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 Hermitian symplectic forms.

The main objective of the present work is to study the negative spectrum of (differential) Laplace operators on metric graphs as well as their resolvents and associated heat semigroups. We prove an upper bound on the number of negative eigenvalues and a lower bound on the spectrum of Laplace operators. Also we provide a sufficient condition for the associated heat semigroup to be positivity preserving.2000 Mathematics Subject Classification. 34B45, 34L15, 47D03.

In this article we continue our investigations of one particle quantum scattering theory for Schrödinger operators on a set of connected (idealized one‐dimensional) wires forming a graph with an arbitrary number of open ends. The Hamiltonian is given as minus the Laplace operator with suitable linear boundary conditions at the vertices (the local Kirchhoff law). In “Kirchhoff's rule for quantum wires” [J. Phys. A: Math. Gen. 32, 595–630 (1999)] we provided an explicit algebraic expression for the resulting (on‐shell) S‐matrix in terms of the boundary conditions and the lengths of the internal lines and we also proved its unitarity. Here we address the inverse problem in the simplest context with one vertex only but with an arbitrary number of open ends. We provide an explicit formula for the boundary conditions in terms of the S‐matrix at a fixed, prescribed energy. We show that any unitary n × n matrix may be realized as the S‐matrix at a given energy by choosing appropriate (unique) boundary conditions. This might possibly be used for the design of elementary gates in quantum computing. As an illustration we calculate the boundary conditions associated to the unitary operators of some elementary gates for quantum computers and raise the issue whether in general the unitary operators associated to quantum gates should rather be viewed as scattering operators instead of time evolution operators for a given time associated to a quantum mechanical Hamiltonian. We also suggest an approach by which the S‐matrix in our context may be obtained from “scattering experiments”, another aspect of the inverse problem. Finally we extend our previous discussion, how our approach is related to von Neumann's theory of selfadjoint extensions.