arXiv subjects

O. Ogulcan Tuncer

Publications and source records attributed to O. Ogulcan Tuncer.

7 recordsLinked to original sources

arXiv2026

Nested polynomial algebras and the exact spectrum of a three-dimensional two-spin oscillator

We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-$\tfrac12$ degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular--spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series $su(1,1)$ representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional $j=0$ sector, complete energy-shell multiplicities, and the effect of particle interchange.

math-ph↗

arXiv2026

Superintegrable systems of two interacting spin-$\frac12$ particles with first-order vector integrals of motion

We study quantum superintegrability for two interacting non-relativistic spin-$\frac12$ particles in three-dimensional Euclidean space. The Hamiltonian contains a central potential together with spin-orbit, spin-spin, tensor, and quadratic spin-orbit interaction terms, all depending only on the relative distance. We restrict the classification to the $V_4=0$ case, so that the spin-momentum interaction term is not included. We determine all such systems admitting non-trivial first-order vector integrals of motion. For this purpose, we construct the most general Hermitian first-order vector operator built from the relative position, momentum, orbital angular momentum, and the two spin vectors. The commutativity condition with the Hamiltonian leads to an overdetermined system of radial determining equations, whose solution gives the complete list of admissible potentials and the corresponding vector integrals within this class. The results extend the previous classifications of scalar and pseudo-scalar first-order integrals for two particles with spin. We also discuss selected symmetry algebras generated by the vector integrals and show, in one representative case, how a scalar reduction leads to exact Coulomb- and oscillator-type solutions.

math-ph↗

arXiv2026

Superintegrability in the interaction of two particles with spin: First-order pseudo-scalar integrals of motion

In recent work, we initiated a research program aimed at the systematic investigation of quantum superintegrable systems describing the interaction of two non-relativistic spin-$1/2$ particles in three-dimensional Euclidean space. In that study, we classified all such superintegrable systems admitting additional first-order scalar integrals of motion. In the present paper, we continue this program by focusing on systems that admit additional pseudo-scalar integrals of motion. Starting from the most general rotationally invariant Hamiltonian for two interacting spin-$1/2$ particles, we construct the most general first-order pseudo-scalar operator in the form of a matrix polynomial in the momenta. Imposing the commutativity of this operator with the Hamiltonian leads to a system of determining equations. By solving these equations, we obtain a complete classification of such superintegrable systems and determine the corresponding pseudo-scalar integrals of motion. The resulting classification provides new families of superintegrable systems with spin-dependent interactions. These systems enrich the class of integrable models relevant to nucleon--nucleon interactions and contribute to the broader program of classifying superintegrable quantum systems with spin. For selected cases, we further construct the associated polynomial symmetry algebras generated by the integrals of motion, providing additional insight into the algebraic structure of the systems.

math-ph↗

arXiv2025

Superintegrability in the interaction of two particles with spin

We initiate a research program for the systematic investigation of quantum superintegrable systems involving the interaction of two non-relativistic particles with spin $1/2$ moving in the three-dimensional Euclidean space. In this paper, we focus specifically on such superintegrable systems that allow additional scalar integrals of motion, linear in the momenta. We first identify specific potentials appearing in the Hamiltonian that should be excluded from the analysis, as they can be immediately derived through gauge transformations of the natural Hamiltonian. Next, we construct the most general symmetric scalar operator and derive the determining equations from its commutation with the Hamiltonian. Solving these equations, we obtain 30 new superintegrable systems with spin, along with their corresponding first-order scalar integrals of motion.

math-ph↗

arXiv2023

Null surfaces of pseudo-spherical spacelike framed curves in the anti-de Sitter 3-space

We introduce null surfaces (or nullcone fronts) of pseudo-spherical spacelike framed curves in the three-dimensional anti-de Sitter space. These surfaces are formed by the light rays emitted from points on anti-de Sitter spacelike framed curves. We then classify singularities of the nullcone front of a pseudo-spherical spacelike framed curve and show how these singularities are related to the singularities of the associated framed curve. We also define a family of functions called the Anti-de Sitter distance-squared functions to explain the nullcone front of a pseudo-spherical spacelike framed curve as a wavefront from the viewpoint of the Legendrian singularity theory. We finally provide some examples to illustrate the results of this paper.

math.DG↗

arXiv2023

Singularities of focal sets of pseudo-spherical framed immersions in the three-dimensional anti-de Sitter space

We introduce pseudo-spherical non-null framed curves in the three-dimensional anti-de Sitter spacetime and establish the existence and uniqueness of these curves. We then give moving frames along pseudo-spherical framed curves, which are well-defined even at singular points of the curve. These moving frames enable us to define evolutes and focal surfaces of pseudo-spherical framed immersions. We investigate the singularity properties of these evolutes and focal surfaces. We then reveal that the evolute of a pseudo-spherical framed immersion is the set of singular points of its focal surface. We also interpret evolutes and focal surfaces as the discriminant and the secondary discriminant sets of certain height functions, which allows us to explain evolutes and focal surfaces as wavefronts from the viewpoint of Legendrian singularity theory. Examples are provided to flesh out our results, and we use the hyperbolic Hopf map to visualize these examples.

math.DG↗

arXiv2023

Pedal curves of hyperbolic frontals and their singularities

This paper introduces pedal curves of spacelike frontals in the hyperbolic 2-space. We mainly investigate the singularities of these hyperbolic pedal curves of spacelike frontals for non-singular and singular dual curve germs. We then show that for non-singular dual curve germs, the singularities of a pedal curve are dependent on the singularities of the first hyperbolic Legendrian curvature germ and the location of the pedal point, while for singular dual curve germs, they depend upon the singularities of both hyperbolic Legendrian curvature germs and also the location of the pedal point. We provide several examples with figures.

math.DG↗