We study the competitive action of magnetic field, Coulomb repulsion and space curvature on the motion of a charged particle. The three types of interaction are characterized by three basic lengths: l_{B} the magnetic length, l_{0} the Bohr radius and R the radius of the sphere. The energy spectrum of the particle is found by solving a Schrödinger equation of the Heun type, using the technique of continued fractions. It displays a rich set of functioning regimes where ratios \frac{R}{l_{B}} and \frac{R}{l_{0}} take definite values.
The Heun's equation with its four regular singularities emerges in many applications in science. Despite the growing interest of the scientific community, the literature has many gaps in conceptual mathematical aspects of this equation. Moreover, the translation of the mathematical language for non-mathematicians in making contemporary ideas applied in physical research is not also well developed. In this paper, Maier's Heun-to-hypergeometric reduction cases are studied in detail for four problems in quantum mechanics: (1) The Schrodinger's equation for the Coulomb problem on a 3-sphere, (2) t…
In this work, we study of the (2+1)-dimensional Dirac oscillator in the presence of a homogeneous magnetic field in an Aharonov-Bohm-Coulomb system. To solve our system, we apply the $left$-$handed$ and $right$-$handed$ projection operators in the Dirac oscillator to obtain a biconfluent Heun equation. Next, we explicitly determine the energy spectrum for the bound states of the system and their exact dependence on the cyclotron frequency $ω_c$ and on the parameters $Z$ and $Φ_{AB}$ that characterize the Aharonov-Bohm-Coulomb system. As a result, we observe that by adjusting the frequency of t…
We show that the oscillators on a sphere and pseudosphere are related, by the so-called Bohlin transformation, with the Coulomb systems on the pseudosphere: the even states of an oscillator yields the conventional Coulomb system on pseudosphere, while the odd states yield the Coulomb system on pseudosphere in the presence of magnetic flux tube generating half spin. In the higher dimensions the oscillator and Coulomb(-like) systems are connected in the similar way. In particular, applying the Kustaanheimo-Stiefel transformation to the oscillators on sphere and pseudosphere, we obtained the preu…
With the use of the stereographic projection of momentum space into the four-dimensional sphere of unit radius. the possibility of the analytical solution of the three-dimensional two-body Lippmann-Schwinger equation with the Coulomb interaction at negative energy has been studied. Simple analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction and positive integer values of the Coulomb parameter have been obtained. The worked out method has been also applied for the generalized three-dimensional Coulomb transition matrix in th…
We consider the motion of electrons confined to a two dimensional plane with an externally applied perpendicular inhomogeneous magnetic field, both with and without a Coulomb potential. We find that as long as the magnetic field is slowly-decaying, bound states in magnetic quantum dots are indeed possible. Several example cases of such magnetic quantum dots are considered in which one can find the eigenvalues and eigenfunctions in closed form, including two hitherto unknown quasi-exactly solvable models treated with confluent and biconfluent Heun polynomials. It is shown how a modulation of th…
Comment on "Anomalies in electrostatic calibration for the measurement of the Casimir force in a sphere-plane geometry"
最近W。J. Kim，M. Brown-Hayes，D. A. R. Dalvit，J. H. Brownell和R. Onofrio [Phys.Rev.A，v.78，036102（R）（2008）]对厘米大小的球面透镜上方的平板进行了静电校准，分离度低至20-30 nm，并观察到“异常行为”。据发现，梯度的静电力不依赖于分离的基础上预测的纯库仑贡献。一些假设，可能会解释偏离预期的行为被认为是，有利于补丁表面电位的影响的定性参数。我们证明，对于大透镜在几十纳米的板分离，作者使用的静电力法是不适用的，由于可能的偏差的机械抛光和地面透镜表面从一个完美的球形。提出了一个模型，它解释了所观察到的“异常行为”，使用标准的库仑力。
