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The wave functions for two dimensional Bloch electrons in a uniform magnetic field at the mid-band points are studied with the help of the algebraic structure of the quantum group $U_q(sl_2)$. A linear combination of its generators gives…
The energy spectrum of an electron confined to an arbitrary surface of revolution in an external magnetic field, parallel to the symmetry axis, is studied analitycally and numerically. The problem is reduced via conformal mapping to one on…
We study the Schroedinger equation for an off-diagonal tight-binding hamiltonian, as well as the equations of motion for out-of-plane vibrations, on the separable square Fibonacci quasicrystal. We discuss the nature of the spectra and wave…
We present a simple algebraic procedure that can be applied to solve a range of quantum eigenvalue problems without the need to know the solution of the Schr\"odinger equation. The procedure, presented with a pedagogical purpose, is based…
The description of interacting many-electron systems in external magnetic fields is considered in the framework of the optimized effective potential method extended to current-spin-density functional theory. As a case study, a…
The Hamiltonian of a pure hydrogen atom possesses the SO(4) symmetry group generated by the integrals of motion: the angular momentum and the Runge-Lenz vector. The pure hydrogen atom is a supersymmetric and superintegrable system, since…
We develop an algebraic approach to studying the spectral properties of the stationary Schr\"odinger equation in one dimension based on its high order conditional symmetries. This approach makes it possible to obtain in explicit form…
The intertwining technique has been widely used to study the Schr\"odinger equation and to generate new Hamiltonians with known spectra. This technique can be adapted to find the bound states of certain Dirac Hamiltonians. In this paper the…
In this paper, we study the Schr\"odinger equation with a new quasi-exactly solvable double-well potential. Exact expressions for the energies, the corresponding wave functions and the allowed values of the potential parameters are obtained…
An exactly solvable position-dependent mass Schr\"odinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems…
A new non-Hermitian E2-quasi-exactly solvable model is constructed containing two previously known models of this type as limits in one of its three parameters. We identify the optimal finite approximation to the double scaling limit to the…
We have constructed the quasi-exactly-solvable two-mode bosonic realizations of su(2) and su(1, 1) algebra. We derive the relations leading to the conditions for quasi-exact solvability of two-boson Hamiltonians by determining a general…
The magnetization for electrons on a two-dimensional sphere, under a spherically symmetrical normal magnetic field has been studied in the large field limit. This allows us to use an Euclidean approximation for low energies electron states…
In quasi-exactly solvable problems partial analytic solution (energy spectrum and associated wavefunctions) are obtained if some potential parameters are assigned specific values. We introduce a new class in which exact solutions are…
The properties of a two-dimensional electron are investigated in the presence of a circular step magnetic field profile. Both electrons with parabolic dispersion as well as Dirac electrons with linear dispersion are studied. We found that…
We construct quantum models of two particles on a compact metric graph with singular two-particle interactions. The Hamiltonians are self-adjoint realisations of Laplacians acting on functions defined on pairs of edges in such a way that…
We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives…
Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schr\"{o}dinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer,…
The electron motion in rather strong magnetic fields (when only the lowest Landau level is populated) is considered. In this case the electron kinetic energy is frozen out and the electrons are guided by slowly varied potential. Using the…
Two integrable cases of two-dimensional Schr\"odinger equation with a magnetic field are proposed. Using the polar coordinates and the symmetrical gauge, we will obtain solutions of these equation through Biconfluent and Confluent Heun…