Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix‐functions

Abstract

AbstractWe consider the spectral problem for the Schrödinger operator ℒ︁ on a quantum network, represented as a union of a finite number of quantum wells and finite or semi‐infinite straight quantum wires connecting them to each other and to infinity. The absolutely continuous spectrum of the Schrödinger operator, with real piecewisecontinuous potential on the wells and constant potential on the wires, has a band structure defined by the geometry and the potentials on the semi‐infinite wires. We split the Schrödinger operator, depending on the Fermi level Ë, into an orthogonal sum of two operators ℒ︁ → lΛ ⊕ LΛ, which have the complementary branches of the absolutely continuous spectra coincident, counting multiplicity, respectively with the unionsΣ = ∪σopen , Σ = ∪σclosed of absolutely continuous branches σ = [λσ , ∞) of open and closed channels in the semi‐infinite wires, with thresholds λσ < Λ, λσ > Λ, respectively. This enables us to reduce the problem of evaluation of the scattering matrix on the network to the solution of a finite linear system, involving the Dirichlet‐to‐Neumann map of the intermediate Hamiltonian LΛ. We calculate the intermediate Dirichlet‐to‐Neumann map approximately, for thin networks, based on the classical DN‐map of the relevant Schrödinger operator on the compact part of the network. To this end we develop a modified analytic perturbation procedure for meromorphic operatorfunctions with small one‐dimensional residues. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)