Guiding structures with multiply connected cross sections: Evolution of propagation in external fields at complex Robin parameters

Properties of the two-dimensional ring and three-dimensional infinitely long straight hollow waveguide with unit width and inner radius ρ0ρ0 in the superposition of the longitudinal uniform magnetic field B and the Aharonov–Bohm flux are analyzed within the framework of the scalar Helmholtz equation under the assumption that the Robin boundary conditions at the inner and outer confining walls contain extrapolation lengths ΛinΛin and ΛoutΛout, respectively, with nonzero imaginary parts. As a result of this complexity, the self-adjointness of the Hamiltonian is lost, its eigenvalues EE, in general, become complex too and the discrete bound states of the annulus, which are characteristic for the real ΛΛ, turn into the corresponding quasibound states with their lifetime defined by the imaginary parts EiEi of the eigenenergies while the current along the wire exponentially amplifies/attenuates with the distance depending on the sign of EiEi. It is shown that, compared to the disk geometry, the annulus opens up additional possibilities of varying magnetization and currents by tuning imaginary components of the Robin parameters on each confining circumference; in particular, the possibility of restoring a lossless longitudinal flux by zeroing EiEi is discussed from mathematical and physical points of view. For each ρ0ρ0, the energy EE turns real under a special correlation between the imaginary parts of ΛinΛin and ΛoutΛout with the opposite signs being what physically corresponds to the equal transverse fluxes through the inner and outer interfaces of the annulus. In the asymptotic case of the very large radius, ρ0→∞ρ0→∞, simple expressions are derived and applied to the analysis of the dependence of the real energy EE on ΛinΛin and ΛoutΛout. New features also emerge in the magnetic field influence; for example, if, for the quantum disk, the imaginary energy EiEi is quenched by the strong intensities BB, then for the annulus this takes place only when the inner Robin distance ΛinΛin is real; otherwise, it almost quadratically depends on BB with the corresponding enhancement of the reactive scattering. The closely related problem of the hole in the otherwise uniform medium is also addressed for the real and complex extrapolation lengths with the emphasis on the comparative analysis with its dot counterpart.

► Annular geometry with complex Robin boundary conditions offers new features as compared to the solid disk.
► Conditions for the existence of the real energies in such a non-Hermitian system are derived and analyzed.
► Wide variety of possibilities of controlling properties of the ring by the external electric and magnetic fields are predicted.