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PT-symmetric branched optical lattices: Spectral properties and stability of solitons

We investigate branched PT-symmetric optical lattices. We consider both the linear and nonlinear Schrödinger equations with a PT-symmetric periodic potential on the graph and solve them by imposing weighted vertex boundary conditions. A constraint derived from these vertex conditions determines the exceptional point of the system. In the PT unbroken phase, this constraint enforces PT-symmetric boundary conditions at the vertices, ensuring a purely real spectrum; its violation leads to the emergence of complex eigenvalues in the linear regime. In the nonlinear regime, the same constraint determines the linear stability of solitons: satisfying the constraint yields stable solitons, whereas violating it corresponds to unstable solitons.