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Constraints on conductances for Y-junction of quantum wires

We consider the Y-junction, connecting three quantum wires, in the scattering states formalism. In the absence of fermionic interaction we analyze the restrictions on the values of conductance matrix, imposed by the unitarity of scattering S-matrix. Using the combination of numerical and analytical results, we describe the four-dimensional body of values of reduced conductance matrix. We show that this body touches the unit sphere at six points only, in accordance with Birkhoff-von Neumann theorem. It implies that the Abelian bosonization analysis for the vanishing interaction strength can be performed only in the vicinity of these six points.