Boundary condition at the junction

摘要

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_0 = 2m~* E_f h~(-2) of the Schroedinger 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_0ψ_0 and geometry of the leads.