Hurwitz Transformation Research Articles

Eigen-potentials in the Schrödinger equation

In this paper, we extend the number of potentials where the Schrödinger equation can be solved analytically. The technique generalizes the well-known Frobenius trial series method used successfully on, e.g., the hydrogen atom and the Simple Harmonic Oscillator problems, where each of these potentials have one term. In these simple systems, the energy is quantized by the requirement of normalizability of the wave function. Our method uses this same idea, but is applied to potentials with an arbitrary number of terms, where here these terms are monomials of integer powers of r. Upon applying our technique to these more complicated potentials, we find quantization and relationships between not only the energy but also the various coupling constants. We apply our method to the two-dimensional exciton potential and to a family of two-dimensional potentials related by the Hurwitz transformation. We also suggest families of potentials functionally invariant, or periodic, to Hurwitz transformations. Finally, we treat a three-dimensional potential of interest in nuclear physics, and the three-dimensional sextic double-well potential.

Normed Division Algebras Application to the Monopole Physics

We review some normed division algebras (R, C, H, O) applications to the monopole physics and MICZ-Kepler problems. More specifically, we will briefly review some results in applying the normed division algebras to interpret the existence of Dirac, Yang, and SO(8) monopoles. These monopoles also appear during the examination of the duality between isotropic harmonic oscillators and the MICZ-Kepler problems. We also revisit some of our newest results in the nine-dimensional MICZ-Kepler problem using the generalized Hurwitz transformation.

Hidden symmetry of the 16D oscillator and its 9D coulomb analogue

We present the quadratic Hahn algebra QH(3) as an algebra of the hidden symmetry for a certain class of exactly solvable potentials, generalizing the 16D oscillator and its 9D coulomb analogue to the generalized version of the Hurwitz transformation based on SU (1,1)⊕ SU (1,1) . The solvability of the Schrodinger equation of these problems by the variables separation method are discussed in spherical and parabolic (cylindrical) coordinates. The overlap coefficients between wave functions in these coordinates are shown to coincide with the Clebsch – Gordan coefficients for the SU(1,1) algebra.

Variables separation and superintegrability of the nine-dimensional MICZ-Kepler problem

The nine-dimensional MICZ-Kepler problem is of recent interest. This is a system describing a charged particle moving in the Coulomb field plus the field of a SO(8) monopole in a nine-dimensional space. Interestingly, this problem is equivalent to a 16-dimensional harmonic oscillator via the Hurwitz transformation. In the present paper, we report on the multiseparability, a common property of superintegrable systems, and the superintegrability of the problem. First, we show the solvability of the Schrödinger equation of the problem by the variables separation method in different coordinates. Second, based on the SO(10) symmetry algebra of the system, we construct explicitly a set of seventeen invariant operators, which are all in the second order of the momentum components, satisfying the condition of superintegrability. The found number 17 coincides with the prediction of (2n − 1) law of maximal superintegrability order in the case n = 9. Until now, this law is accepted to apply only to scalar Hamiltonian eigenvalue equations in n-dimensional space; therefore, our results can be treated as evidence that this definition of superintegrability may also apply to some vector equations such as the Schrödinger equation for the nine-dimensional MICZ-Kepler problem.

Dirac equation with anisotropic oscillator, quantum E3′ and Holt superintegrable potentials and relativistic generalized Yang–Coulomb monopole system

The two-dimensional Dirac equation with spin and pseudo-spin symmetries is investigated in the presence of the maximally superintegrable potentials. The integrals of motion and the quadratic algebras of the superintegrable quantum [Formula: see text], anisotropic oscillator and the Holt potentials are studied. The corresponding Casimir operators and the structure functions of the mentioned superintegrable systems are found. Also, we obtain the relativistic energy spectra of the corresponding superintegrable systems. Finally, the relativistic energy eigenvalues of the generalized Yang–Coulomb monopole (YCM) superintegrable system (a [Formula: see text] non-Abelian monopole) are calculated by the energy spectrum of the eight-dimensional oscillator which is dual to the former system by Hurwitz transformation.

Exact analytical solutions of the Schrödinger equation for the nine-dimensional MICZ-Kepler problem

The nine-dimensional MICZ-Kepler problem has been established recently as a system describing the motion of a nine-dimensional charged particle in the Coulomb potential with the presence of the SO(8) monopole. Interestingly, this is the last case of dimension in which the MICZ-Kepler problem is equivalent to a harmonic oscillator via generalized Hurwitz transformation. In this work, exact analytical solutions of the Schrödinger equation for the nine-dimensional MICZ-Kepler problem are successfully built for the first time and the degeneration degree of the energy is also calculated.

Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation

The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a su(2) non-Abelian monopole interaction (Yang-Coulomb monopole). This system is also related to an 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this system. They generate a quadratic algebra with structure constants involving the Casimir operator of a so(4) Lie algebra. We also show that this system remains superintegrable with a su(2) non-Abelian monopole (generalized Yang-Coulomb monopole). We study this system using parabolic coordinates and obtain from Hurwitz transformation its dual that is an 8D singular oscillator. This 8D singular oscillator is also a new superintegrable system and multiseparable. We obtained its quadratic algebra that involves two Casimir operators of so(4) Lie algebras. This correspondence is used to obtain algebraically the energy spectrum of the generalized Yang-Coulomb monopole.

A non-Abelian SO(8) monopole as generalization of Dirac-Yang monopoles for a 9-dimensional space

We establish an explicit form of a non-Abelian SO(8) monopole in a 9-dimensional space and show that it is indeed a direct generalization of Dirac and Yang monopoles. Using the generalized Hurwitz transformation, we have found a connection between a 16-dimensional harmonic oscillator and a 9-dimensional hydrogenlike atom in the field of the SO(8) monopole (MICZ-Kepler problem). Using the built connection the group of dynamical symmetry of the 9-dimensional MICZ-Kepler problem is found as SO(10, 2).

A hidden non-Abelian monopole in a 16-dimensional isotropic harmonic oscillator

We suggest one variant of generalization of the Hurwitz transformation by adding seven extra variables that allow an inverse transformation to be obtained. Using this generalized transformation we establish the connection between the Schrödinger equation of a 16-dimensional isotropic harmonic oscillator and that of a nine-dimensional hydrogen-like atom in the field of a monopole described by a septet of potential vectors in a non-Abelian model of 28 operators. The explicit form of the potential vectors and all the commutation relations of the algebra are given.

Coulomb Problem in a One-Dimensional Space with Constant Positive Curvature

We construct a complex transformation \(S_{1\mathbb{C}} \to S_1 \) generalizing the known Hurwitz transformation in the Euclidean space for one-dimensional quantum mechanics. As in the case of the flat space, this transformation allows establishing the connection between the Coulomb problem and the oscillator problem with the Calogero–Sutherland potential added. We fully describe the motion in a Coulomb field in S1 and determine the energy spectrum and the wave functions with the correct normalizing constant.

Hurwitz transformation and oscillator representation of a 5D “isospin” particle

The correspondence between 5D quantized Kepler problem and 8D harmonic oscillator in the case of topologically nontrivial constraints imposed on the wavefunction of the latter is considered. It is shown that both the “isospinor” and the “isovector” particles on the SU(2) instanton plus 1R background in 5D can be described in terms of an 8D singular oscillator.

The Hurwitz transformation: Nonbilinear version

An alternative approach to the Hurwitz (H) transformation reducing Euclidean space E8 to Euclidean space E5 is developed. It is shown how refusal of the bilinearity condition leads to the replacement of the H-transformation by its modified nonbilinear (left and right) version: HL- and HR-transformations. The HL(HR)-transformation of the specific type eight-dimensional hyperspherical coordinates is investigated. The radial coordinate u and the polar angle θ/2, in this approach, transform into u2 and θ, respectively, like in the H-case. Action of these transformations on the remaining hyperspherical coordinates, unlike in the H-case, is equivalent to the invariance (shutting) of one angular triplet and the shutting (invariance) of another triplet. The connection of HL and HR with H is established and the structure of the H-transformation itself is revealed on this basis.

Geometric quantization of the five-dimensional Kepler problem

An extension of the Hurwitz transformation to a canonical transformation between phase spaces allows conversion of the five-dimensional Kepler problem into that of a constrained harmonic oscillator problem in eight dimensions. Thus a new regularization of the Kepler problem is established. Then, following Dirac, we quantize the extended phase space, imposing constraint conditions as superselection rules. In that way the interchangeability of the reduction and the quantization procedures is proved.