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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.

##### MSC:
 46N99 Miscellaneous applications of functional analysis 81R30 Coherent states
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##### References:
 [1] DOI: 10.1063/1.1727528 [2] DOI: 10.1016/0003-4916(71)90031-5 [3] DOI: 10.1007/BF02754884 [4] DOI: 10.1007/BF02906835 [5] DOI: 10.1007/BF02906835 [6] DOI: 10.1007/BF02906835 [7] DOI: 10.1016/0003-4916(77)90351-7 [8] DOI: 10.1016/0003-4916(77)90351-7 [9] DOI: 10.1063/1.98066 [10] DOI: 10.1109/TMTT.1983.1131585 [11] DOI: 10.1103/PhysRevLett.60.2081
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