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 σ _d ^h of the problem contains only one eigenvalur λ₁ ^h , but σ _d ^h = ⊘ at h ≥ h _*. The asymptotics of the eigenvalue λ₁ ^h at h → + 0 is found.