Vadim Kostrykin, Robert Schrader
Kirchhoff's Rule for Quantum Wires
(137K, LATeX 2e)
ABSTRACT. 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.