Resonant alteration of propagation in guiding structures with complex Robin parameter and its magnetic-field-induced restoration

Solutions of the scalar Helmholtz wave equation are derived for the analysis of the transport and thermodynamic properties of the two-dimensional disk and three-dimensional infinitely long straight wire in the external uniform longitudinal magnetic field $\bf B$ under the assumption that the Robin boundary condition contains extrapolation length $\Lambda$ with nonzero imaginary part $\Lambda_i$. As a result of this complexity, the self-adjointness of the Hamiltonian is lost, its eigenvalues $E$ become complex too and the discrete bound states of the disk characteristic for the real $\Lambda$ turn into the corresponding quasibound states with their lifetime defined by the eigenenergies imaginary parts $E_i$. Accordingly, the longitudinal flux undergoes an alteration as it flows along the wire with its attenuation/amplification being $E_i$-dependent too. It is shown that, for zero magnetic field, the component $E_i$ as a function of the Robin imaginary part exhibits a pronounced sharp extremum with its magnitude being the largest for the zero real part $\Lambda_r$ of the extrapolation length. Increasing magnitude of $\Lambda_r$ quenches the $E_i-\Lambda_i$ resonance and at very large $\Lambda_r$ the eigenenergies $E$ approach the asymptotic real values independent of $\Lambda_i$. The extremum is also wiped out by the magnetic field when, for the large $B$, the energies tend to the Landau levels. Mathematical and physical interpretations of the obtained results are provided; in particular, it is shown that the finite lifetime of the disk quasibound states stems from the $\Lambda_i$-induced currents flowing through the sample boundary. Possible experimental tests of the calculated effect are discussed; namely, it is argued that it can be observed in superconductors by applying to them the external electric field $\cal E$ normal to the surface.