Time scale symplectic systems with analytic dependence on spectral parameter

Abstract

This paper is devoted to the study of time scale symplectic systems with polynomial and analytic dependence on the complex spectral parameter . We derive fundamental properties of these systems (including the Lagrange identity) and discuss their connection with systems known in the literature, in particular with linear Hamiltonian systems. In analogy with the linear dependence on , we present a construction of the Weyl disks and determine the number of linearly independent square integrable solutions. These results extend the discrete time theory considered recently by the authors. To our knowledge, in the continuous time case this concept is new. We also establish the invariance of the limit circle case for a special quadratic dependence on and its extension to two (generally nonsymplectic) time scale systems, which yields new results also in the discrete case. The theory is illustrated by several examples.