Quantum bound states in open geometries

Trapped modes in an infinite or semi-infinite tube with a local enlargement

Trapped modes in a hard cylindrical tube with a local axisymmetric enlargement or bulge and filled with a uniform acoustic medium is studied. The governing Helmholtz equation in the cylindrical coordinate system is employed to deal with this problem through the domain decomposition method and matching technique. The trapped modes and the corresponding frequencies less than the threshold frequency or cut-off frequency are derived. It is found that in addition to the fundamental mode, the second- and higher-order trapped modes exist and depend on the geometry parameters of the local bulge. The effects of the bulge radius and width on the frequencies are discussed. The local bulge leads to a decrease of the frequencies and the corresponding vibration mode is localized near the bulge. A multimodal analysis is made and frequency band gap of generalized trapped modes is also studied. A frequency band gap depends on the radius of a bulge and is independent of its width. The obtained results can be extended to analyze bound states in quantum wires.

Bound states in elastic waveguides

We consider numerically the L-, T-, and X-shaped elastic waveguides with the Dirichlet boundary conditions for in-plane deformations (displacements) which obey the vectorial Navier-Cauchy equation. In the X-shaped waveguide we show the existence of a doubly degenerate bound state with frequency below the first symmetrical cutoff frequency, which belongs to the two-dimensional irreducible representation E of symmetry group C(4upsilon). Moreover the next bound state is below the next antisymmetric cutoff frequency. This bound state belongs to the irreducible representation A2. The T-shaped waveguide has only one bound state while the L-shaped one has no bound states.