Kirchhoff's rule for quantum wires

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 Hermitian symplectic forms.

Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

“…An extensive discussion of other boundary conditions which yield self adjointness can be found in [7,12]. Among those is the class of symmetric BC; the adaptation of the argument to this case is discussed in Section 6.…”

Approximations by graphs and emergence of global structures

“…The particular case α = 0 represents the most simple boundary conditions, called usually Kirchhoff [4], which we will employ in the example of Sec. IV, however, for the moment it is useful to consider the more general situation (2.3).…”

“…Nowadays, there is an extensive literature devoted to the subject; for recent reviews see [4,5] and also [6].…”

Factorization of the transition matrix for the general Jacobi system

“…Corresponding to the fragmentation (1.8), we have the factorization formula [2,4,17,19,[25][26][27] 9) where the right-hand side consists of the product of the 2 × 2 transition matrices in the order indicated.…”