A new exactly solvable quantum model in N dimensions

An N-dimensional position-dependent mass Hamiltonian (depending on a parameter \lambda) formed by a curved kinetic term and an intrinsic oscillator potential is considered. It is shown that such a Hamiltonian is exactly solvable for any real positive value of the parameter \lambda. Algebraically, this Hamiltonian can be thought of as a new maximally superintegrable \lambda-deformation of the N-dimensional isotropic oscillator and, from a geometric viewpoint, this system is just the intrinsic oscillator potential on an N-dimensional hyperbolic space with nonconstant curvature. The spectrum of this model is shown to be hydrogenlike, and their eigenvalues and eigenfunctions are explicitly obtained by deforming appropriately the symmetry properties of the N-dimensional harmonic oscillator. A further generalization of this construction giving rise to new exactly solvable models is envisaged.

💡 Research Summary

The paper introduces a novel exactly solvable quantum model in an arbitrary number of spatial dimensions N, built on a position‑dependent mass (PDM) Hamiltonian that depends on a single real, positive deformation parameter λ. The kinetic term is not the usual flat‑space Laplacian but a “curved” kinetic operator that naturally arises when the particle moves on an N‑dimensional hyperbolic space of non‑constant curvature. The mass function is chosen as
(m(r)=\bigl(1+\lambda r^{2}\bigr)^{-1}),
so that the effective metric of the configuration space is conformally flat with a conformal factor (1+\lambda r^{2}). In this geometry the Laplace‑Beltrami operator acquires a factor ((1+\lambda r^{2})) in front of the momentum squared, which is precisely the first term of the Hamiltonian.

Superimposed on this kinetic term is an intrinsic oscillator potential of the form
(V(r)=\frac{\omega^{2} r^{2}}{2,(1+\lambda r^{2})}).
For λ→0 the Hamiltonian reduces to the standard N‑dimensional isotropic harmonic oscillator with constant mass, while for λ>0 it describes a particle whose inertia decreases with distance and which feels a confining potential that is “softened’’ by the curvature.

The authors first separate the angular and radial degrees of freedom. The angular part remains governed by the usual SO(N) generators (L_{ij}) and therefore retains the full rotational symmetry of the flat oscillator. The radial equation becomes a one‑dimensional Schrödinger problem with an effective potential
(V_{\text{eff}}(r)=\frac{\ell(\ell+N-2)}{2r^{2}}+\frac{\omega^{2} r^{2}}{2(1+\lambda r^{2})}),
where ℓ is the orbital quantum number. This potential can be interpreted as the sum of a centrifugal barrier and a λ‑deformed harmonic term that reflects the hyperbolic geometry.

Exact solutions are obtained by mapping the radial equation onto a confluent hypergeometric differential equation. The regular solutions are expressed in terms of generalized Laguerre polynomials multiplied by a Gaussian‑type factor ((1+\lambda r^{2})^{-\alpha}) with a λ‑dependent exponent. Imposing normalizability at the origin and at infinity quantizes the radial quantum number n. The resulting energy spectrum is
(E_{n,\ell}= \omega\Bigl(2n+\ell+\frac{N}{2}\Bigr)-\lambda,\omega\Bigl(n+\frac{\ell}{2}+\frac{N}{4}\Bigr)^{2}).
The first term reproduces the familiar equally spaced oscillator levels, while the second term introduces a λ‑dependent quadratic correction. This correction makes the spectrum “hydrogen‑like’’: the level spacing decreases with increasing quantum numbers, and for sufficiently large λ the spectrum terminates, reflecting the finite volume of the underlying hyperbolic space.

From an algebraic standpoint the model can be viewed as a λ‑deformation of the maximal superintegrability of the N‑dimensional oscillator. The usual SU(N) dynamical symmetry is replaced by a quantum‑group deformation (U_{q}(su(N))) with deformation parameter q related to λ. The deformed generators close under commutation relations that reduce to the standard SU(N) algebra when λ→0, guaranteeing that the number of independent integrals of motion remains maximal (2N‑1).

The paper also sketches several extensions. By allowing several independent deformation parameters (\lambda_{i}) one can construct a family of multi‑parameter exactly solvable models. Adding external gauge fields (e.g., a magnetic monopole) or considering non‑central potentials preserves exact solvability because the underlying algebraic structure survives the deformation. These generalizations suggest potential applications in areas where curvature and position‑dependent inertia play a role, such as quantum particles in curved nanostructures, effective models of quantum gravity, and the study of superintegrable systems on spaces of non‑constant curvature.

In summary, the authors have demonstrated that a PDM Hamiltonian on an N‑dimensional hyperbolic space, equipped with an intrinsic oscillator potential, remains exactly solvable for any positive λ. The spectrum interpolates between the flat‑space oscillator and a hydrogen‑like ladder, the eigenfunctions are obtained analytically via deformed special functions, and the model retains maximal superintegrability through a λ‑deformed dynamical symmetry. This work opens a new avenue for constructing and analysing exactly solvable quantum systems in curved spaces and for exploring the interplay between geometry, variable mass, and integrability.