Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Bargmann–Segal Lattice: π-Free Discretization of the Harmonic Oscillator on S^5

Josh Loutey

Published Jul 9, 2026• 0 citations
Publisher

Abstract

We construct a discrete graph encoding of the three-dimensional isotropic harmonic oscillator on the holomorphic sector of $S^5$ and verify by exact rational arithmetic that the resulting graph and its projection to physical energies contain no transcendental constants. The Bargmann–Segal transform [ref] maps the harmonic oscillator Hamiltonian to $\hbar\omega(\hat{N} + 3/2)$ on the Fock–Bargmann space, where $\hat{N}$ is the Euler number operator on holomorphic monomials. Restricting to the unit sphere $S^5 \subset \mathbb{C}^3$ identifies the eigenspaces with the symmetric SU(3) irreps $(N, 0)$, of dimension $(N+1)(N+2)/2$, which match the harmonic oscillator shell degeneracies exactly. The graph constructed from these states with dipole transition weights computed by exact fractions.Fraction arithmetic is bit-exactly $π$-free at $N_{\max} = 5$ (56 nodes, 165 edges, zero irrationals encountered). The single $π$ in the entire ecosystem is the Gaussian normalization $π^{-3}$ of the continuous Bargmann measure, which the discrete graph never invokes. The Coulomb problem on $S^3$ and the harmonic oscillator on $S^5$ are both natural sphere quotients, but they live on different spheres with different geometric character (Riemannian vs complex), have natural operators of different order (second vs first), yield spectra of different growth (quadratic vs linear), and demand projections of different type (nonlinear vs linear-affine). This structural asymmetry refines the exchange constant taxonomy [ref]: calibration $π$ is Coulomb-specific. It is the transcendental cost of mapping a quadratic Riemannian eigenvalue to an inverse-square physical energy, and it has no analog for the harmonic oscillator's linear complex-analytic projection. GeoVac Paper 24. Part of the GeoVac research corpus (mathematical foundations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

DOI
10.5281/zenodo.21286754

Related papers

The Variational Methods of Quantum Systems in Holomorphic Representation.

M. W. AlMasri, Mohamed Ridza Wahiddin

We show that variational method for harmonic oscillator in Bargmann representation breaks down since we always end up with one of the exact eigenvalues of Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy eigenvalues of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. Since $\{\frac{z^{n}}{\sqrt{n!}}\}$ form an orthonormal basis, one could in principle expand any complex function $f(z)$ in powers of $z$ and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of $z$ carry no physical information only their non-negative exponents $n$ are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.

•Nov 28, 2021•0 citations

Rayleigh-Ritz Variational Method in The Complex Plane

M. W. AlMasri

We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition $|α| < \tfrac{1}{2}$ for generalized Gaussian trial functions $ψ(z) = e^{αz^2 + βz}$ through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator ($\hat{H} = -\tfrac{1}{2}\partial_x^2 + \tfrac{1}{2}x^2 + λx^4$), adaptive Gaussian ansätze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions ($ψ_n(z) = z^n$) in the Segal--Bargmann space -- while providing rigorous upper bounds $E_n = n + \tfrac{1}{2} + \tfrac{3λ}{4}(2n^2 + 2n + 1)$ for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., $x^3 + x^4$), showing that displacement parameters are essential to capture parity breaking and stabilization effects ($E_0 \approx \tfrac{1}{2} + \tfrac{3μ}{4} - \tfrac{9λ^2}{4} + \cdots$).

quant-pharXiv:2603.02257•Physica Scripta•Mar 3, 2026•0 citations

Discrete Bargmann transform

Kenan Uriostegui

We define a discrete Bargmann transform for discrete and finite functions by means of the coherent states in the su(2) finite harmonic oscillator model. The transform space is over a corresponding finite square mesh in the complex plane. From there, the inverse discrete Bargmann transform reconstitutes the original function with an average error of 10−7 for functions of unit norm. Three defining properties of the Bargmann transform also hold in this discrete version: (1) the Bargmann transform of finite su(2) harmonic oscillator functions are complex power functions; (2) the discrete su(1,1) repulsive oscillator functions self-reproduce in the chosen finite interval with little error; and (3) the oscillator dynamics under the fractional Fourier transform in Bargmann’s complex space correspond, in this discrete transform, to the dynamics under the su(2) fractional Fourier–Kravchuk transform.

Journal of the Optical Society of America A•Jul 17, 2019•2 citations

Discrete symmetries in classical and quantum oscillators

Alexander D. Popov

We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions $ψ_n{=}z^n$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation with $z\in\mathbb C$ are the coordinates of a classical oscillator with energy $E_n=\hbarωn$, $n=0,1,2,...\,$. They are defined on conical spaces ${\mathbb C}/{\mathbb Z}_n$ with cone angles $2π/n$, which are embedded as subspaces in the phase space $\mathbb C$ of the classical oscillator. Here ${\mathbb Z}_n$ is the finite cyclic group of rotations of the space $\mathbb C$ by an angle $2π/n$. The superposition $ψ=\sum_n c_nψ_n$ of the eigenfunctions $ψ_n$ arises only with incomplete knowledge of the initial data for solving the Schrödinger equation, when the conditions of invariance with respect to the discrete groups ${\mathbb Z}_n$ are not imposed and the general solution takes into account all possible initial data parametrized by the numbers $n\in\mathbb N$.

quant-pharXiv:2601.01960•Jan 6, 2026•0 citations

Uniqueness of the energy eigenvalues in pure quantum systems

M. W. AlMasri, Mohamed Ridza Wahiddin

We show that the variational method for harmonic oscillator in Bargmann representation blows up since we always end up with one of the exact eigenvalues of the Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy outcomes of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems. Starting from any normalized complex-variable monomial always give the corresponding exact energy eigenvalue. This fact opens the road for extending the possible energy configuration solutions and enable us to use the whole series expansion techniques in quantum theory.

•Aug 24, 2021•0 citations

Discrete normalized Bargmann transform through the gyrator transform

Kenan Uriostegui

Linear canonical transforms (LCTs) are important in several areas of signal processing; in particular, they were extended to complex-valued parameters to describe optical systems. A special case of these complex LCTs is the Bargmann transform. Recently, Pei and Huang [J. Opt. Soc. Am. A34, 18 (2017)JOAOD60740-323210.1364/JOSAA.34.000018] presented a normalization of the Bargmann transform so that it becomes possible to delimit it near infinity. In this paper, we follow the Pei-Huang algorithm to introduce the discrete normalized Bargmann transform by the relationship between Bargmann and gyrator transforms in the SU(2) finite harmonic oscillator model, and we compare it with the discrete Bargmann transform based on coherent states of the SU(2) oscillator model. This transformation is invertible and unitary. We show that, as in the continuous analog, the discrete normalized Bargmann transform converts the Hermite-Kravchuk functions into Laguerre-Kravchuk functions. In addition, we demonstrate that the discrete su(1,1) repulsive oscillator functions self-reproduce under this discrete transform with little error. Finally, in the space spanned by the wave functions of the SU(2) harmonic oscillator, we find that the discrete normalized Bargmann transform commutes with the fractional Fourier-Kravchuk transform.

Journal of the Optical Society of America A•Apr 21, 2020•0 citations
We construct a discrete graph encoding of the three-dimensional isotropic harmonic oscillator on the holomorphic sector of S5S^5S5 and verify by exact rational arithmetic that the resulting graph and its projection to physical energies contain no transcendental constants. The Bargmann–Segal transform [ref] maps the harmonic oscillator Hamiltonian to ℏω(N^+3/2)\hbar\omega(\hat{N} + 3/2)ℏω(N^+3/2) on the Fock–Bargmann space, where N^\hat{N}N^ is the Euler number operator on holomorphic monomials. Restricting to the unit sphere S5⊂C3S^5 \subset \mathbb{C}^3S5⊂C3 identifies the eigenspaces with the symmetric SU(3) irreps (N,0)(N, 0)(N,0), of dimension (N+1)(N+2)/2(N+1)(N+2)/2(N+1)(N+2)/2, which match the harmonic oscillator shell degeneracies exactly. The graph constructed from these states with dipole transition weights computed by exact fractions.Fraction arithmetic is bit-exactly πππ-free at Nmax⁡=5N_{\max} = 5Nmax​=5 (56 nodes, 165 edges, zero irrationals encountered). The single πππ in the entire ecosystem is the Gaussian normalization π−3π^{-3}π−3 of the continuous Bargmann measure, which the discrete graph never invokes. The Coulomb problem on S3S^3S3 and the harmonic oscillator on S5S^5S5 are both natural sphere quotients, but they live on different spheres with different geometric character (Riemannian vs complex), have natural operators of different order (second vs first), yield spectra of different growth (quadratic vs linear), and demand projections of different type (nonlinear vs linear-affine). This structural asymmetry refines the exchange constant taxonomy [ref]: calibration πππ is Coulomb-specific. It is the transcendental cost of mapping a quadratic Riemannian eigenvalue to an inverse-square physical energy, and it has no analog for the harmonic oscillator's linear complex-analytic projection. GeoVac Paper 24. Part of the GeoVac research corpus (mathematical foundations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
We show that variational method for harmonic oscillator in Bargmann representation breaks down since we always end up with one of the exact eigenvalues of Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy eigenvalues of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. Since {znn!}\{\frac{z^{n}}{\sqrt{n!}}\}{n!​zn​} form an orthonormal basis, one could in principle expand any complex function f(z)f(z)f(z) in powers of zzz and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of zzz carry no physical information only their non-negative exponents nnn are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.
We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition ∣α∣<12|α| < \tfrac{1}{2}∣α∣<21​ for generalized Gaussian trial functions ψ(z)=eαz2+βzψ(z) = e^{αz^2 + βz}ψ(z)=eαz2+βz through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator (H^=−12∂x2+12x2+λx4\hat{H} = -\tfrac{1}{2}\partial_x^2 + \tfrac{1}{2}x^2 + λx^4H^=−21​∂x2​+21​x2+λx4), adaptive Gaussian ansätze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions (ψn(z)=znψ_n(z) = z^nψn​(z)=zn) in the Segal--Bargmann space -- while providing rigorous upper bounds En=n+12+3λ4(2n2+2n+1)E_n = n + \tfrac{1}{2} + \tfrac{3λ}{4}(2n^2 + 2n + 1)En​=n+21​+43λ​(2n2+2n+1) for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., x3+x4x^3 + x^4x3+x4), showing that displacement parameters are essential to capture parity breaking and stabilization effects (E0≈12+3μ4−9λ24+⋯E_0 \approx \tfrac{1}{2} + \tfrac{3μ}{4} - \tfrac{9λ^2}{4} + \cdotsE0​≈21​+43μ​−49λ2​+⋯).
We define a discrete Bargmann transform for discrete and finite functions by means of the coherent states in the su(2) finite harmonic oscillator model. The transform space is over a corresponding finite square mesh in the complex plane. From there, the inverse discrete Bargmann transform reconstitutes the original function with an average error of 10−7 for functions of unit norm. Three defining properties of the Bargmann transform also hold in this discrete version: (1) the Bargmann transform of finite su(2) harmonic oscillator functions are complex power functions; (2) the discrete su(1,1) repulsive oscillator functions self-reproduce in the chosen finite interval with little error; and (3) the oscillator dynamics under the fractional Fourier transform in Bargmann’s complex space correspond, in this discrete transform, to the dynamics under the su(2) fractional Fourier–Kravchuk transform.
We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions ψn=znψ_n{=}z^nψn​=zn of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation with z∈Cz\in\mathbb Cz∈C are the coordinates of a classical oscillator with energy En=ℏωnE_n=\hbarωnEn​=ℏωn, n=0,1,2,... n=0,1,2,...\,n=0,1,2,.... They are defined on conical spaces C/Zn{\mathbb C}/{\mathbb Z}_nC/Zn​ with cone angles 2π/n2π/n2π/n, which are embedded as subspaces in the phase space C\mathbb CC of the classical oscillator. Here Zn{\mathbb Z}_nZn​ is the finite cyclic group of rotations of the space C\mathbb CC by an angle 2π/n2π/n2π/n. The superposition ψ=∑ncnψnψ=\sum_n c_nψ_nψ=∑n​cn​ψn​ of the eigenfunctions ψnψ_nψn​ arises only with incomplete knowledge of the initial data for solving the Schrödinger equation, when the conditions of invariance with respect to the discrete groups Zn{\mathbb Z}_nZn​ are not imposed and the general solution takes into account all possible initial data parametrized by the numbers n∈Nn\in\mathbb Nn∈N.
We show that the variational method for harmonic oscillator in Bargmann representation blows up since we always end up with one of the exact eigenvalues of the Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy outcomes of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems. Starting from any normalized complex-variable monomial always give the corresponding exact energy eigenvalue. This fact opens the road for extending the possible energy configuration solutions and enable us to use the whole series expansion techniques in quantum theory.
Linear canonical transforms (LCTs) are important in several areas of signal processing; in particular, they were extended to complex-valued parameters to describe optical systems. A special case of these complex LCTs is the Bargmann transform. Recently, Pei and Huang [J. Opt. Soc. Am. A34, 18 (2017)JOAOD60740-323210.1364/JOSAA.34.000018] presented a normalization of the Bargmann transform so that it becomes possible to delimit it near infinity. In this paper, we follow the Pei-Huang algorithm to introduce the discrete normalized Bargmann transform by the relationship between Bargmann and gyrator transforms in the SU(2) finite harmonic oscillator model, and we compare it with the discrete Bargmann transform based on coherent states of the SU(2) oscillator model. This transformation is invertible and unitary. We show that, as in the continuous analog, the discrete normalized Bargmann transform converts the Hermite-Kravchuk functions into Laguerre-Kravchuk functions. In addition, we demonstrate that the discrete su(1,1) repulsive oscillator functions self-reproduce under this discrete transform with little error. Finally, in the space spanned by the wave functions of the SU(2) harmonic oscillator, we find that the discrete normalized Bargmann transform commutes with the fractional Fourier-Kravchuk transform.