On the gap between the complex structured singular value and its convex upper bound

Published versio

Grassmann-Gaussian integrals and generalized star products

In quantum scattering on networks there is a non-linear composition rule for on-shell scattering matrices which serves as a replacement for the multiplicative rule of transfer matrices valid in other physical contexts. In this article, we show how this composition rule is obtained using Berezin integration theory with Grassmann variables.Comment: 14 pages, 2 figures. In memory of Al.B. Zamolodichiko

Communications and Related Projects

Contains reports on three research projects.Office of Scientific Research and Development (OSRD) OEMsr-26

Kirchhoff's Rule for Quantum Wires

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with $n$ open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with $n$ channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy $E>0$ is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms.Comment: 40 page

Superposition rules for higher-order systems and their applications

Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations and analyse their properties. Several results concerning the existence of various types of superposition rules for higher-order systems are proved and illustrated with examples extracted from the physics and mathematics literature. In particular, two new superposition rules for second- and third-order Kummer--Schwarz equations are derived.Comment: (v2) 33 pages, some typos corrected, added some references and minor commentarie

The Generalized Star Product and the Factorization of Scattering Matrices on Graphs

In this article we continue our analysis of Schr\"odinger operators on arbitrary graphs given as certain Laplace operators. In the present paper we give the proof of the composition rule for the scattering matrices. This composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs. We perform a detailed analysis of the generalized star product for arbitrary unitary matrices. The relation to the theory of transfer matrices is also discussed

Miscellaneous Problems

Contains reports on eight research projects

Miscellaneous Problems

Contains reports on eleven research projects