Some Exactly-Solvable Quantum Problems and their Applications to Hetero- and Nano-Structures with Nontrivial Topology

Analytical calculations based on a Landau Level (LL) picture are reported for a many-electron system moving in an interface (with a finite-width Quantum Well (QW)) and in the presence of an external perpendicular magnetic field. They lead to a sequence of previously unnoticed singular features in global magnetization and magnetic susceptibility that give rise to nontrivial corrections to the standard de Haas-van Alphen periods. Additional features due to Zeeman splitting are also reported (such as new energy minima that originate from the interplay of QW, Zeeman and LL Physics). A corresponding calculation in a Composite Fermion picture leads to new predictions on magnetic response properties of a fully-interacting electron liquid in a finite-width interface. Exact solutions for the energetics of a fully three-dimensional system of many noninteracting electrons in a magnetic field are also presented (a system mostly discussed in astrophysical applications), with Hurwitz zeta functions playing an important role on thermodynamic properties. Finally, exact solutions for the energetics of an electron gas on a cylindrical surface (and in the presence of an Aharonov-Bohm flux threading the cylinder) are presented, in two cases: when the radius R of the cylinder is microscopically small and the length of the cylinder is macrocopically large (an electron gas nanotube), and vice versa. Inclusion in the above systems of a radial magnetic field and also of Zeeman splitting gives rise to curvature-induced spin-orbit coupling, that leads to interesting behaviors that can also be dealt with analytically. A corresponding SU(2) formulation and a number of diagonalization results for the energy of such curved systems have, in the limit of vanishing curvature, the correct behaviors (namely, the standard spin-Physics in flat space, that is decoupled from the standard orbital-Physics of Landau Levels).