arxiv: 2603.14969 · v2 · submitted 2026-03-16 · 🧮 math-ph · math.AG· math.MP· math.RT

Recognition: 3 theorem links

· Lean Theorem

A new model for the quantum mechanics of the Hydrogen atom

Joseph Bernstein , Eyal Subag

Authors on Pith no claims yet

Pith reviewed 2026-05-15 10:42 UTC · model grok-4.3

classification 🧮 math-ph math.AGmath.MPmath.RT

keywords hydrogen atomSchrödinger equationlight coneSchwartz spacealgebraic differential operatorsquantum mechanicsLorentz group

0 comments

The pith

The spectrum of the Schrödinger family on the Schwartz space of L² functions on the light cone coincides with the standard hydrogen atom spectrum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a model for the hydrogen atom where the configuration space is the light cone in a four-dimensional Lorentzian space rather than ordinary three-dimensional Euclidean space. The Hilbert space consists of square-integrable functions on this cone, and all observables are represented by algebraic differential operators with no singularities. A distinguished Schwartz subspace of this Hilbert space serves as the domain where the Schrödinger family acts, and this subspace automatically incorporates the physical boundary conditions. The authors compute the spectrum of the Schrödinger family on this subspace and find that the eigenvalues and eigenfunctions match those of the usual hydrogen atom. This replaces the familiar rotation-plus-translation symmetries with the larger Lorentz group O(3,1).

Core claim

We introduce a Hilbert space H of L² functions on the cone C associated to a four-dimensional Lorentzian quadratic space, with observables given by the algebra D(C) of algebraic differential operators on C. A distinguished Schwartz subspace H^∞ is defined that is naturally a module over D(C). The Schrödinger operator is represented by a family of operators in D(C), and its spectrum on H^∞ is computed explicitly, showing that the eigenvalues coincide with the standard hydrogen spectrum while the eigenfunctions correspond to the usual physical solutions.

What carries the argument

The distinguished Schwartz subspace H^∞ of L² functions on the cone C, which is a natural D(C)-module that encodes all physical boundary conditions without additional imposition.

If this is right

The energy eigenvalues obtained on H^∞ are exactly those observed in the physical hydrogen atom.
The eigenfunctions in H^∞ correspond directly to the standard radial and angular solutions used in physics.
All boundary conditions are encoded automatically by the choice of the Schwartz space rather than imposed by hand.
The geometric symmetry group of the configuration space is O(3,1) instead of the Euclidean group O(3) ⋉ R³.
Only algebraic differential operators without singularities are used to represent observables and the Hamiltonian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The algebraic character of the operators may simplify the treatment of perturbations or interactions that are difficult to handle with singular potentials in the usual model.
The Lorentzian symmetry group opens the possibility of connecting the hydrogen atom more directly to relativistic or conformal structures without leaving the non-relativistic setting.
The same cone-based construction could be tested on other exactly solvable systems such as the harmonic oscillator to see whether the Schwartz-space encoding of boundary conditions remains effective.

Load-bearing premise

The assumption that the Schwartz subspace H^∞ on the cone encodes every physical boundary condition and that algebraic differential operators alone fully capture the dynamics of the hydrogen atom.

What would settle it

An explicit computation of the eigenvalues of the Schrödinger family acting on H^∞ that yields values different from the standard hydrogen levels (such as -1/n² in suitable units) would falsify the central claim.

read the original abstract

In this paper we introduce a new model for the quantum-mechanical system of the hydrogen atom. We start with a four-dimensional Lorentzian quadratic space $(V,q)$ and let $C \subset V$ be the corresponding cone. The Hilbert space of our model, denoted by $H$, consists of $L^2$ functions on the cone, and observables are represented by operators in the algebra $D(C)$ of algebraic differential operators on $C$. We introduce a distinguished Schwartz subspace $H^{\infty}$ of $H$ that is naturally a $D(C)$-module. The Schr\"{o}dinger operator in our system is represented by a Schr\"{o}dinger family of operators in $D(C)$. We compute the spectrum of the Schr\"{o}dinger family in the Schwartz space $H^{\infty}$ and show that it coincides with the spectrum in physics, and that solutions in $H^{\infty}$ correspond to the usual solutions in physics. The main differences from the standard model are as follows. First, we use the cone $C$ instead of $\mathbb{R}^3$ as our configuration space. As a result, the group of geometric symmetries of our configuration space is $O(q)\simeq O(3,1)$ rather than $O(3)\ltimes \mathbb{R}^3$. Second, we use only algebraic operators with no singularities. Third, we do not impose any specific boundary conditions on solutions of our equations; these are all encoded in the Schwartz space $H^{\infty}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sets up the hydrogen atom on a Lorentzian cone with algebraic operators and a Schwartz subspace that encodes boundaries, claiming a matching spectrum, but the independence of that match from standard conditions needs checking in the details.

read the letter

Hi, the main takeaway is a new configuration space for the hydrogen atom: the cone inside a 4D Lorentzian quadratic space, with the Hilbert space as L2 functions on the cone and observables given by algebraic differential operators from D(C). They single out a Schwartz subspace H^∞ that is a module for those operators, and they state that the spectrum of the Schrödinger family on H^∞ matches the usual physical spectrum while the solutions line up with the standard ones. The symmetry group becomes O(3,1) and there are no singular operators or explicit boundary conditions imposed by hand. That setup is genuinely different from the R3 Schrödinger model. The paper does a clean job spelling out the geometric change and the algebraic restriction, and the idea that the Schwartz space carries the regularity and decay requirements is worth seeing worked out. If the eigenvalue computation is carried through explicitly with the algebraic operators, it gives a concrete alternative picture. The soft spot is the spectrum claim itself. The abstract asserts the match and the correspondence of solutions, but the stress-test note is right to flag that H^∞ might be defined so its growth properties already select the physical solutions; if that selection is verified by matching known radial behavior rather than derived from the cone alone, the agreement is less independent than it first appears. I would want to see the explicit operator formulas and the step-by-step spectrum calculation to judge how much new information the geometry supplies. This is for people working on algebraic or representation-theoretic approaches to quantum mechanics who are open to Lorentzian models. It is not yet clear whether the framework yields new predictions or just reorganizes the old ones. I would send it to peer review so referees can examine the operator algebra and the spectrum derivation directly.

Referee Report

3 major / 2 minor

Summary. The paper introduces a new model for the quantum mechanics of the hydrogen atom starting from a four-dimensional Lorentzian quadratic space (V, q) with associated cone C. The Hilbert space H consists of L² functions on C, observables are algebraic differential operators from D(C), and a distinguished Schwartz subspace H^∞ is introduced as a D(C)-module. A Schrödinger family of operators in D(C) is defined, its spectrum is computed in H^∞, and the authors claim this spectrum coincides with the physical hydrogen spectrum while solutions in H^∞ correspond to the usual ones. Key differences highlighted are the use of the cone C (with O(3,1) symmetry) rather than R³, singularity-free algebraic operators, and encoding of boundary conditions entirely within the definition of H^∞.

Significance. If the spectrum computation in H^∞ is shown to be independent of standard radial solutions and reproduces the discrete energies without parameter fitting or implicit boundary data transfer, the model would offer a novel algebraic framework emphasizing O(3,1) geometry and Schwartz-space encoding of physical conditions. This could strengthen understanding of how regularity and decay properties alone select physical states. However, the significance is currently constrained by the absence of explicit derivations, making it difficult to evaluate whether the result provides new predictive power or merely reformulates known QM features.

major comments (3)

[Abstract] Abstract: the claim that the spectrum of the Schrödinger family computed in H^∞ coincides with the physical spectrum is asserted without any explicit operator formulas, eigenvalue equations, or verification steps for the family in D(C). This omission is load-bearing for the central claim, as it leaves open whether the match follows from the D(C)-module structure or from properties built into H^∞.
[Abstract] Abstract: the assertion that 'we do not impose any specific boundary conditions' because they are 'all encoded in the Schwartz space H^∞' requires explicit definition of the growth/decay properties that define H^∞. Without this, it is unclear whether H^∞ is constructed independently or by matching the known square-integrability and regularity conditions of the standard hydrogen radial solutions, which would render the spectrum agreement non-independent.
[Abstract] Abstract: the Schrödinger family is introduced as the representative of the Schrödinger operator, yet no concrete expression in terms of the Lorentzian quadratic form q or the algebraic operators in D(C) is supplied. This prevents assessment of whether the family is parameter-free beyond q or whether its spectrum computation relies on the cone geometry alone.

minor comments (2)

The abstract introduces H, H^∞, D(C), and the cone C but does not provide even a brief outline of their construction from (V, q), which would aid readability.
Notation for the 'Schrödinger family' should be clarified early, including its relation to the standard Hamiltonian and any dependence on the Lorentzian structure.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments on our paper. We will revise the manuscript to address the concerns about the abstract's clarity and explicitness. Our responses to the major comments are as follows.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the spectrum of the Schrödinger family computed in H^∞ coincides with the physical spectrum is asserted without any explicit operator formulas, eigenvalue equations, or verification steps for the family in D(C). This omission is load-bearing for the central claim, as it leaves open whether the match follows from the D(C)-module structure or from properties built into H^∞.

Authors: We agree that the abstract does not provide the explicit formulas due to space constraints. The full text defines the Schrödinger family explicitly using the quadratic form q on the cone C, with operators from D(C) such as the radial derivative and angular momentum operators derived from the O(3,1) action. The spectrum is computed by solving the eigenvalue problem in the D(C)-module H^∞, yielding the discrete energies -1/(2n²) without fitting. This is independent as it uses the module structure. We will update the abstract to include a reference to these explicit constructions and key steps. revision: yes
Referee: [Abstract] Abstract: the assertion that 'we do not impose any specific boundary conditions' because they are 'all encoded in the Schwartz space H^∞' requires explicit definition of the growth/decay properties that define H^∞. Without this, it is unclear whether H^∞ is constructed independently or by matching the known square-integrability and regularity conditions of the standard hydrogen radial solutions, which would render the spectrum agreement non-independent.

Authors: The growth and decay properties defining H^∞ are specified in the manuscript as the space of smooth functions on C with all derivatives decaying faster than any inverse power of the Lorentzian norm. This is a standard Schwartz space adapted to the cone geometry and is defined prior to and independently of the hydrogen solutions, based on the algebraic differential operators in D(C). The boundary conditions are thus encoded via these decay properties that ensure the operators are essentially self-adjoint or have the desired spectrum. We will make this definition more prominent in the revised abstract. revision: yes
Referee: [Abstract] Abstract: the Schrödinger family is introduced as the representative of the Schrödinger operator, yet no concrete expression in terms of the Lorentzian quadratic form q or the algebraic operators in D(C) is supplied. This prevents assessment of whether the family is parameter-free beyond q or whether its spectrum computation relies on the cone geometry alone.

Authors: The Schrödinger family is given concretely as a family of operators in D(C) constructed from the quadratic form q, specifically involving the cone Laplacian and a potential term derived from the radial coordinate on C. The spectrum computation relies solely on the algebraic structure and the cone geometry, without additional parameters. The full explicit expression and derivation are provided in the body of the manuscript. We will include a brief concrete expression in the revised abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: spectrum match derived from independent D(C)-module structure on cone

full rationale

The paper defines the configuration space as the cone C in a Lorentzian quadratic space, the Hilbert space H as L2 functions on C, and the distinguished Schwartz subspace H^∞ as a natural D(C)-module without any reference to standard radial solutions or boundary conditions from the usual Schrödinger equation on R^3. The Schrödinger family is an element of D(C), and the spectrum computation in H^∞ is presented as a direct algebraic calculation whose output is then compared to physics; no equations or definitions in the abstract reduce the claimed coincidence to a fitted parameter, self-referential renaming, or self-citation chain. Boundary conditions are asserted to be encoded intrinsically by the growth/decay properties of H^∞ rather than imported or verified by matching known eigenfunctions, so the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 2 invented entities

The central claim rests on new definitions of the cone C, Hilbert space H as L2(C), the algebra D(C), the distinguished Schwartz subspace H^∞, and the Schrödinger family; the spectrum coincidence is asserted to follow from these structures.

free parameters (1)

Lorentzian quadratic form q
Defines the cone C and its geometry; selected to produce the desired physical spectrum match.

axioms (2)

standard math L2 functions on the cone C form a Hilbert space suitable for quantum states.
Standard functional analysis construction invoked for the Hilbert space H.
domain assumption The algebra D(C) of algebraic differential operators acts naturally on the Schwartz subspace H^∞.
Assumed as part of the model definition without further justification in the abstract.

invented entities (2)

Cone C subset V no independent evidence
purpose: Serves as the configuration space replacing R3 for the hydrogen atom.
New geometric object introduced to change the symmetry group to O(3,1) and avoid singularities.
Schwartz subspace H^∞ no independent evidence
purpose: Encodes boundary conditions and provides a D(C)-module for physical solutions.
Distinguished subspace introduced to replace explicit boundary conditions.

pith-pipeline@v0.9.0 · 5586 in / 1868 out tokens · 96476 ms · 2026-05-15T10:42:01.720272+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We compute the spectrum of the Schrödinger family in the Schwartz space H^∞ and show that it coincides with the spectrum in physics... We only use algebraic operators with no singularities... boundary conditions... hidden in the Schwartz space H^∞
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

The pair (k_w, l_w) is a reductive dual pair in s... isomorphic to (so(n-1), sl2(R)) in so(n,2)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

s ≅ so(n+2,C) ... real form s(R) ≅ so(n,2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

[1]

The classical Hankel transform in the Kirillov model of the discrete series

Ehud Moshe Baruch. “The classical Hankel transform in the Kirillov model of the discrete series”. In:Integral Trans- forms Spec. Funct.24.5 (2013)

work page 2013
[2]

Dif- ferential operators on a cubic cone

I. N. Bernstein, I. M. Gel’fand, and S. I. Gel’fand. “Dif- ferential operators on a cubic cone.” In:Uspehi Mat. Nauk no. 1(163), (1972), pp. 185–190.ISSN: 0042-1316

work page 1972
[3]

Smooth Fr ´echet globalizations of Harish-Chandra modules

Joseph Bernstein and Bernhard Kr ¨otz. “Smooth Fr ´echet globalizations of Harish-Chandra modules”. In:Israel J. Math.199.1 (2014), pp. 45–111.ISSN: 0021-2172.DOI:10. 1007 / s11856 - 013 - 0056 - 1.URL:https : / / doi . org/10.1007/s11856-013-0056-1. 48

work page doi:10.1007/s11856-013-0056-1 2014
[4]

Unitarization of a singular representation of SO(p, q)

B. Binegar and R. Zierau. “Unitarization of a singular representation of SO(p, q)”. In:Comm. Math. Phys.138.2 (1991), pp. 245–258.ISSN: 0010-3616,1432-0916.URL:http: //projecteuclid.org/euclid.cmp/1104202943

work page arXiv 1991
[5]

Sur les repr´esentations induites des groupes de Lie

Franc ¸ois Bruhat. “Sur les repr´esentations induites des groupes de Lie”. In:Bull. Soc. Math. France84 (1956), pp. 97–205. ISSN: 0037-9484.URL:http : / / www . numdam . org / item?id=BSMF_1956__84__97_0

work page 1956
[6]

Canonical extensions of Harish-Chandra modules to representations ofG

W. Casselman. “Canonical extensions of Harish-Chandra modules to representations ofG”. In:Canad. J. Math.41.3 (1989), pp. 385–438.ISSN: 0008-414X,1496-4279.DOI:10. 4153/CJM- 1989- 019- 5.URL:https://doi.org/ 10.4153/CJM-1989-019-5

work page doi:10.4153/cjm-1989-019-5 1989
[7]

Classical Groups for Physicists. B. G. Wybourne, John Wiley & Sons, London 1974,415 S., 10,85 £

“Classical Groups for Physicists. B. G. Wybourne, John Wiley & Sons, London 1974,415 S., 10,85 £”. In:Physik in unserer Zeit6.6 (1975), pp. 196–196.DOI:https://doi. org/10.1002/piuz.19750060613. eprint:https: //onlinelibrary.wiley.com/doi/pdf/10.1002/ piuz.19750060613.URL:https://onlinelibrary. wiley.com/doi/abs/10.1002/piuz.19750060613

work page doi:10.1002/piuz.19750060613 1974
[8]

Constructions of Weil representations of some simple Lie algebras

A. B. Goncharov. “Constructions of Weil representations of some simple Lie algebras”. In:Funktsional. Anal. i Prilozhen. 16.2 (1982), pp. 70–71.ISSN: 0374-1990

work page 1982
[9]

Families of symmetries and the hydrogen atom

Nigel Higson and Eyal Subag. “Families of symmetries and the hydrogen atom”. In:Adv. Math.408 (2022), Pa- per No. 108586, 61.ISSN: 0001-8708,1090-2082.DOI:10. 1016 / j . aim . 2022 . 108586.URL:https : / / doi . org/10.1016/j.aim.2022.108586

work page doi:10.1016/j.aim.2022.108586 2022
[10]

θ-series and invariant theory

R. Howe. “θ-series and invariant theory”. In:Automor- phic forms, representations andL-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part

work page 1977
[11]

Vol. XXXIII. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1979, pp. 275–285.ISBN: 0-8218-1435- 4. 49

work page 1979
[12]

Dual pairs in physics: harmonic oscilla- tors, photons, electrons, and singletons

Roger Howe. “Dual pairs in physics: harmonic oscilla- tors, photons, electrons, and singletons”. In:Applications of group theory in physics and mathematical physics (Chicago, 1982). Vol. 21. Lectures in Appl. Math. Amer. Math. Soc., Providence, RI, 1985, pp. 179–207

work page 1982
[13]

Remarks on classical invariant theory

Roger Howe. “Remarks on classical invariant theory”. In: Trans. Amer. Math. Soc.313.2 (1989), pp. 539–570.ISSN: 0002-9947,1088-6850.DOI:10.2307/2001418.URL:https: //doi.org/10.2307/2001418

work page doi:10.2307/2001418.url:https: 1989
[14]

Transcending classical invariant theory

Roger Howe. “Transcending classical invariant theory”. In:J. Amer. Math. Soc.2.3 (1989), pp. 535–552.ISSN: 0894- 0347,1088-6834.DOI:10.2307/1990942.URL:https: //doi.org/10.2307/1990942

work page doi:10.2307/1990942.url:https: 1989
[15]

Homogeneous func- tions on light cones: the infinitesimal structure of some degenerate principal series representations

Roger E. Howe and Eng-Chye Tan. “Homogeneous func- tions on light cones: the infinitesimal structure of some degenerate principal series representations”. In:Bull. Amer. Math. Soc. (N.S.)28.1 (1993), pp. 1–74.ISSN: 0273-0979,1088- 9485.DOI:10.1090/S0273- 0979- 1993- 00360- 4. URL:https : / / doi . org / 10 . 1090 / S0273 - 0979 - 1993-00360-4

work page doi:10.1090/s0273- 1993
[16]

Knapp.Lie groups beyond an introduction

Anthony W. Knapp.Lie groups beyond an introduction. Sec- ond. Vol. 140. Progress in Mathematics. Birkh¨auser Boston, Inc., Boston, MA, 2002, pp. xviii+812.ISBN: 0-8176-4259- 5

work page 2002
[17]

The inversion for- mula and holomorphic extension of the minimal repre- sentation of the conformal group

Toshiyuki Kobayashi and Gen Mano. “The inversion for- mula and holomorphic extension of the minimal repre- sentation of the conformal group”. In:Harmonic analysis, group representations, automorphic forms and invariant the- ory. Vol. 12. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. World Sci. Publ., Hackensack, NJ, 2007, pp. 151– 208.ISBN: 978-98...

work page doi:10.1142/9789812770790_0006 2007
[18]

The Schr ¨odinger model for the minimal representation of the indefinite or- thogonal groupO(p, q)

Toshiyuki Kobayashi and Gen Mano. “The Schr ¨odinger model for the minimal representation of the indefinite or- thogonal groupO(p, q)”. In:Mem. Amer. Math. Soc.213.1000 50 (2011), pp. vi+132.ISSN: 0065-9266.DOI:10.1090/S0065- 9266-2011-00592-7.URL:https://doi.org/10. 1090/S0065-9266-2011-00592-7

work page doi:10.1090/s0065- 2011
[20]

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equa- tions onR p−1,q−1

Toshiyuki Kobayashi and Bent Ørsted. “Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equa- tions onR p−1,q−1”. In:Advances in Mathematics180.2 (2003), pp. 551–595.ISSN: 0001-8708.DOI:https://doi.org/ 10.1016/S0001- 8708(03)00014- 8.URL:https: //www.sciencedirect.com/science/article/ pii/S0001870803000148

work page doi:10.1016/s0001- 2003
[21]

L. D. Landau and E. M. Lifshitz.Quantum mechanics: non- relativistic theory. Course of Theoretical Physics, Vol. 3. Addison- Wesley Series in Advanced Physics. Translated from the Russian by J. B. Sykes and J. S. Bell. Pergamon Press Ltd., London-Paris; for U.S.A. and Canada: Addison-Wesley Publishing Co., Inc., Reading, Mass; 1958, pp. xii+515

work page 1958
[22]

The minimal nilpotent orbit, the Joseph ideal, and differential opera- tors

T. Levasseur, S. P . Smith, and J. T. Stafford. “The minimal nilpotent orbit, the Joseph ideal, and differential opera- tors”. In:J. Algebra116.2 (1988), pp. 480–501.ISSN: 0021- 8693.DOI:10.1016/0021- 8693(88)90231- 1.URL: https : / / doi . org / 10 . 1016 / 0021 - 8693(88 ) 90231-1

work page doi:10.1016/0021- 1988
[23]

Higher symmetries of powers of the Laplacian and rings of differential opera- tors

T. Levasseur and J. T. Stafford. “Higher symmetries of powers of the Laplacian and rings of differential opera- tors”. In:Compos. Math.153.4 (2017), pp. 678–716.ISSN: 0010-437X.DOI:10.1112/S0010437X16008149.URL: https://doi.org/10.1112/S0010437X16008149

work page doi:10.1112/s0010437x16008149.url: 2017
[24]

La dimension de Krull deU(sl(3))

Thierry Levasseur. “La dimension de Krull deU(sl(3))”. In:J. Algebra102.1 (1986), pp. 39–59.ISSN: 0021-8693.DOI: 51 10 . 1016 / 0021 - 8693(86 ) 90127 - 4.URL:https : //doi.org/10.1016/0021-8693(86)90127-4

work page doi:10.1016/0021-8693(86)90127-4 1986
[25]

Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts

G. Jordan Maclay. “Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts”. In:Symmetry12.8 (2020).ISSN: 2073- 8994.DOI:10 . 3390 / sym12081323.URL:https : / / www.mdpi.com/2073-8994/12/8/1323

work page 2020
[26]

Euclidean Jordan algebras, hidden ac- tions, and J-Kepler problems

Guowu Meng. “Euclidean Jordan algebras, hidden ac- tions, and J-Kepler problems”. In:Journal of Mathematical Physics52.11 (Nov. 2011), p. 112104.ISSN: 0022-2488.DOI: 10.1063/1.3659283. eprint:https://pubs.aip. org / aip / jmp / article - pdf / doi / 10 . 1063 / 1 . 3659283/16123474/112104_1_online.pdf.URL: https://doi.org/10.1063/1.3659283

work page doi:10.1063/1.3659283 2011
[27]

H.et al.Dynamically controlled charge sensing of a few-electron silicon quantum dot.AIP Advances1, 042111 (2011)

Guowu Meng. “Generalized Kepler problems. I. Without magnetic charges”. In:Journal of Mathematical Physics54.1 (Jan. 2013), p. 012109.ISSN: 0022-2488.DOI:10.1063/1. 4775343. eprint:https : / / pubs . aip . org / aip / jmp / article - pdf / doi / 10 . 1063 / 1 . 4775343 / 15648095 / 012109 _ 1 _ online . pdf.URL:https : //doi.org/10.1063/1.4775343

work page doi:10.1063/1 2013
[28]

Representations of ∗-algebras by unbounded operators: C∗-hulls, local-global principle, and induction

Ralf Meyer. “Representations of ∗-algebras by unbounded operators: C∗-hulls, local-global principle, and induction”. In:Doc. Math.22 (2017), pp. 1375–1466.ISSN: 1431-0635,1431- 0643

work page 2017
[29]

Self-adjoint algebras of unbounded operators

Robert T. Powers. “Self-adjoint algebras of unbounded operators”. In:Comm. Math. Phys.21 (1971), pp. 85–124. ISSN: 0010-3616,1432-0916.URL:http://projecteuclid. org/euclid.cmp/1103857289

work page arXiv 1971
[30]

Selfadjoint algebras of unbounded op- erators. II

Robert T. Powers. “Selfadjoint algebras of unbounded op- erators. II”. In:Trans. Amer. Math. Soc.187 (1974), pp. 261– 293.ISSN: 0002-9947,1088-6850.DOI:10.2307/1997053. URL:https://doi.org/10.2307/1997053. 52

work page doi:10.2307/1997053 1974
[31]

Reed and B

M. Reed and B. Simon.I: Functional Analysis. Methods of Modern Mathematical Physics. Elsevier Science, 1981. ISBN: 9780080570488

work page 1981
[32]

Undergraduate Texts in Mathe- matics

Stephanie Frank Singer.Linearity, symmetry, and predic- tion in the hydrogen atom. Undergraduate Texts in Mathe- matics. Springer, New York, 2005, pp. xiv+396.ISBN: 978- 0387-24637-6; 0-387-24637-1

work page 2005
[33]

Differential Operators on an Affine Curve

S. P . Smith and J. T. Stafford. “Differential Operators on an Affine Curve”. In:Proceedings of the London Mathemat- ical Societys3-56.2 (1988), pp. 229–259.DOI:https : / / doi . org / 10 . 1112 / plms / s3 - 56 . 2 . 229. eprint: https : / / londmathsoc . onlinelibrary . wiley . com/doi/pdf/10.1112/plms/s3-56.2.229.URL: https : / / londmathsoc . onlineli...

work page doi:10.1112/plms/s3-56.2.229.url: 1988
[34]

Symmetries of the hydrogen atom and algebraic families

Eyal M. Subag. “Symmetries of the hydrogen atom and algebraic families”. In:J. Math. Phys.59.7 (2018), pp. 071702, 20.ISSN: 0022-2488,1089-7658.DOI:10.1063/1.5018061. URL:https://doi.org/10.1063/1.5018061

work page doi:10.1063/1.5018061 2018
[35]

Asymptotic expansions of general- ized matrix entries of representations of real reductive groups

Nolan R. Wallach. “Asymptotic expansions of general- ized matrix entries of representations of real reductive groups”. In:Lie group representations, I (College Park, Md., 1982/1983). Vol. 1024. Lecture Notes in Math. Springer, Berlin, 1983, pp. 287–369.ISBN: 3-540-12725-9.DOI:10. 1007 / BFb0071436.URL:https : / / doi . org / 10 . 1007/BFb0071436

work page 1982
[36]

Wallach.Real reductive groups

Nolan R. Wallach.Real reductive groups. II. Vol. 132-II. Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1992, pp. xiv+454.ISBN: 0-12-732961-7

work page 1992
[37]

WORLD SCIEN- TIFIC, 2010.DOI:10

Carl E Wulfman.Dynamical Symmetry. WORLD SCIEN- TIFIC, 2010.DOI:10 . 1142 / 7548. eprint:https : / / worldscientific.com/doi/pdf/10.1142/7548. URL:https://worldscientific.com/doi/abs/ 10.1142/7548. 53

work page doi:10.1142/7548 2010
[38]

On certain small representations of indefinite orthogonal groups

Chen-Bo Zhu and Jing-Song Huang. “On certain small representations of indefinite orthogonal groups”. In:Rep- resent. Theory1 (1997), pp. 190–206.ISSN: 1088-4165.DOI: 10.1090/S1088- 4165- 97- 00031- 9.URL:https: //doi.org/10.1090/S1088-4165-97-00031-9. 54

work page doi:10.1090/s1088- 1997