TY - JOUR

T1 - Boundary condition at the junction

AU - Harmer, Mark

AU - Pavlov, Boris

AU - Yafyasov, Adil

PY - 2007/9/1

Y1 - 2007/9/1

N2 - When modeling a 2-d quantum network by a 1-d quantum graph one usually substitutes the 2-d vertex domains by the point-wise junctions with appropriate boundary conditions imposed on the boundary values ψ(a) = (ψ 1 (a), ψ 2 (a), ψ 3 (a), ...ψ n (a)), ψ′ = ψ′ 1 (a), ψ′ 2 (a), ψ′ 3 (a),... ψ′ n (a)) of the wave-function on the leads ω 1 , ω 2 ,...ω n at the junction a. In particular Datta proposed parametrization of the boundary condition, for symmetric T-junction, by some orthogonal 1-d projection P 0 : R n → R n P 0 ⊥ ψ(a) = 0, P 0 ψ′(a) = 0. We consider an arbitrary junction, n ≥ 3 of 2-d leads attached to a 2-d vertex domain Ω int , in case, when there exist a resonance eigenvalue λ = 2m* E∫ ℏ -2 of the Schrödinger operator L int . We derive, from the first principles, energy-dependent boundary conditions for thin, quasi-1-d, network, and obtain from it, in the limit of zero temperature, Datta-type boundary condition, interpreting the projection P 0 in terms of the resonance eigenfunction ψ 0 : L int ψ 0 = λ 0 ψ 0 and geometry of the leads.

AB - When modeling a 2-d quantum network by a 1-d quantum graph one usually substitutes the 2-d vertex domains by the point-wise junctions with appropriate boundary conditions imposed on the boundary values ψ(a) = (ψ 1 (a), ψ 2 (a), ψ 3 (a), ...ψ n (a)), ψ′ = ψ′ 1 (a), ψ′ 2 (a), ψ′ 3 (a),... ψ′ n (a)) of the wave-function on the leads ω 1 , ω 2 ,...ω n at the junction a. In particular Datta proposed parametrization of the boundary condition, for symmetric T-junction, by some orthogonal 1-d projection P 0 : R n → R n P 0 ⊥ ψ(a) = 0, P 0 ψ′(a) = 0. We consider an arbitrary junction, n ≥ 3 of 2-d leads attached to a 2-d vertex domain Ω int , in case, when there exist a resonance eigenvalue λ = 2m* E∫ ℏ -2 of the Schrödinger operator L int . We derive, from the first principles, energy-dependent boundary conditions for thin, quasi-1-d, network, and obtain from it, in the limit of zero temperature, Datta-type boundary condition, interpreting the projection P 0 in terms of the resonance eigenfunction ψ 0 : L int ψ 0 = λ 0 ψ 0 and geometry of the leads.

KW - Eigenfunction

KW - Eigenvalue

KW - Junction

KW - Scattering matrix

UR - http://www.scopus.com/inward/record.url?scp=34247340417&partnerID=8YFLogxK

U2 - 10.1007/s10825-006-0085-7

DO - 10.1007/s10825-006-0085-7

M3 - Article

AN - SCOPUS:34247340417

VL - 6

SP - 153

EP - 157

JO - Journal of Computational Electronics

JF - Journal of Computational Electronics

SN - 1569-8025

IS - 1-3

ER -