The Y‐junction of quantum waveguides

A planar quantum waveguide is composed from two semi‐strips of width 1 and a semi‐strip of width H and the angle between strips is 2π/3. The spectrum of the Dirichlet problem in this waveguide is investigated. In particular, it is proved that there exists a critical width H * for which the discrete spectrum is non‐empty for any H ≤ H * and conversely is empty for any H ≥ H * . The critical width is estimated to be approximately H * ≈︁ 1.25. Furthermore, the eigenvalue in the discrete spectrum is unique for 0 ≤ H ≤ H * .