Multiple bound states in scissor-shaped waveguides

We study bound states of the two-dimensional Helmholtz equations with Dirichlet boundary conditions in an open geometry given by two straight leads of the same width which cross at an angle $\ensuremath{\theta}.$ Such a four-terminal junction with a tunable $\ensuremath{\theta}$ can realized experimentally if a right-angle structure is filled by a ferrite. It is known that for $\ensuremath{\theta}=90\ifmmode^\circ\else\textdegree\fi{}$ there is one proper bound state and one eigenvalue embedded in the continuum. We show that the number of eigenvalues becomes larger with increasing asymmetry and the bound-state energies are increasing as functions of $\ensuremath{\theta}$ in the interval $(0,90\ifmmode^\circ\else\textdegree\fi{}).$ Moreover, states which are sufficiently strongly bound exist in pairs with a small energy difference and opposite parities. Finally, we discuss how the bound states transform with increasing $\ensuremath{\theta}$ into quasibound states with a complex wave vector.