimal restrictions on these terms.