Bound states in curved quantum waveguides. (English) Zbl 0693.46066

Summary: A free quantum particle living on a curved planar strip \(\Omega\) of a fixed width d with Dirichlet boundary conditions is studied. It can serve as a model for electrons in thin films on a cylinder-type substrate, or in a curved quantum wire. Assuming that the boundary of \(\Omega\) is infinitely smooth and its curvature decays fast enough at infinity, it is proved that a bound state with energy below the first transversal mode exists for all sufficiently small d. A lower bound on the critical width is obtained using the Birman-Schwinger technique.


46N99 Miscellaneous applications of functional analysis
81R30 Coherent states
Full Text: DOI


[1] DOI: 10.1063/1.1727528 · doi:10.1063/1.1727528
[2] DOI: 10.1016/0003-4916(71)90031-5 · doi:10.1016/0003-4916(71)90031-5
[3] DOI: 10.1007/BF02754884 · doi:10.1007/BF02754884
[4] DOI: 10.1007/BF02906835 · doi:10.1007/BF02906835
[5] DOI: 10.1007/BF02906835 · doi:10.1007/BF02906835
[6] DOI: 10.1007/BF02906835 · doi:10.1007/BF02906835
[7] DOI: 10.1016/0003-4916(77)90351-7 · doi:10.1016/0003-4916(77)90351-7
[8] DOI: 10.1016/0003-4916(77)90351-7 · doi:10.1016/0003-4916(77)90351-7
[9] DOI: 10.1063/1.98066 · doi:10.1063/1.98066
[10] DOI: 10.1109/TMTT.1983.1131585 · doi:10.1109/TMTT.1983.1131585
[11] DOI: 10.1103/PhysRevLett.60.2081 · doi:10.1103/PhysRevLett.60.2081
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