Boundary condition at the junction

Abstract

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 \( \vec{\psi}(a) = (\psi_1 (a),\, \psi_2 (a),\, \psi_3 (a), \ldots \psi_n (a) ),\,\, \vec{\psi}' = (\psi'_1 (a),\, \psi'_2 (a),\, \psi'_3 (a), \ldots \psi'_n (a) ) \) of the wave-function on the leads \( \omega_1,\, \omega_2,\ldots \omega_n\) at the junction a. In particular Datta proposed parametrization of the boundary condition, for symmetric T-junction, by some orthogonal 1-d projection \( P_0: R_n \to R_n \) $$ P_0^{\bot}\,\,\vec{\psi}(a) = 0,\quad P_0\,\,\vec{\psi}'(a) = 0. $$ We consider an arbitrary junction, \( n \geq 3\), of 2-d leads attached to a 2-d vertex domain \(\Omega_{int}\), in case, when there exist a resonance eigenvalue \(\lambda_0 = 2 m^* E_f \,\hbar^{-2}\) 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 \(\psi_0: L_{int} \psi_0 = \lambda_0 \psi_0\) and geometry of the leads.