Tennis-racket instability of twisted electrons

S.S. Baturin

arxiv: 2604.05089 · v1 · submitted 2026-04-06 · 🪐 quant-ph · physics.acc-ph· physics.optics

Tennis-racket instability of twisted electrons

S.S. Baturin This is my paper

Pith reviewed 2026-05-10 19:17 UTC · model grok-4.3

classification 🪐 quant-ph physics.acc-phphysics.optics

keywords twisted electronstennis-racket instabilityDzhanibekov effectorbital pseudospinLaguerre-Gaussian beamssolenoidal fieldelectron microscopyHermite-Gaussian states

0 comments

The pith

A weak nonlinear magnetic edge at the entrance of a uniform field triggers tennis-racket instability in the orbital pseudospin of twisted electrons.

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

The paper establishes that a small nonlinear distortion in the magnetic entrance field can destabilize the pseudospin dynamics of twisted electrons even inside a perfectly uniform solenoidal region. This produces periodic reversals of the mean orbital pseudospin that show up as repeated switches between vortex and multi-lobed transverse beam shapes. A sympathetic reader would care because the effect is predicted to occur at microscope-scale energies and distances using only standard correction magnets, turning an abstract instability into an observable beam phenomenon without requiring exotic apparatus.

Core claim

A weak nonlinear magnetic entrance edge induces a tennis-racket (Dzhanibekov) instability in the shell-resolved orbital pseudospin dynamics of twisted electrons propagating in a nominally uniform solenoidal field. Starting from a Maxwell-consistent thin-edge extension of the entrance field, an effective fixed-shell Hamiltonian is obtained in which the linear Schwinger pseudospin precession acquires an anisotropic quadratic correction. In the symmetric aligned limit an exact linear eigenstate becomes a hyperbolic fixed point of the large-shell dynamics, producing recurrent reversals of the mean pseudospin projection that appear in real space as repeated conversions of the transverse profile.

What carries the argument

The Maxwell-consistent thin-edge extension of the entrance field, which supplies an anisotropic quadratic correction to the linear Schwinger pseudospin precession while holding the shell number fixed.

If this is right

Recurrent reversals of the mean pseudospin projection occur for large shell numbers.
Real-space beam profiles repeatedly convert between Laguerre-Gaussian vortex and Hermite-Gaussian multi-lobed states.
Lewis-Ermakov breathing modulates the nonlinear strength and sets the growth time scale without introducing a separate instability mechanism.
The required field strengths and distances fall within the range of standard octupole correctors in a transmission electron microscope.

Where Pith is reading between the lines

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

The same thin-edge mechanism could be engineered to control orbital angular momentum transfer in electron beams for lithography or microscopy applications.
Analogous instabilities may appear in other charged-particle systems that carry orbital angular momentum through mildly nonuniform magnetic regions.
The effect supplies a concrete, laboratory-accessible test of whether pseudospin dynamics in paraxial wave packets obey the same geometric rules as rigid-body rotations.

Load-bearing premise

The entrance field can be extended as a thin nonlinear Maxwell-consistent edge whose quadratic term remains anisotropic and does not mix shells.

What would settle it

Direct imaging of the electron beam profile that shows repeated, periodic switching between a single Laguerre-Gaussian vortex spot and a multi-lobed Hermite-Gaussian pattern as the packet travels along a uniform solenoidal field.

Figures

Figures reproduced from arXiv: 2604.05089 by S.S. Baturin.

read the original abstract

We demonstrate that a weak nonlinear magnetic entrance edge induces a tennis-racket (Dzhanibekov) instability in the shell-resolved orbital pseudospin dynamics of twisted electrons propagating in a nominally uniform solenoidal field. Starting from a Maxwell-consistent thin-edge extension of the entrance field, we derive an effective fixed-shell Hamiltonian in which linear Schwinger pseudospin precession acquires an anisotropic quadratic correction. In the symmetric aligned limit, an exact linear eigenstate (a Laguerre-Gaussian vortex state) becomes a hyperbolic fixed point of the large-shell dynamics, producing recurrent reversals of the mean pseudospin projection. These reversals appear in real space as repeated conversions of the transverse profile between Laguerre-Gaussian vortex and Hermite-Gaussian multi-lobed states. The unavoidable Lewis-Ermakov breathing of realistic wave packets does not generate a separate mechanism; it naturally modulates the nonlinear strength and sets the growth time scale. Microscope-scale estimates show that the required regime is accessible with standard octupole correctors in a transmission electron microscope.

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 maps Dzhanibekov instability onto shell-resolved orbital pseudospin of twisted electrons via a thin-edge magnetic correction, producing predicted profile reversals in solenoids.

read the letter

The main takeaway is that a weak nonlinear entrance edge in a solenoid can turn a Laguerre-Gaussian vortex state into a hyperbolic fixed point for large-shell dynamics, driving recurrent reversals between vortex and multi-lobed Hermite-Gaussian profiles. The work starts from a Maxwell-consistent thin-edge field model, adds an anisotropic quadratic term to the linear Schwinger precession, and keeps the shell fixed to derive the effective Hamiltonian. This produces the tennis-racket reversals without needing separate breathing mechanisms, though the breathing modulates the growth rate. The microscope-scale estimates with octupole correctors are concrete and show the regime is reachable in a TEM. What is new is the explicit mapping of the instability to orbital pseudospin in this beam setting and the real-space profile conversions that follow. The derivation stays within standard paraxial quantum mechanics for vortex electrons, and the fixed-shell assumption lets the hyperbolic character appear cleanly in the symmetric limit. The soft spot is the load-bearing field model itself. The thin-edge extension must remain Maxwell-consistent at quadratic order, induce no appreciable shell mixing, and generate exactly the stated anisotropy; any deviation in a real solenoidal edge would remove the hyperbolic point and the reversals. The paper asserts this holds but the abstract leaves the explicit verification steps implicit. Lewis-Ermakov breathing is dismissed as non-interfering, yet its modulation of the nonlinear strength still needs to be shown not to wash out the effect over the relevant distances. This is a specialized beam-physics result aimed at electron microscopy and accelerator groups working with vortex beams. A reader already familiar with Schwinger pseudospin and paraxial vortex states will get the most out of it. The central claim is a theoretical prediction rather than a fitted result, so the work deserves a serious referee to check the field expansion and the fixed-shell numerics. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The paper claims that a weak nonlinear magnetic entrance edge induces a tennis-racket (Dzhanibekov) instability in the shell-resolved orbital pseudospin dynamics of twisted electrons propagating in a nominally uniform solenoidal field. Starting from a Maxwell-consistent thin-edge extension of the entrance field, it derives an effective fixed-shell Hamiltonian in which linear Schwinger pseudospin precession acquires an anisotropic quadratic correction. In the symmetric aligned limit, an exact linear eigenstate (Laguerre-Gaussian vortex state) becomes a hyperbolic fixed point of the large-shell dynamics, producing recurrent reversals of the mean pseudospin projection that appear in real space as repeated conversions between Laguerre-Gaussian vortex and Hermite-Gaussian multi-lobed states. The Lewis-Ermakov breathing modulates the nonlinear strength and sets the growth timescale, with microscope-scale estimates indicating accessibility using standard octupole correctors in a transmission electron microscope.

Significance. If the central derivation holds, the result would be significant for connecting classical rigid-body instabilities to quantum paraxial electron beam dynamics in inhomogeneous magnetic fields, offering a mechanism for controlled orbital pseudospin reversals observable in electron microscopy. The use of a Maxwell-consistent field model and the fixed-shell approximation (without introducing fitted parameters beyond the edge strength) are strengths that ground the claim in standard quantum mechanics for twisted electrons.

major comments (1)

[Derivation of the effective fixed-shell Hamiltonian] The load-bearing step is the thin-edge extension of the entrance field: it must be shown to remain Maxwell-consistent at quadratic order, induce no appreciable shell mixing, and generate precisely the stated anisotropic quadratic correction to the linear Schwinger term. Without this explicit verification, the transformation of the Laguerre-Gaussian eigenstate into a hyperbolic fixed point (and the resulting recurrent reversals) does not necessarily follow.

minor comments (1)

[Abstract and estimates] The abstract and estimates section could clarify the precise order of the nonlinear correction and the specific octupole parameters assumed for the microscope-scale regime.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work's significance, and for highlighting the central derivation that requires explicit verification. We address the major comment below and will strengthen the presentation in the revised manuscript.

read point-by-point responses

Referee: The load-bearing step is the thin-edge extension of the entrance field: it must be shown to remain Maxwell-consistent at quadratic order, induce no appreciable shell mixing, and generate precisely the stated anisotropic quadratic correction to the linear Schwinger term. Without this explicit verification, the transformation of the Laguerre-Gaussian eigenstate into a hyperbolic fixed point (and the resulting recurrent reversals) does not necessarily follow.

Authors: We agree that explicit verification of these properties is essential for the claim. In Sec. II of the manuscript we construct the thin-edge extension by supplementing the uniform solenoidal field with a quadratic transverse correction to the vector potential chosen to satisfy both ∇·B=0 and ∇×B=0 identically at quadratic order. Within the fixed-shell subspace the perturbation is diagonal in the Laguerre-Gaussian basis because the quadratic term preserves azimuthal symmetry and commutes with the shell-number operator; off-diagonal matrix elements that would induce shell mixing therefore vanish identically. The resulting effective Hamiltonian (Eq. 12) contains the anisotropic quadratic correction to the linear Schwinger precession term, with the anisotropy coefficients obtained directly from the edge-strength parameter without additional fitting. To make the verification fully transparent we will add a dedicated appendix that recomputes the Maxwell conditions, lists the explicit matrix elements, and confirms the absence of inter-shell coupling at the working order. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from external field model via standard QM

full rationale

The paper begins with an externally chosen Maxwell-consistent thin-edge magnetic field extension and applies standard paraxial quantum mechanics to obtain the effective fixed-shell Hamiltonian and the resulting pseudospin instability. No equation or claim reduces the output to a parameter fitted from the target result itself, nor does any load-bearing step rely on a self-citation chain that is unverified or self-defining. The tennis-racket reversals are shown to follow from the quadratic anisotropy term under the stated assumptions, keeping the analysis self-contained against external benchmarks such as Maxwell consistency and Laguerre-Gaussian mode properties.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on standard electromagnetic theory and quantum beam dynamics with one key domain assumption about fixed shells and a thin-edge field approximation.

free parameters (1)

nonlinear edge field strength
The amplitude of the weak nonlinear component at the magnetic entrance edge is introduced as a tunable parameter that sets the instability growth rate.

axioms (2)

standard math Maxwell equations govern the magnetic field at the thin entrance edge
Invoked to justify the thin-edge extension used to derive the quadratic correction.
domain assumption Electron dynamics remain within a fixed shell with no inter-shell transitions
The effective Hamiltonian is constructed under the fixed-shell approximation stated in the abstract.

pith-pipeline@v0.9.0 · 5473 in / 1394 out tokens · 49845 ms · 2026-05-10T19:17:09.275241+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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

effective shell Hamiltonian Ĥ(j)_eff = 2 L̂₃ + μ₀ (L̂₁² − L̂₂²) ... linearized near positive-ℓ₃ pole yields ¨ℓ_{1,2} = (16μ² − 4)ℓ_{1,2}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

orbital Poincaré sphere ... Schwinger su(2) algebra

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

35 extracted references · 35 canonical work pages

[1]

1 4(2ˆnx + 1)(2ˆny + 1) + 1 4 ˆa†2 x ˆa2 y + ˆa†2 y ˆa2 x # ˆPj. 7 Hence ˆPj ˆV (n) 4 ˆPj = (B15) µ0 6 ˆPj

= 2, ℓ 2 1 +ℓ 2 2 +ℓ 2 3 = 1,(38) one finds that the minimumℓ 3 reached on the separatrix is ℓmin 3 = 1 µ −1.(39) Therefore the separatrix reachesℓ 3 <0 iffµ >1. This is the direct analogue of the Dzhanibekov instabil- ity in the reduced orbital pseudospin dynamics. While this simplified model already establishes the existence of the instability at the co...

work page
[2]

Mardeˇ si´ c, G

P. Mardeˇ si´ c, G. J. G. Guillen, L. Van Damme, and D. Sugny, Phys. Rev. Lett.125, 064301 (2020)

work page 2020
[3]

Van Damme, D

L. Van Damme, D. Leiner, P. Mardeˇ si´ c, S. J. Glaser, and D. Sugny, Scientific Reports7, 3998 (2017)

work page 2017
[4]

Y. Ma, K. E. Khosla, B. A. Stickler, and M. S. Kim, Phys. Rev. Lett.125, 053604 (2020)

work page 2020
[5]

Uchida and A

M. Uchida and A. Tonomura, Nature464, 737 (2010)

work page 2010
[6]

Verbeeck, H

J. Verbeeck, H. Tian, and P. Schattschneider, Nature 467, 301 (2010)

work page 2010
[7]

B. J. McMorran, A. Agrawal, I. M. Ander- son, A. A. Herzing, H. J. Lezec, J. J. McClel- land, and J. Unguris, Science331, 192 (2011), https://www.science.org/doi/pdf/10.1126/science.1198804

work page doi:10.1126/science.1198804 2011
[8]

Grillo, E

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, Phys. Rev. X4, 011013 (2014)

work page 2014
[9]

A. H. Tavabi, P. Rosi, A. Roncaglia, E. Rotunno, M. Be- leggia, P.-H. Lu, L. Belsito, G. Pozzi, S. Frabboni, P. Tiemeijer, R. E. Dunin-Borkowski, and V. Grillo, Ap- plied Physics Letters121, 073506 (2022)

work page 2022
[10]

Guzzinati, P

G. Guzzinati, P. Schattschneider, K. Y. Bliokh, F. Nori, and J. Verbeeck, Phys. Rev. Lett.110, 093601 (2013)

work page 2013
[11]

Schattschneider, M

P. Schattschneider, M. St¨ oger-Pollach, and J. Verbeeck, Phys. Rev. Lett.109, 084801 (2012)

work page 2012
[12]

N. V. Filina and S. S. Baturin, Phys. Rev. A110, 022204 (2024)

work page 2024
[13]

van Enk, Optics Communications102, 59 (1993)

S. van Enk, Optics Communications102, 59 (1993)

work page 1993
[14]

M. J. Padgett and J. Courtial, Opt. Lett.24, 430 (1999)

work page 1999
[15]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett.90, 203901 (2003)

work page 2003
[16]

G. F. Calvo, Opt. Lett.30, 1207 (2005)

work page 2005
[17]

Cisowski, J

C. Cisowski, J. B. G¨ otte, and S. Franke-Arnold, Rev. Mod. Phys.94, 031001 (2022)

work page 2022
[18]

K. Y. Bliokh, P. Schattschneider, J. Verbeeck, and F. Nori, Phys. Rev. X2, 041011 (2012)

work page 2012
[19]

Melkani and S

A. Melkani and S. J. van Enk, Phys. Rev. Res.3, 033060 (2021)

work page 2021
[20]

L. Zou, P. Zhang, and A. J. Silenko, Phys. Rev. A103, L010201 (2021)

work page 2021
[21]

N. V. Filina and S. S. Baturin, Phys. Rev. A108, 012219 (2023)

work page 2023
[22]

N. V. Filina and S. S. Baturin, Phys. Rev. A113, L021302 (2026)

work page 2026
[23]

A. J. Silenko, Journal of Mathematical Physics44, 2952 (2003)

work page 2003
[24]

A. J. Silenko, Modern Physics Letters A37, 2250097 (2022)

work page 2022
[25]

Schwinger,Quantum Theory of Angular Momentum: A Collection of Reprints and Original Papers, edited by L

J. Schwinger,Quantum Theory of Angular Momentum: A Collection of Reprints and Original Papers, edited by L. Biedenharn and H. Van Dam, Perspectives in Physics: a Series of Reprint Collections (Academic Press, 1965)

work page 1965
[26]

G. S. Agarwal and J. Banerji, Journal of Physics A: Mathematical and General39, 11503 (2006)

work page 2006
[27]

Y. F. Chen, Phys. Rev. A83, 032124 (2011)

work page 2011
[28]

B. J. McMorran, A. Agrawal, P. A. Ercius, V. Grillo, A. A. Herzing, T. R. Harvey, M. Linck, and J. S. Pierce, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences375, 20150434 (2017), https://royalsocietypublishing.org/rsta/article- pdf/doi/10.1098/rsta.2015.0434/1384369/rsta.2015.0434.pdf

work page doi:10.1098/rsta.2015.0434/1384369/rsta.2015.0434.pdf 2017
[29]

M. S. Epov, I. E. Shenderovich, and S. S. Baturin, Geometry-enabled radiation from structured paraxial electrons (2026), arXiv:2602.07858 [quant-ph]

work page arXiv 2026
[30]

J. H. R. Lewis and W. B. Riesenfeld, Journal of Mathe- matical Physics10, 1458 (1969)

work page 1969
[31]

H. R. Lewis, Phys. Rev. Lett.18, 510 (1967)

work page 1967
[32]

Aldaya, F

V. Aldaya, F. Coss´ ıo, J. Guerrero, and F. F. L´ opez-Ruiz, Journal of Physics A: Mathematical and Theoretical44, 065302 (2011)

work page 2011
[33]

Guerrero and F

J. Guerrero and F. L´ opez-Ruiz, Nuovo Cimento- Societa Italiana di Fisica Sezione C36, 127 (2013)

work page 2013
[34]

L´ opez-Ruiz and J

F. L´ opez-Ruiz and J. Guerrero, Journal of Physics Con- ference Series538, 012015 (2014)

work page 2014
[35]

Fern´ andez Guasti and H

M. Fern´ andez Guasti and H. M. Moya-Cessa, Journal of Physics A: Mathematical and General36, 2069 (2003)

work page 2069

[1] [1]

1 4(2ˆnx + 1)(2ˆny + 1) + 1 4 ˆa†2 x ˆa2 y + ˆa†2 y ˆa2 x # ˆPj. 7 Hence ˆPj ˆV (n) 4 ˆPj = (B15) µ0 6 ˆPj

= 2, ℓ 2 1 +ℓ 2 2 +ℓ 2 3 = 1,(38) one finds that the minimumℓ 3 reached on the separatrix is ℓmin 3 = 1 µ −1.(39) Therefore the separatrix reachesℓ 3 <0 iffµ >1. This is the direct analogue of the Dzhanibekov instabil- ity in the reduced orbital pseudospin dynamics. While this simplified model already establishes the existence of the instability at the co...

work page

[2] [2]

Mardeˇ si´ c, G

P. Mardeˇ si´ c, G. J. G. Guillen, L. Van Damme, and D. Sugny, Phys. Rev. Lett.125, 064301 (2020)

work page 2020

[3] [3]

Van Damme, D

L. Van Damme, D. Leiner, P. Mardeˇ si´ c, S. J. Glaser, and D. Sugny, Scientific Reports7, 3998 (2017)

work page 2017

[4] [4]

Y. Ma, K. E. Khosla, B. A. Stickler, and M. S. Kim, Phys. Rev. Lett.125, 053604 (2020)

work page 2020

[5] [5]

Uchida and A

M. Uchida and A. Tonomura, Nature464, 737 (2010)

work page 2010

[6] [6]

Verbeeck, H

J. Verbeeck, H. Tian, and P. Schattschneider, Nature 467, 301 (2010)

work page 2010

[7] [7]

B. J. McMorran, A. Agrawal, I. M. Ander- son, A. A. Herzing, H. J. Lezec, J. J. McClel- land, and J. Unguris, Science331, 192 (2011), https://www.science.org/doi/pdf/10.1126/science.1198804

work page doi:10.1126/science.1198804 2011

[8] [8]

Grillo, E

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, Phys. Rev. X4, 011013 (2014)

work page 2014

[9] [9]

A. H. Tavabi, P. Rosi, A. Roncaglia, E. Rotunno, M. Be- leggia, P.-H. Lu, L. Belsito, G. Pozzi, S. Frabboni, P. Tiemeijer, R. E. Dunin-Borkowski, and V. Grillo, Ap- plied Physics Letters121, 073506 (2022)

work page 2022

[10] [10]

Guzzinati, P

G. Guzzinati, P. Schattschneider, K. Y. Bliokh, F. Nori, and J. Verbeeck, Phys. Rev. Lett.110, 093601 (2013)

work page 2013

[11] [11]

Schattschneider, M

P. Schattschneider, M. St¨ oger-Pollach, and J. Verbeeck, Phys. Rev. Lett.109, 084801 (2012)

work page 2012

[12] [12]

N. V. Filina and S. S. Baturin, Phys. Rev. A110, 022204 (2024)

work page 2024

[13] [13]

van Enk, Optics Communications102, 59 (1993)

S. van Enk, Optics Communications102, 59 (1993)

work page 1993

[14] [14]

M. J. Padgett and J. Courtial, Opt. Lett.24, 430 (1999)

work page 1999

[15] [15]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, Phys. Rev. Lett.90, 203901 (2003)

work page 2003

[16] [16]

G. F. Calvo, Opt. Lett.30, 1207 (2005)

work page 2005

[17] [17]

Cisowski, J

C. Cisowski, J. B. G¨ otte, and S. Franke-Arnold, Rev. Mod. Phys.94, 031001 (2022)

work page 2022

[18] [18]

K. Y. Bliokh, P. Schattschneider, J. Verbeeck, and F. Nori, Phys. Rev. X2, 041011 (2012)

work page 2012

[19] [19]

Melkani and S

A. Melkani and S. J. van Enk, Phys. Rev. Res.3, 033060 (2021)

work page 2021

[20] [20]

L. Zou, P. Zhang, and A. J. Silenko, Phys. Rev. A103, L010201 (2021)

work page 2021

[21] [21]

N. V. Filina and S. S. Baturin, Phys. Rev. A108, 012219 (2023)

work page 2023

[22] [22]

N. V. Filina and S. S. Baturin, Phys. Rev. A113, L021302 (2026)

work page 2026

[23] [23]

A. J. Silenko, Journal of Mathematical Physics44, 2952 (2003)

work page 2003

[24] [24]

A. J. Silenko, Modern Physics Letters A37, 2250097 (2022)

work page 2022

[25] [25]

Schwinger,Quantum Theory of Angular Momentum: A Collection of Reprints and Original Papers, edited by L

J. Schwinger,Quantum Theory of Angular Momentum: A Collection of Reprints and Original Papers, edited by L. Biedenharn and H. Van Dam, Perspectives in Physics: a Series of Reprint Collections (Academic Press, 1965)

work page 1965

[26] [26]

G. S. Agarwal and J. Banerji, Journal of Physics A: Mathematical and General39, 11503 (2006)

work page 2006

[27] [27]

Y. F. Chen, Phys. Rev. A83, 032124 (2011)

work page 2011

[28] [28]

B. J. McMorran, A. Agrawal, P. A. Ercius, V. Grillo, A. A. Herzing, T. R. Harvey, M. Linck, and J. S. Pierce, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences375, 20150434 (2017), https://royalsocietypublishing.org/rsta/article- pdf/doi/10.1098/rsta.2015.0434/1384369/rsta.2015.0434.pdf

work page doi:10.1098/rsta.2015.0434/1384369/rsta.2015.0434.pdf 2017

[29] [29]

M. S. Epov, I. E. Shenderovich, and S. S. Baturin, Geometry-enabled radiation from structured paraxial electrons (2026), arXiv:2602.07858 [quant-ph]

work page arXiv 2026

[30] [30]

J. H. R. Lewis and W. B. Riesenfeld, Journal of Mathe- matical Physics10, 1458 (1969)

work page 1969

[31] [31]

H. R. Lewis, Phys. Rev. Lett.18, 510 (1967)

work page 1967

[32] [32]

Aldaya, F

V. Aldaya, F. Coss´ ıo, J. Guerrero, and F. F. L´ opez-Ruiz, Journal of Physics A: Mathematical and Theoretical44, 065302 (2011)

work page 2011

[33] [33]

Guerrero and F

J. Guerrero and F. L´ opez-Ruiz, Nuovo Cimento- Societa Italiana di Fisica Sezione C36, 127 (2013)

work page 2013

[34] [34]

L´ opez-Ruiz and J

F. L´ opez-Ruiz and J. Guerrero, Journal of Physics Con- ference Series538, 012015 (2014)

work page 2014

[35] [35]

Fern´ andez Guasti and H

M. Fern´ andez Guasti and H. M. Moya-Cessa, Journal of Physics A: Mathematical and General36, 2069 (2003)

work page 2069