Annemarie Luger

Sum rules and constraints on passive systems

A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. A system on convolution form is fully described by its transfer function, and the class of Herglotz functions, holomorphic functions mapping the open upper half plane to the closed upper half plane, is closely related to the transfer functions of passive systems. Following a well-known representation theorem, Herglotz functions can be represented by means of positive measures on the real line. This fact is exploited in this paper in order to rigorously prove a set of integral identities for Herglotz functions that relate weighted integrals of the function to its asymptotic expansions at the origin and infinity. The integral identities are the core of a general approach introduced here to derive sum rules and physical limitations on various passive physical systems. Although similar approaches have previously been applied to a wide range of specific applications, this paper is the first to deliver a general procedure together with the necessary proofs. This procedure is described thoroughly, and exemplified with examples from electromagnetic theory. (Less)

On the number of negative eigenvalues of the Laplacian on a metric graph

The number of negative eigenvalues of self-adjoint Laplacians on metric graphs is calculated in terms of the boundary conditions and the underlying geometric structure. This extends and complements earlier results by Kostrykin and Schrader (2006 Contemp. Math. 415 201-25).

Analytic characterizations of Jordan chains by pole cancellation functions of higher order

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of the representing relation of length l, we show that there exists a pole cancellation function of order at least l, and, moreover, the construction shows that it is of surprisingly simple form.

Generalized Nevanlinna Functions : Operator Representations, Asymptotic Behavior

This article gives an introduction and short overview on generalized Nevanlinna functions, with special focus on asymptotic behavior and its relation to the operator representation.

Asymptotics of the Weyl function for Schrödinger operators with measure-valued potentials

We derive an asymptotic expansion for the Weyl function of a one-dimensional Schrödinger operator which generalizes the classical formula by Atkinson. Moreover, we show that the asymptotic formula can also be interpreted in the sense of distributions.

Sum rules and constraints on passive systems with applications in electromagnetics

A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. In this paper there is a review of some results on linear, time translational invariant, continuous, causal and passive systems, where it turns out that Herglotz functions are related to the Fourier transform of the impulse response of such systems. Some well known facts of this function class is considered, and a set of integral identities and an outline of the proof of these are presented. The identities may be used to derive sum rules and constraints on various physical systems. The theory is illuminated with two examples from electromagnetics: the first revisits Fanos maching equations, while the latter makes a link to the Kramers-Kronig relations and discusses physical limitations on metamaterials.