Surface plasmon spectrum of a metallic hyperbola can be found analytically with the separation of variables in elliptic coordinates. The spectrum consists of two branches: symmetric, low-frequency branch, $omega <omega_0/sqrt{2}$, and antisymmetric h
igh-frequency branch, $omega >omega_0/sqrt{2}$, where $omega_0$ is the bulk plasmon frequency. The frequency width of the plasmon band increases with decreasing the angle between the asymptotes of the hyperbola. For the simplest multi-connected geometry of two hyperbolas separated by an air spacer the plasmon spectrum contains two low-frequency branches and two high-frequency branches. Most remarkably, the lower of two low-frequency branches exists at $omega rightarrow 0$, i.e., unlike a single hyperbola, it is thresholdless. We study how the complex structure of the plasmon spectrum affects the energy transfer between two emitters located on the surface of the same hyperbola and on the surfaces of different hyperbolas.
We investigate the spin and charge densities of surface states of the three-dimensional topological insulator $Bi_2Se_3$, starting from the continuum description of the material [Zhang {em et al.}, Nat. Phys. 5, 438 (2009)]. The spin structure on sur
faces other than the 111 surface has additional complexity because of a misalignment of the contributions coming from the two sublattices of the crystal. For these surfaces we expect new features to be seen in the spin-resolved ARPES experiments, caused by a non-helical spin-polarization of electrons at the individual sublattices as well as by the interference of the electron waves emitted coherently from two sublattices. We also show that the position of the Dirac crossing in spectrum of surface states depends on the orientation of the interface. This leads to contact potentials and surface charge redistribution at edges between different facets of the crystal.
