Kirchhoff's Rule for Quantum Wires

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dc.creatorKostrykin, Vadim
dc.creatorSchrader, Robert
dc.date1998-06-18
dc.date1999-10-15
dc.date.accessioned2026-07-07T04:32:26Z
dc.date.available2026-07-07T04:32:26Z
dc.descriptionIn 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.
dc.description40 pages
dc.identifierhttps://arxiv.org/abs/math-ph/9806013
dc.identifierhttp://arxiv.org/abs/math-ph/9806013
dc.identifierJ. Phys. A: Math. Gen. Vol.32 (1999) 595-630
dc.identifierdoi:10.1088/0305-4470/32/4/006
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/58188
dc.subjectMathematical Physics
dc.subjectRings and Algebras
dc.subject81U20; 47A40; 34L25; 82D20; 82D60
dc.titleKirchhoff's Rule for Quantum Wires
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