Trapped modes in a T-shaped waveguide

We study the discrete spectrum of the Laplace operator with the Dirichlet boundary conditions in a plane waveguide made of a band of unit width and half-band of width h perpendicular to it. The existence of critical width h* ∈ (1,2] is established: in the case of h ∈ (0, h*), the discrete spectrum σdh of the problem contains only one eigenvalur λ1h, but σdh = ⊘ at h ≥ h*. The asymptotics of the eigenvalue λ1h at h → + 0 is found.