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Structures composed of classical dipoles in higher-dimensional space present a unique opportunity to venture beyond the conventional paradigm of few-body or cluster physics. In this work, we consider the six convex regular polychora that exist in an Euclidean four-dimensional space as a theoretical benchmark for hte investigation of dipolar systems in higher dimensions. The structures under consideration represent the four-dimensional counterparts of the well-known Platonic solids in three-dimensions. A dipole is placed in each vertex of the structure and is allowed to interact with the rest of the system via the usual dipole–dipole interaction generalized to the higher dimension. We use numerical tools to minimize the total interaction energy of the systems and observe that all six structures represent dipole clusters with a zero net dipole moment. The minimum energy is achieved for dipoles arranging themselves with orientations whose angles are commensurate or irrational fractions of the number π. 

We study systems of two and three electrons confined to circular rings. The electrons are considered spinless, and we assume that one electron occupies a single ring. We use the framework of such a model to calculate the linear entropy and, thus, the spatial entanglement between the confined electrons. The geometry of the problem for the case of two electrons incorporates situations in which the planes of the two rings form an arbitrary angle with each other. The resulting Schrödinger’s equation is solved numerically with very high accuracy by means of the exact diagonalization method. We compute the ground state energy and entanglement for all configurations under consideration. We also study the case of three electrons confined to identical, parallel and concentric rings which are located in three different equidistant planes. The vertically separated system of rings is allowed to gradually merge into a single ring geometry, which would represent the equivalent system of a ring with three electrons. It is observed that the system of three electrons gives rise to a richer structure, as the three rings merge into a single one. 

Systems composed of several multi-layer compounds have been extremely useful in tailoring different quantum physical properties of nanomaterials. This is very much true when it comes to semiconductor materials and, in particular, to heterostructures and heterojunctions. The formalism of a position-dependent effective mass has proved to be a very efficient tool in those cases where quantum wells emerge either in one or two dimensions. In this work, we use a variety of mathematical theorems, as well as numerical computations, to study different scenarios pertaining to choices of a specific piecewise constant effective mass for a particle that causes its energy eigenvalues to reach an extremum. These results are relevant when it comes to practical technological applications such as modifying the optical energy gap between the first excited state and the ground state energy of the system. At the end of our contribution, we also question the physical validity of some approximations for systems with particles that possess a position-dependent mass especially for those cases in which the mass distribution is divergent.