Trigonometric Rosen–Morse potential

non-negative integer, and is due to Schrödinger[3] who intended to formulate the hydrogen atom problem on Albert Einstein's closed universe,

, the direct product of a time line with a three-dimensional closed space of positive constant curvature, the hypersphere

, and introduced it on this geometry in his celebrated equation as the counterpart to the Coulomb potential, a mathematical problem briefly highlighted below.

case: Four-dimensional rigid rotator in inertial quantum motion on the three dimensional hypersphere

Stated differently, the equation (10), and its version (14) describe inertial (free) quantum motion of a rigid rotator in the four-dimensional Euclidean space,

, a reason for which Schrödinger considered it as the counterpart to the Coulomb potential in flat space, by itself a fundamental solution to the

Due to this analogy, the cotangent function is frequently termed to as "curved Coulomb" potential.

[4] Such an interpretation ascribes the cotangent potential to a single charge source, and here lies a severe problem.

In order to understand this decisive issue, one needs to focus attention to the necessity of ensuring validity on

according to The electric field to this dipole is obtained in the standard way through differentiation as and coincides with the precise expression prescribed by the Gauss theorem on

, in which case the special notation of is introduced for the so-called fundamental coupling constant of electrodynamics.

in combination with the superposition principle, and could be interpreted as a dipole potential generated by a system consisting of two opposite fundamental charges.

The result has been concluded from fitting magnetic dipole elements to hydrogen hyper-fine structure effects (see [11]} and reference therein).

The aforementioned radius is sufficiently large to allow approximating the hyper-sphere locally by plane space in which case the existence of single charge still could be justified.

In cases in which the hyper spherical radius becomes comparable to the size of the system, the charge neutrality takes over.

[12] [13] [14] Coulomb fluids consist of dipolar particles and are modelled by means of direct numerical simulations.

It is commonly used to choose cubic cells with periodic boundary conditions in conjunction with Ewald summation techniques.

In a more efficient alternative method pursued by,[15][16] one employs as a simulation cell the hyper spherical surface

The solution found, equals the potential in (30), modulo conventions regarding the charge signs and units.

The confining nature of the cotangent potential in (28) finds an application in a phenomenon known from the physics of strong interaction which refers to the non-observability of free quarks, the constituents of the hadrons.

Quarks are considered to possess three fundamental internal degree of freedom, conditionally termed to as "colors", red

Quark "colors" are the fundamental degrees of freedom of the Quantum Chromodynamics (QCD), the gauge theory of strong interaction.

, of the transferred momentum, giving rise to the so-called, running of the strong coupling constant,

[18] However, at low momentum transfer, near the so-called infrared regime, the momentum dependence of the color charge significantly weakens,[19] and in starting approaching a constant value, drives the Gauss law back to the standard form known from Abelian theories.

geometry could be viewed as the unique closed space-like geodesic of a four-dimensional hyperboloid of one sheet,

, foliating outside of the causal Minkowski light-cone the space-like region, assumed to have one more spatial dimension, this in accord with the so-called de Sitter Special Relativity,

[21] Indeed, potentials, in being instantaneous and not allowing for time orderings, represent virtual, i.e. acausal processes and as such can be generated in one-dimensional wave equations upon proper transformations of virtual quantum motions on surfaces located outside the causal region marked by the Light Cone.

[7] An illustrative example for the application of the color confining dipole potential in (39) to meson spectroscopy is given in Fig.

It should be pointed out that the potentials in the above equations (23) and (24) have been alternatively derived in,[22][23] from Wilson loops with cusps, predicting their magnitude as

parametrization of interest to QCD from the previous section, qualifies it to studies of phase transitions in systems with electromagnetic or strong interactions on hyperspherical "boxes" of finite volumes [25] .

Figure 1: The charge neutrality of the sphere. The lines pouring out of a source placed on the sphere necessarily intersect at the anti-podal point whether or not a real charge of opposite sign is placed there. Consistent formulation of charge-static on the sphere requires a real charge at the anti-podal point and thereby charge-dipoles as fundamental configurations. Consequently, the sphere supports only poles with an integer .
Figure 2: Schematic presentation of the shape of the harge ipole potential in ( 25 ) on . This potential is confining in the sense that with growing distance from the equator, it becomes infinite near the poles, thus preventing the "escape" of the charges, and keeping them "confined" on . Instead, with the decrease of the distances to the equator, it becomes gradually vanishing, and leaves the charges in this region "asymptotically free". In this way, charge-statics on closed spaces provide convenient templates for simulations of confinement phenomena, the most prominent being the confinement of color charges known from the quantum chromodynamics (QCD).
Figure 3: Schematic presentation of the internal meson structure.
Figure 4: The mass distributions of the mesons with spin and CP parities calculated by the aids of and the mass formula in eq. ( 33 ) modulo the replacement of by .