The spectral structure of two parameter unbounded operator pencils of waveguide type is studied. Theorems on discreteness of the spectrum for a fixed parameter are proved. Variational principles for real eigenvalues in some parts of the root zones are established. In the case of n = 1 (quadratic pencils) domains containing the spectrum are described (see Fig. 1-3). Conditions in the definition of the pencils of waveguide type arise naturally from physical problems and each of them has a physical meaning. In particular a connection between the energetic stability condition and a perturbation problem for the coefficients is given.
Mathematics Subject Classification (2000). Primary 47A75; Secondary 49R50, 34L15.
We characterize disjoint hypercyclic and supercyclic tuples of unilateral Rolewicz-type operators on c 0 (N) and ℓ p (N), p ∈ [1, ∞), which are a generalization of the unilateral backward shift operator. We show that disjoint hypercyclicity and disjoint supercyclicity are equivalent among a subfamily of these operators and disjoint hypercyclic unilateral Rolewicz-type operators always satisfy the Disjoint Hypercyclicity Criterion. We also characterize simultaneous hypercyclic unilateral Rolewicz-type operators on c 0 (N) and ℓ p (N), p ∈ [1, ∞).
Abstract. This is a largely expository paper in which a finite dimensional model for gyroscopic/waveguiding systems is studied. Properties of the spectrum that play an important role when computing with such models are studied. The notion of "waveguide-type" is explored in this context. The main theorem provides a form of the central result (due to Abramov) concerning the existence of real spectrum for such systems. The roles of semisimple/defective eigenvalues are discussed, as well as the roles played by eigenvalue "types" (or "Krein signatures"). The theory is illustrated with examples.
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