1. Introduction
The study of quantum systems subject to combinations of fundamental interactions, in particular the harmonic oscillator and the Coulomb potential, lies at the core of theoretical and mathematical physics. Their superposition, often enriched by a singular 1/r2 term, arises in a variety of realistic physical settings: atoms trapped in optical lattices [1], strongly correlated plasmas [2], nuclear confinement models [3], or electrons in curved nanostructures such as deformed graphene [4]. Such systems, sometimes referred to as generalized Kratzer potentials, are well known for their spectral richness; they are generally not exactly solvable in flat Euclidean space.
Indeed, in flat space the isotropic oscillator is endowed with a U(N) dynamical symmetry, whereas the hydrogen atom is characterized by a hidden SO(N + 1) symmetry associated with the Runge–Lenz vector [5, 6]. The direct superposition
$\begin{eqnarray}\begin{array}{r}V(r)\propto {r}^{2}-\frac{1}{r}+\frac{\alpha }{{r}^{2}},\end{array}\end{eqnarray}$
breaks these competing symmetries, thus preventing any closed-form analytical solution. One is then led to rely on variational [7], perturbative [8], generalized pseudospectral [9], semiclassical (Wentzel–Kramers–Brillouin (WKB)) [10], or numerical methods [11]. While effective, such approaches do not provide access to an exact spectral structure.A significant conceptual advance emerged with the introduction of geometric deformations. Placing the system in a space of constant curvature modifies not only the metric, but also the Laplacian and the volume element. This deformation induces effective terms in the quantum potential, which may restore algebraic properties lost in flat space. In this framework, it has been shown that the harmonic and Coulomb potentials, considered separately, become superintegrable on the sphere or hyperbolic space [5, 6]. Subsequent studies [12–16] have deepened these insights, unveiling exact dualities as well as supersymmetric structures intrinsic to curved geometries.
However, to the best of our knowledge, no study has so far established the exact coexistence—that is, the simultaneous solvability—of both potentials within a single Schrödinger equation defined on a curved space of arbitrary dimension. The present work aims precisely to fill this gap.
To this end, we adopt a conformal metric of the form
$\begin{eqnarray}\begin{array}{r}{\rm{d}}{s}^{2}=\frac{1}{{(1+\tau {r}^{2})}^{2}}\left({\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{N-1}^{2}\right),\end{array}\end{eqnarray}$
which allows us to construct a radial Hamiltonian incorporating the harmonic, Coulomb, and 1/r2 terms simultaneously. By applying a supersymmetric factorization adapted to this geometry [16], we impose shape invariance, leading to an exact integrability condition that links the curvature τ, the frequency ω, the angular momentum ℓ, and the dimension N. More specifically, the analysis of the constraint $\begin{eqnarray}\begin{array}{r}{\ell }({\ell }+N-2)\lt \frac{1}{4}\left(-{N}^{2}+8N-15\right),\end{array}\end{eqnarray}$
shows that only the dimension N = 4 admits a physical coexistence, even for ℓ = 0. Importantly, this dimension is not merely a mathematical curiosity: it already occupies a privileged place in several physical contexts. For instance, it has been demonstrated [17] that an exactly solvable oscillator on the four-dimensional sphere, coupled to a Yang monopole, possesses a degenerate ground state, suggesting a possible connection with the four-dimensional quantum Hall effect.These observations show that curvature is not merely a kinematical parameter but acts as an active geometric mediator capable of restoring hidden symmetries that are broken in flat space. The purpose of this article is to make this geometric mechanism explicit and to highlight the emergence of the critical dimension.
The organization of the paper is as follows. Section 2 introduces the geometric framework based on a conformal metric, including an explicit verification of its constant curvature, and presents the construction of the corresponding radial Hamiltonian. Section 3 is devoted to the spectral analysis within the supersymmetric approach: we first study the generalized harmonic oscillator, then extend the analysis to include the Kepler–Coulomb interaction. A numerical validation of the ground-state energy is also provided to support the analytical results. We then discuss the precise condition for the exact coexistence of the two potentials and its physical and geometric implications, highlighting the emergence of a critical dimension. Finally, section 4 concludes with perspectives on the broader role of geometry in quantum dynamics.
2. Construction of the radial Hamiltonian in conformal space
The study of quantum systems in spaces of nonzero curvature has witnessed remarkable development in recent decades, both in fundamental physics (such as quantum cosmology and lower-dimensional gravity) and in applied contexts including trapped atoms and curved two-dimensional materials such as deformed graphene [5, 6, 13, 14]. In such settings, geometry plays an active role: it modifies the structure of the Laplacian, the volume element, and consequently the very nature of quantum dynamics.
To capture these effects while retaining an analytically tractable framework, we adopt a metric conformal to the Euclidean one in the radial direction. This approach unifies spherical (τ > 0), hyperbolic (τ < 0), and Euclidean (τ = 0) geometries within a single formalism, while preserving constant curvature, a key ingredient for potential superintegrability [12, 15].
We consider the N-dimensional conformal metric 6 ), we obtain 4 ) has constant scalar curvature. Since a space of constant sectional curvature κ satisfies R = N(N − 1)κ, we identify κ = 4τ.
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & {{\rm{\Omega }}}_{c}^{2}(r)\left({\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{N-1}^{2}\right),\\ {{\rm{\Omega }}}_{c}(r) & = & \frac{1}{1+\tau {r}^{2}},\end{array}\end{eqnarray}$
where $\tau \in {\mathbb{R}}$ is the effective curvature parameter and ${\rm{d}}{{\rm{\Omega }}}_{N-1}^{2}$ denotes the standard metric on the unit sphere SN−1. This metric is conformally related to the flat Euclidean metric $\begin{eqnarray*}\bar{g}={\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}_{N-1}^{2},\qquad g={{\rm{e}}}^{2u}\bar{g},\end{eqnarray*}$
with conformal factor $\begin{eqnarray}\begin{array}{r}{{\rm{e}}}^{2u}=\frac{1}{{(1+\tau {r}^{2})}^{2}},\quad u(r)=-{\mathrm{ln}}(1+\tau {r}^{2}).\end{array}\end{eqnarray}$
Since $\bar{g}$ is flat, its scalar curvature vanishes. For a conformal transformation $g={{\rm{e}}}^{2u}\bar{g}$ in dimension N, the scalar curvature is given by $\begin{aligned}R[g]= & \mathrm{e}^{-2 u}\left(-2(N-1) \Delta_{\bar{g}} u\right. \\& \left.-(N-1)(N-2)|\nabla u|_{\bar{g}}^{2}\right) .\end{aligned}$
The required derivatives, computed with respect to the flat metric $\bar{g}$, are $\begin{eqnarray}\begin{array}{r}{u}^{{\prime} }(r)=-\frac{2\tau r}{1+\tau {r}^{2}},\quad {u}^{{\prime\prime} }(r)=-\frac{2\tau (1-\tau {r}^{2})}{{(1+\tau {r}^{2})}^{2}},\end{array}\end{eqnarray}$
leading to $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Delta }}}_{\bar{g}}u & = & -\frac{2\tau (N+(N-2)\tau {r}^{2})}{{(1+\tau {r}^{2})}^{2}},\\ | {\rm{\nabla }}u{| }_{\bar{g}}^{2} & = & \frac{4{\tau }^{2}{r}^{2}}{{(1+\tau {r}^{2})}^{2}}.\end{array}\end{eqnarray}$
Substituting into equation ( $\begin{eqnarray}\begin{array}{r}R[g]=4\tau \,N(N-1),\end{array}\end{eqnarray}$
which shows that the metric (The Ricci (Ric) and Riemann tensors (Rijkl) therefore take their standard constant-curvature forms,
$\begin{eqnarray}\begin{array}{rcl}{\rm{Ric}} & = & 4\tau (N-1)\,g,\\ {R}_{ijkl} & = & \kappa \,({g}_{ik}{g}_{jl}-{g}_{il}{g}_{jk}),\end{array}\end{eqnarray}$
confirming the geometric consistency of the conformal ansatz.The Laplace–Beltrami operator associated with the metric g reads
$\begin{eqnarray}\begin{array}{r}{{\rm{\Delta }}}_{g}f=\frac{1}{\sqrt{| g| }}{\partial }_{\mu }(\sqrt{| g| }\,{g}^{\mu \nu }{\partial }_{\nu }f),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{r}\sqrt{| g| }={{\rm{\Omega }}}_{c}^{N}(r)\,{r}^{N-1}\sqrt{{\rm{\det }}({g}_{{S}^{N-1}})}.\end{array}\end{eqnarray}$
Owing to spherical symmetry, the Laplacian separates into radial and angular parts. After straightforward calculation, one finds
$\begin{eqnarray}\begin{array}{rcl}{({{\rm{\Delta }}}_{g}f)}_{{\rm{rad}}} & = & {(1+\tau {r}^{2})}^{2}\frac{{{\rm{d}}}^{2}f}{{\rm{d}}{r}^{2}}+(1+\tau {r}^{2})\\ & & \times \left[\frac{N-1}{r}(1+\tau {r}^{2})-2\tau r(N-2)\right]\frac{{\rm{d}}f}{{\rm{d}}r},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{r}{({{\rm{\Delta }}}_{g}f)}_{{\rm{ang}}}=-{(1+\tau {r}^{2})}^{2}\frac{{\ell }({\ell }+N-2)}{{r}^{2}}\,f,\end{array}\end{eqnarray}$
where ${\ell }\in {\mathbb{N}}$ is the orbital quantum number.The resulting radial Hamiltonian operator reads
$\begin{eqnarray}\begin{array}{rcl}{H}_{{\rm{R}}} & = & -\frac{{\hslash }^{2}}{2m}\left[{(1+\tau {r}^{2})}^{2}\frac{{{\rm{d}}}^{2}}{{\rm{d}}{r}^{2}}+(1+\tau {r}^{2})\right.\\ & & \times \left(\frac{N-1}{r}(1+\tau {r}^{2})-2\tau r(N-2)\right)\frac{{\rm{d}}}{{\rm{d}}r}\\ & & \left.-{(1+\tau {r}^{2})}^{2}\frac{{\ell }({\ell }+N-2)}{{r}^{2}}\right].\end{array}\end{eqnarray}$
We supplement this kinetic Hamiltonian with three physically relevant contributions:
a harmonic potential $\frac{1}{2}m{\omega }^{2}{r}^{2}$, modeling quadratic confinement;
a singular inverse-square term $\frac{{\hslash }^{2}}{2m}\,\frac{\alpha }{{r}^{2}}$, which frequently appears in conformal symmetry contexts or in the presence of topological defects [18, 19];
a Kepler–Coulomb potential $-\frac{k}{r}$, chosen specifically to preserve the hidden dynamical symmetry associated with the Runge–Lenz vector, a key condition for superintegrability [5, 6, 14].
It is important to distinguish here the potential $-\frac{k}{r}$ from the electrostatic Coulomb potential in N dimensions. The latter, obtained as the solution of Poisson’s equation ∇2V = −ρ in flat N-dimensional space, behaves as ${V}_{{\rm{Coul}}}(r)\propto \frac{1}{{r}^{N-2}}$ for N ≥ 3. In contrast, the Kepler–Coulomb potential $-\frac{k}{r}$ is retained precisely because it preserves the Runge–Lenz symmetry independently of N. This makes it the natural choice in studies of superintegrable systems on curved spaces [12, 13].
The full Hamiltonian studied in this work is therefore
$\begin{eqnarray}\begin{array}{r}H={H}_{{\rm{R}}}+\frac{1}{2}m{\omega }^{2}{r}^{2}-\frac{k}{r}+\frac{{\hslash }^{2}}{2m}\frac{\alpha }{{r}^{2}}.\end{array}\end{eqnarray}$
This construction provides a nontrivial generalization of classical systems such as the harmonic oscillator and the Kepler problem to a curved space of arbitrary dimension N. Whereas most existing approaches treat these potentials independently, the framework developed here enables their exact superposition within a single Schrödinger equation, provided a geometric constraint arising from shape invariance is satisfied. The resulting coexistence, made possible through the mediating action of curvature, constitutes the central conceptual novelty of this work.
3. Spectral analysis of the system via the supersymmetric method
The supersymmetric factorization method (SUSY-QM) provides a powerful tool for the exact solution of Schrödinger equations, both in standard quantum mechanics and in curved geometric settings. Unlike the traditional Hermitian approach, it relies on the introduction of scaling operators that are not necessarily adjoint, thereby enabling the construction of partner Hamiltonians and the exploitation of shape invariance properties. This generalization is particularly suited to our framework: it allows one to capture simultaneously the combined effects of curvature, the singular term, and the fundamental harmonic and Coulomb interactions.
In what follows, we apply this method to two distinct configurations. The first corresponds to the generalized harmonic oscillator, which serves as a reference case to analyze the joint influence of curvature and the singular term. The second extends this system by incorporating the Kepler–Coulomb potential, thereby introducing a long-range attractive interaction. Finally, we determine the conditions under which the exact coexistence of these two potentials is possible, which manifest themselves as a strict geometric constraint.
3.1. Generalized harmonic oscillator in curved space
The corresponding radial Hamiltonian takes the form 15 ). This system models a generalized harmonic oscillator in a space of constant curvature, supplemented by a singular term, a feature characteristic of conformal symmetry models and of media with topological defects.
$\begin{eqnarray}\begin{array}{r}{H}_{{\rm{osc}}}={H}_{{\rm{R}}}+\frac{1}{2}m{\omega }^{2}{r}^{2}+\frac{{\hslash }^{2}}{2m}\frac{\alpha }{{r}^{2}},\end{array}\end{eqnarray}$
where HR is defined in equation (The associated eigenvalue equation for (17 ) reads
$\begin{eqnarray}\begin{array}{r}H\psi (r)=(\varepsilon -2\tau \beta )\,\psi (r),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}H & = & -{(1+\tau {r}^{2})}^{2}\frac{{{\rm{d}}}^{2}}{{\rm{d}}{r}^{2}}-(1+\tau {r}^{2})\\ & & \times \left[\frac{N-1}{r}(1+\tau {r}^{2})-2\tau r(N-2)\right]\frac{{\rm{d}}}{{\rm{d}}r}\\ & & +\frac{\alpha +\beta }{{r}^{2}}+({a}^{2}+{\tau }^{2}\beta ){r}^{2},\end{array}\end{eqnarray}$
and the dimensionless parameters $\begin{eqnarray}\begin{array}{rcl}a & = & \frac{m\omega }{\hslash },\quad \beta ={\ell }({\ell }+N-2),\\ \varepsilon & = & \frac{2m{E}^{{\rm{osc}}}}{{\hslash }^{2}}.\end{array}\end{eqnarray}$
The Hamiltonian is factorized as H ≡ H− = BA, with scaling operators defined by
$\begin{eqnarray}\begin{array}{rcl}A & = & F(r)\frac{{\rm{d}}}{{\rm{d}}r}+W(r)+{\rm{\Omega }}(r),\\ B & = & -F(r)\frac{{\rm{d}}}{{\rm{d}}r}+W(r)-{\rm{\Omega }}(r),\end{array}\end{eqnarray}$
where F(r) = 1 + τr2. The superpotentials W(r) and Ω(r) are chosen in rational form, adapted to curved geometry: $\begin{eqnarray}\begin{array}{rcl}W(r) & = & {c}_{1}r+\frac{{d}_{1}}{r},\\ {\rm{\Omega }}(r) & = & {c}_{2}r+\frac{{d}_{2}(1+\tau {r}^{2})}{r}.\end{array}\end{eqnarray}$
The associated partner Hamiltonians are
$\begin{eqnarray}\begin{array}{r}{H}^{\mp }=-{\left[F(r)\frac{{\rm{d}}}{{\rm{d}}r}\right]}^{2}-2F(r){\rm{\Omega }}(r)\frac{{\rm{d}}}{{\rm{d}}r}+{V}^{\mp }(r),\end{array}\end{eqnarray}$
with partner potentials given by $\begin{eqnarray}\begin{array}{rcl}{V}^{\mp }(r) & = & \mp F(r)\left(\frac{{\rm{d}}W}{{\rm{d}}r}\pm \frac{{\rm{d}}{\rm{\Omega }}}{{\rm{d}}r}\right)\\ & & +{W}^{2}(r)-{{\rm{\Omega }}}^{2}(r).\end{array}\end{eqnarray}$
Identification with the effective potential imposes the following relations:
$\begin{eqnarray}\begin{array}{r}{c}_{2}=-\tau (N-2),\qquad {d}_{2}=\frac{1}{2}(N-1),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{c}_{1}=\frac{\tau }{2}+\frac{1}{2}\sqrt{4{a}^{2}+4{\tau }^{2}\beta +{\tau }^{2}{(N-4)}^{2}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}{d}_{1}=-\frac{1}{2}-\frac{1}{2}\sqrt{{(N-2)}^{2}+4(\alpha +\beta )}.\end{array}\end{eqnarray}$
A central property of SUSY-QM is shape invariance, which guarantees the exact solvability of the spectrum. In our case, one verifies that
$\begin{eqnarray}\begin{array}{r}\begin{array}{l}{V}^{+}(r;{c}_{1},{c}_{2},{d}_{1},{d}_{2})\\ \quad =\,{V}^{-}(r;{c}_{1}+\tau ,{c}_{2},{d}_{1}-1,{d}_{2})\\ \qquad +\,4[{c}_{1}+\tau (1-{d}_{1})].\end{array}\end{array}\end{eqnarray}$
This shape invariance allows the spectrum to be constructed recursively. The energy levels are then given by
$\begin{eqnarray}\begin{array}{r}{\varepsilon }_{n}^{-}=4n({c}_{1}-\tau {d}_{1})+4{n}^{2}\tau ,\quad n=0,1,2,\ldots \end{array}\end{eqnarray}$
from which the total energy of the system follows: $\begin{eqnarray}\begin{array}{rcl}\frac{2m{E}_{n}^{{\rm{osc}}}}{{\hslash }^{2}} & = & (4n+1)({c}_{1}-\tau {d}_{1})+2\tau (2{n}^{2}+\beta )\\ & & -{\tau }^{2}({N}^{2}-2N-1)-2{c}_{1}{d}_{1}.\end{array}\end{eqnarray}$
In the limit τ → 0, one recovers the well-known spectrum of the N-dimensional harmonic oscillator with a 1/r2 term:
$\begin{eqnarray}\begin{array}{rcl}{E}_{n}^{{\rm{osc}}}(\tau =0) & = & \hslash \omega \left(2n+1\right.\\ & & \left.+\frac{1}{2}\sqrt{{(N-2)}^{2}+4[\alpha +{\ell }({\ell }+N-2)]}\right).\end{array}\end{eqnarray}$
The ground-state wavefunction is defined by the condition Aψ0 = 0, which leads to
$\begin{eqnarray}\begin{array}{r}{\psi }_{0}(r)={ \mathcal N }\,{r}^{-({d}_{1}+{d}_{2})}{(1+\tau {r}^{2})}^{\frac{{d}_{2}}{2}-\frac{{c}_{1}}{2\tau }-\frac{{c}_{2}}{2\tau }},\end{array}\end{eqnarray}$
which in the limit τ = 0 reduces to $\begin{eqnarray}\begin{array}{r}{\psi }_{0}(r)={ \mathcal N }\,{r}^{-({d}_{1}+{d}_{2})}{{\rm{e}}}^{\left(-\frac{a}{2}{r}^{2}\right)}.\end{array}\end{eqnarray}$
Excited states are obtained algebraically by repeated application of the creation operator:
$\begin{eqnarray}\begin{array}{r}{\psi }_{n}(r)={C}_{n}{B}^{n}{\psi }_{0}(r),\end{array}\end{eqnarray}$
with Cn a normalization constant.This analysis establishes a solid basis for subsequently addressing the system enriched by a Kepler–Coulomb term, which is treated in the next subsection.
3.2. Harmonic oscillator extended by the generalized Kepler–Coulomb potential
In this case, the Hamiltonian of the system is given by equation (16 ). This model corresponds to a generalized hydrogen-like atom evolving in a space of constant curvature, where the Kepler–Coulomb potential is supplemented by a harmonic confining term and a singular 1/r2 contribution. While in flat space such a superposition is generally not exactly solvable, the conformal geometry associated with the metric (4 ) makes an analytic resolution possible, provided certain geometric constraints are satisfied.
The associated spectral equation reads
$\begin{eqnarray}\begin{array}{r}\widetilde{H}\,{\rm{\Phi }}(r)=(E-2\tau \beta )\,{\rm{\Phi }}(r),\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}\widetilde{H} & = & -{(1+\tau {r}^{2})}^{2}\frac{{{\rm{d}}}^{2}}{{\rm{d}}{r}^{2}}-(1+\tau {r}^{2})\\ & & \times \left[\frac{N-1}{r}(1+\tau {r}^{2})-2\tau r(N-2)\right]\frac{{\rm{d}}}{{\rm{d}}r}\\ & & +\frac{\alpha +\beta }{{r}^{2}}+({a}^{2}+{\tau }^{2}\beta ){r}^{2}-\frac{2\zeta }{r},\end{array}\end{eqnarray}$
where the dimensionless parameters are $\begin{eqnarray}\begin{array}{rcl}a & = & \frac{m\omega }{\hslash },\qquad \beta ={\ell }({\ell }+N-2),\\ \zeta & = & \frac{mk}{{\hslash }^{2}},\qquad E=\frac{2m{E}^{{kc}}}{{\hslash }^{2}}.\end{array}\end{eqnarray}$
We apply a supersymmetric factorization analogous to the previous case:
$\begin{eqnarray}\begin{array}{r}{\widetilde{H}}^{-}=\widetilde{B}\widetilde{A},\qquad {\widetilde{H}}^{+}=\widetilde{A}\widetilde{B},\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}\widetilde{A} & = & F(r)\frac{{\rm{d}}}{{\rm{d}}r}+\widetilde{W}(r)+{\rm{\Omega }}(r),\\ \widetilde{B} & = & -F(r)\frac{{\rm{d}}}{{\rm{d}}r}+\widetilde{W}(r)-{\rm{\Omega }}(r),\end{array}\end{eqnarray}$
where F(r) = 1 + τr2. The superpotentials are chosen as $\begin{eqnarray}\begin{array}{rcl}\widetilde{W}(r) & = & {\widetilde{c}}_{1}+\frac{{\widetilde{d}}_{1}}{r},\\ {\rm{\Omega }}(r) & = & {c}_{2}+\frac{{d}_{2}(1+\tau {r}^{2})}{r}.\end{array}\end{eqnarray}$
The associated partner Hamiltonians take the form
$\begin{eqnarray}\begin{array}{rcl}{\widetilde{H}}^{\mp } & = & -{\left[F(r)\frac{{\rm{d}}}{{\rm{d}}r}\right]}^{2}-2F(r){\rm{\Omega }}(r)\frac{{\rm{d}}}{{\rm{d}}r}\\ & & +{\widetilde{V}}^{\mp }(r),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{\widetilde{V}}^{\mp }(r) & = & \mp F(r)\left(\frac{{\rm{d}}\widetilde{W}}{{\rm{d}}r}\pm \frac{{\rm{d}}{\rm{\Omega }}}{{\rm{d}}r}\right)\\ & & +{\widetilde{W}}^{2}(r)-{{\rm{\Omega }}}^{2}(r).\end{array}\end{eqnarray}$
By identification with the spectral equation, one obtains the following conditions:
$\begin{eqnarray}\begin{array}{rcl}{\widetilde{c}}_{1} & = & -\frac{\zeta }{{\widetilde{d}}_{1}},\qquad {c}_{2}=-\tau (N-2),\\ {d}_{2} & = & \frac{1}{2}(N-1),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{r}\begin{array}{l}{\widetilde{d}}_{1}=-\frac{1}{2}-\frac{1}{2}\\ \times \sqrt{1+4\alpha +4{\ell }({\ell }+N-2)+(N-1)(N-3)}.\end{array}\end{array}\end{eqnarray}$
The compatibility condition between the harmonic and Coulomb potentials leads to the relation
$\begin{eqnarray}\begin{array}{r}\begin{array}{l}{\tau }^{2}\left[{\ell }({\ell }+N-2)+\frac{1}{4}({N}^{2}-8N+15)\right]\\ \,+\,\frac{{m}^{2}{\omega }^{2}}{{\hslash }^{2}}=0,\end{array}\end{array}\end{eqnarray}$
which explicitly expresses the role of curvature τ as a mediating parameter ensuring the exact balance between the two interactions.Shape invariance plays a central role here: it ensures that the structure of the system is preserved under supersymmetric factorization, thereby making the exact resolution of the Schrödinger equation (35 ) possible. More precisely, it manifests as the following relation between partner Hamiltonians:
$\begin{eqnarray}\begin{array}{rcl}{\widetilde{H}}^{+}(r;{\widetilde{d}}_{1}) & = & {\widetilde{H}}^{-}(r;{\widetilde{d}}_{1}-1)+\frac{{\zeta }^{2}}{{\widetilde{d}}_{1}^{2}}\\ & & -\frac{{\zeta }^{2}}{{({\widetilde{d}}_{1}-1)}^{2}}-\tau (2{\widetilde{d}}_{1}-1),\end{array}\end{eqnarray}$
or, in terms of effective potentials, $\begin{eqnarray}\begin{array}{rcl}{\widetilde{V}}^{+}(r;{\widetilde{d}}_{1}) & = & {\widetilde{V}}^{-}(r;{\widetilde{d}}_{1}-1)+\frac{{\zeta }^{2}}{{\widetilde{d}}_{1}^{2}}\\ & & -\frac{{\zeta }^{2}}{{({\widetilde{d}}_{1}-1)}^{2}}-\tau (2{\widetilde{d}}_{1}-1).\end{array}\end{eqnarray}$
This equation shows that the partner potentials retain their functional structure under an integer shift of the parameter ${\widetilde{d}}_{1}$, which enables the recursive and exact construction of the spectrum.Through this mechanism, the energy levels of the Hamiltonian ${\widetilde{H}}^{-}$ are
$\begin{eqnarray}\begin{array}{rcl}{E}_{n}^{-} & = & \frac{{\zeta }^{2}}{{\widetilde{d}}_{1}^{2}}-\frac{{\zeta }^{2}}{{(n-{\widetilde{d}}_{1})}^{2}}-\tau (2n{\widetilde{d}}_{1}-{n}^{2}),\\ n & = & 0,1,2,\ldots \end{array}\end{eqnarray}$
The total energy of the system then results from adding the ground-state energy E0: $\begin{eqnarray}\begin{array}{r}{E}_{n}={E}_{n}^{-}+{E}_{0},\end{array}\end{eqnarray}$
where E0 is determined directly from the Hamiltonian H−, $\begin{eqnarray}\begin{array}{r}{E}_{0}=-\frac{{\zeta }^{2}}{{\widetilde{d}}_{1}^{2}}-\frac{\tau }{2}\left({N}^{2}-2N-1+2{\widetilde{d}}_{1}\right).\end{array}\end{eqnarray}$
Combining these contributions yields the explicit expression $\begin{eqnarray}\begin{array}{rcl}{E}_{n}^{kc} & = & -\frac{m{k}^{2}}{2{\hslash }^{2}{(n-{\widetilde{d}}_{1})}^{2}}\\ & & +\frac{{\hslash }^{2}}{2m}\,\tau \left[\Space{0ex}{3.25ex}{0ex}{n}^{2}-{\widetilde{d}}_{1}(2n+1)\right.\\ & & \left.-\frac{1}{2}({N}^{2}-2N-1)+2{\ell }({\ell }+N-2)\right].\end{array}\end{eqnarray}$
It is instructive to consider the flat limit τ → 0 (and hence ω = 0): in this case, the conformal geometry disappears and one recovers the well-known spectrum of the N-dimensional Kepler–Coulomb problem supplemented by a $\frac{1}{{r}^{2}}$ term:
$\begin{eqnarray}\begin{array}{l}{E}_{n}^{kc}(\tau =0)\\ =-\frac{m{k}^{2}}{2{\hslash }^{2}{\left(n+\frac{1}{2}+\frac{1}{2}\sqrt{1+4\alpha +4{\ell }({\ell }+N-2)+(N-1)(N-3)}\right)}^{2}}.\end{array}\,\end{eqnarray}$
The ground-state wavefunction is obtained via $\widetilde{A}{{\rm{\Phi }}}_{0}=0$, and depends on the sign of the curvature τ:
for τ > 0 (spherical space):
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{0}(r) & = & { \mathcal N }\,{r}^{-({\widetilde{d}}_{1}+{d}_{2})}{(1+\tau {r}^{2})}^{-\frac{{c}_{2}-\tau {\widetilde{d}}_{1}}{2\tau }}\\ & & \times {{\rm{e}}}^{\left(-\frac{{\widetilde{c}}_{1}}{\sqrt{\tau }}\arctan (\sqrt{\tau }\,r)\right)};\end{array}\end{eqnarray}$
for τ = 0 (flat space):
$\begin{eqnarray}\begin{array}{r}{{\rm{\Phi }}}_{0}(r)={ \mathcal N }\,{r}^{-({\widetilde{d}}_{1}+{d}_{2})}{{\rm{e}}}^{(-{\widetilde{c}}_{1}r)};\end{array}\end{eqnarray}$
for τ < 0 (hyperbolic space, $r\lt 1/\sqrt{| \tau | }$):
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Phi }}}_{0}(r) & = & { \mathcal N }\,{r}^{-({\widetilde{d}}_{1}+{d}_{2})}{(1+\tau {r}^{2})}^{-\frac{{c}_{2}-\tau {\widetilde{d}}_{1}}{2\tau }}\\ & & \times {{\rm{e}}}^{\left(-\frac{{\widetilde{c}}_{1}}{\sqrt{| \tau | }}\arctan (\sqrt{| \tau | }\,r)\right)}.\end{array}\end{eqnarray}$
Excited states are obtained algebraically by repeated application of the creation operator:
$\begin{eqnarray}\begin{array}{r}{{\rm{\Phi }}}_{n}(r)={C}_{n}\,{\widetilde{B}}^{\,n}{{\rm{\Phi }}}_{0}(r),\end{array}\end{eqnarray}$
with Cn a normalization constant.3.3. Numerical validation of the ground-state energy
In order to further substantiate the analytical spectrum obtained from the supersymmetric shape-invariance condition, we have performed a numerical solution of the radial Schrödinger equation using a finite-difference discretization scheme. The computation is based on the full effective radial Hamiltonian introduced in equation (36 ), which explicitly incorporates all curvature-dependent kinetic terms induced by the conformal metric. This ensures that the numerical analysis faithfully reflects the underlying curved-space dynamics. The radial equation is discretized on a uniform grid ri = ih, with i = 0, …, Nr, and with a sufficiently large outer boundary ${r}_{\max }={N}_{r}h$ to guarantee the effective vanishing of the wavefunction at the boundary. Second-order finite-difference formulas are employed to approximate the radial derivatives, resulting in a tridiagonal matrix representation of the Hamiltonian. The ground-state energy is then extracted as the smallest eigenvalue of this discretized operator. Numerical convergence and stability have been carefully checked by varying both the cutoff ${r}_{\max }$ and the number of grid points Nr. The relative deviation between the numerical and analytical ground-state energies of the Kepler–Coulomb sector is defined as 51 ) reads
$\begin{eqnarray}\begin{array}{r}{{\rm{err}}}_{{\rm{rel}}}=\frac{| {E}_{0}^{kc,{\rm{numerical}}}-{E}_{0}^{kc,{\rm{analytical}}}| }{| {E}_{0}^{kc,{\rm{analytical}}}| }.\end{array}\end{eqnarray}$
As a representative and physically relevant example, we consider the critical dimension N = 4 with vanishing angular momentum ℓ = 0, which corresponds precisely to the analytically identified coexistence case. In this situation, the analytical ground-state energy obtained from equation ( $\begin{eqnarray}\begin{array}{r}{E}_{0}^{kc,{\rm{analytical}}}=-\frac{2}{9}-\tau ,\end{array}\end{eqnarray}$
for the parameter choice m = ℏ = k = 1, α = 0, and ω = τ/2.The comparison between analytical and numerical results for several values of the curvature parameter τ is presented in table 1. An excellent agreement is observed in all cases, with relative errors consistently below 10−5. This provides a strong numerical confirmation of the validity and internal consistency of the shape-invariant spectrum derived in the curved-space framework.
Table 1. Comparison between analytical and numerical ground-state energies. The numerical values are obtained from a finite-difference discretization of the radial Schrödinger equation. |
| τ | ${E}_{0}^{kc,{\rm{analytical}}}$ | ${E}_{0}^{kc,{\rm{numerical}}}$ | Relative error |
|---|---|---|---|
| 0 | −0.222 222 2222 | −0.222 221 8847 | 1.52 × 10−6 |
| 0.02 | −0.242 222 2222 | −0.242 221 0315 | 4.92 × 10−6 |
| 0.05 | −0.272 222 2222 | −0.272 220 0448 | 7.98 × 10−6 |
This analysis completes the groundwork for the central discussion, namely the exact coexistence of the harmonic and Kepler–Coulomb potentials, which is addressed in the next subsection.
3.4. Exact coexistence of the harmonic and Kepler–Coulomb potentials
In flat Euclidean space (τ = 0), the superposition of the harmonic potential (∝r2) and the Kepler–Coulomb potential (∝ − 1/r) generically leads to a system that is not exactly solvable. This obstruction originates from the incompatibility of their respective dynamical symmetries: the isotropic oscillator exhibits a U(N) symmetry, whereas the Kepler–Coulomb problem possesses a hidden SO(N + 1) symmetry associated with the Runge–Lenz vector [5, 6]. Their direct superposition therefore destroys the algebraic structure required for exact solvability.
The analysis developed in the previous sections shows that this limitation can be overcome by introducing a conformal metric of constant curvature. The resulting curvature-dependent contributions to the Schrödinger equation modify the radial dynamics in such a way that the competing symmetries of the flat-space problem are effectively compensated. As a consequence, supersymmetric shape invariance imposes a precise compatibility condition, derived from equation (45 ), which can be written as
$\begin{eqnarray}\begin{array}{r}{\ell }({\ell }+N-2)\lt \frac{1}{4}\,(-{N}^{2}+8N-15).\end{array}\end{eqnarray}$
This inequality expresses the requirement that the effective curvature be real (τ2 > 0), ensuring the internal consistency of the geometric framework. To analyze its implications, we define the quadratic function 59 ) can only be satisfied when C(N) > 0, which occurs uniquely at N = 4. In this critical dimension, even the ground state with zero angular momentum (ℓ = 0) fulfills the coexistence condition.
$\begin{eqnarray}\begin{array}{r}C(N)=\frac{1}{4}(-{N}^{2}+8N-15),\end{array}\end{eqnarray}$
which attains its maximum at N = 4, where C(4) = 1/4. Since the left-hand side ℓ(ℓ + N − 2) is nonnegative for all ${\ell }\in {\mathbb{N}}$, the inequality (We therefore conclude that the exact coexistence of the harmonic and Kepler–Coulomb potentials is possible only in the well-defined critical dimension N = 4. This result highlights that curvature should not be regarded merely as a kinematical parameter: it acts as a genuine geometric mediator, restoring a hidden symmetry that is broken in flat space. The delicate balance induced by curvature compensates the quadratic growth of the harmonic confinement and the singular attractive behavior of the Coulomb term, thereby ensuring the exact solvability of the system.
It is important to emphasize that the critical dimension N = 4 emerging in the present work is conceptually distinct from the ‘N = 4’ supersymmetry encountered in one-dimensional superconformal mechanics. In Refs. [14] and [20], the symbol N = 4 denotes the number of supercharges of the superconformal algebra D(2, 1;α), while the dynamics remains strictly one-dimensional and the spatial dimension is fixed to D = 1. By contrast, the framework developed here concerns the quantum radial Hamiltonian on an N-dimensional curved space, where the spatial dimension N enters explicitly through the centrifugal term and the curvature-modified kinetic operator. The supersymmetric shape-invariance conditions thus introduce nontrivial N-dependent coefficients, whose exact cancellation occurs only for N = 4. This geometrically induced critical dimension has no analog in one-dimensional superconformal models and represents a genuinely new compatibility mechanism specific to higher-dimensional quantum systems on curved spaces.
4. Conclusion
We have shown that the exact coexistence of the harmonic and Kepler–Coulomb potentials, supplemented by a singular term, can be achieved in spaces of constant curvature, while it is forbidden in flat Euclidean geometry. This result relies on the introduction of a conformal metric that preserves a supersymmetric structure through shape invariance, thereby ensuring the exact solvability of the system.
A central outcome of our analysis is the emergence of a unique critical dimension, N = 4, in which the geometric and dynamical constraints imposed by supersymmetry are consistently satisfied. In this dimension, curvature plays the role of an effective geometric mediator, restoring a hidden compatibility between the harmonic and Kepler–Coulomb interactions that is otherwise broken in flat space.
More generally, this work highlights the structural role of dimensionality in supersymmetric quantum systems defined on curved backgrounds. The formalism developed here provides a coherent analytical framework for investigating quantum dynamics in effective curved geometries, and it may be extended to relativistic equations, manifolds of non-constant curvature, or to other central and non-central interaction potentials.