Landau levels in a gravitational field: The Schwarzschild spacetime case

Alexandre Landry; Fay\c{c}al Hammad

REVIEW 3 major objections 5 minor 99 references

Landau levels in a gravitational field: The Schwarzschild spacetime case

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A gravitational field splits Landau levels and removes their infinite degeneracy.

desk verdict A serious paper that shows gravity splits Landau levels, but the headline formula (24) rests on an invalid large-l limit that contradicts the boundary condition defining the states. read the letter →

arxiv 1909.01827 v3 pith:M5IKXXDC submitted 2019-09-02 gr-qc quant-ph

Alexandre Landry , Fayçal Hammad This is my paper · ORCID

classification gr-qcquant-ph PACS 04.62.+v03.65.-w

keywords LandaulevelsSchwarzschildspacetimegravitationalsplittingofdegenerateconfluenthypergeometricfunctionsbiconfluentHeunequationinverse-squarelawtestsmagnetizedstarscurvedquantummechanics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the gravitational field of a spherical mass lifts the infinite degeneracy of Landau levels, the quantized energy levels of a charged particle in a uniform magnetic field. The central result is that for the first Landau level with large orbital quantum number $\ell$, the energy is approximately $3\hbar eB/(2m) - GMm\sqrt{eB/(2\hbar\ell)}$; the correction is the product of the gravitational coupling and the square root of the magnetic field, decreasing as $1/\sqrt{\ell}$. The paper derives this from the curved-spacetime wave equation in the Newtonian limit, using two independent methods—time-independent perturbation theory and a harmonic-oscillator approximation—that agree in the large-$\ell$ regime. It also shows that a widely used polynomial-truncation method based on the biconfluent Heun equation cannot give a general consistent quantization, and it extends the formalism to Yukawa-like and power-law departures from Newtonian gravity as a way to test gravity. If the claim holds, the effect could be observable with heavy charged molecules and relevant to the equation of state of magnetized stars.

What carries the argument

The load-bearing objects are the Landau orbitals, expressed through the confluent hypergeometric function ${}_1F_1(-n;\ell+1;\beta\rho^2/2)$ with $\beta=eB/\hbar$, on a reflecting sphere of radius $\rho_0$. The paper evaluates the gravitational matrix element—the integral of $1/\rho$ against two such orbitals—by rewriting the hypergeometric functions as polynomials and using incomplete-gamma-function identities; these identities expose a common factor $\sqrt{eB/2\hbar}$ in the shift. The second method works through a quartic equilibrium condition for the effective potential and a harmonic-oscillator expansion. The biconfluent Heun equation supplies the exact radial equation used to test the polynomial-truncation and asymptotic approaches.

What would settle it

Look for positive integer pairs $(n,\ell)$ satisfying ${}_1F_1(-n;\ell+1;eB\rho_0^2/(2\hbar))=0$ for a chosen field $B$ and sphere radius $\rho_0$; if none exist, the unperturbed basis assumed in the perturbation calculation is unavailable. Alternatively, measure the first-level splitting in a two-dimensional electron gas around a laboratory mass: the paper predicts a shift of $-GMm\sqrt{eB/(2\hbar\ell)}$ at large $\ell$, so a shift that does not scale as $\sqrt{B}/\sqrt{\ell}$, or is independent of the central mass, would refute the central claim.

Watch

Extended reading notes

Core claim

The paper's central discovery is that gravity breaks the degeneracy of Landau levels: orbitals belonging to the same Landau level acquire different energies. Using the unperturbed Landau orbitals as a basis, the first-order shift from the Newtonian potential $-GMm/\rho$ is diagonal in the orbital quantum number $\ell$, so each level splits. For the first level and large $\ell$ the paper finds $E_{1\ell}\approx 3\hbar eB/(2m) - GMm\sqrt{eB/(2\hbar\ell)}$, with the correction small only when $\ell$ exceeds $\beta\rho_0^2/2$. The same conclusion is reached by an independent harmonic-oscillator expansion around the equilibrium radius, with numerical factors coinciding in the large-$\ell$ limit. In the full relativistic treatment, the paper finds an additional curvature–magnetic correction that grows as $\sqrt{\ell}$, in contrast to the Newtonian term that falls as $1/\sqrt{\ell}$. The paper also shows that the polynomial truncation of the biconfluent Heun solution imposes a fine-tuning condition on the central mass and therefore cannot serve as a general quantization rule, while the exact asymptotic condition is consistent but impractical.

Load-bearing premise

The calculation assumes that for a real magnet, mass, and sphere size one can always choose the magnetic field so that the correct Landau wavefunctions vanish at the sphere's surface and remain a complete basis, without proving that such parameter choices exist.

Editorial extensions

If this is right

The infinite degeneracy of each Landau level is removed: orbitals with different $\ell$ in the same level have distinct energies, with a shift that grows as $\sqrt{B}$ and falls as $1/\sqrt{\ell}$ for large $\ell$.
A Yukawa-type departure from the inverse-square law contributes an exponentially suppressed correction proportional to $e^{-\rho_0/\lambda}$, while a power-law departure contributes a factor $(eB L^2/(2\hbar\ell))^{s/2}$; the two are distinguishable in their dependence on $\ell$ and on the length scale.
For heavy charged molecules, the gravitational splitting of the first Landau level can reach about $10^{-3}$ eV at temperatures near $10^{-4}$ K, bringing it in principle within reach of tabletop experiments.
In strongly magnetized stars, the gravity-induced splitting modifies the Landau-quantized equation of state of the surface electron gas, offering an astrophysical observable tied to the star's mass.
The relativistic treatment separates a Newtonian correction proportional to $1/\sqrt{\ell}$ from a curvature–magnetic correction proportional to $\sqrt{\ell}$, so fast particles in high orbitals probe the curved background rather than only Newtonian gravity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the diagonal structure of the Newtonian perturbation follows from rotational symmetry, a rotating or deformed source would generically couple orbitals with different $\ell$; one should then expect avoided crossings and level repulsion in extensions to non-spherical metrics.
The growth of the splitting with $\sqrt{B}$ suggests that ultrastrong fields would amplify the gravitational signal, but the same growth eventually threatens the perturbativity condition $GMm/\rho \ll \hbar eB/m$; the crossover region could itself serve as a diagnostic of strong-field gravity.
The exact asymptotic Heun condition, though impractical for hand calculation, provides a non-perturbative numerical route: solving it for moderate $\ell$ would test whether the large-$\ell$ formulas extrapolate correctly, a check the paper does not perform.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core claim here is that a spherical mass splits Landau levels and removes the infinite degeneracy. The paper does this through two independent Newtonian methods that agree on the qualitative behavior, and it goes on to derive Yukawa and power-law corrections plus a relativistic extension. The comparison of four quantization approaches and the explicit critique of the biconfluent Heun polynomial truncation are genuinely useful—the authors correctly identify why that method fails to give a consistent quantization condition. The appendices with the Kummer-function integrals are also careful and likely reusable.

But there is a load-bearing flaw in the first method. The unperturbed states are required to vanish on the sphere, giving Eq. (19). For n=1 this forces beta*rho0^2/2 = l+1, so l is fixed by the boundary parameter. The large-l evaluation of the matrix elements, Eqs. (A15) and (A19), then discards boundary-dependent incomplete-gamma terms under the condition l > e*beta*rho0^2/2, which is the opposite of what the boundary-adapted state satisfies. The neglected terms are not small; they grow like e^l/sqrt(l). So Eq. (24) is not the large-l limit of those states. The harmonic oscillator approximation independently suggests the same 1/sqrt(l) scaling, so the qualitative conclusion may well be correct, but the paper's primary derivation of the central formula is not valid as written.

The relativistic section also claims an exact Schrödinger-like form in Eq. (60), but the coordinate transformation appears to lead to a result that is only approximate, not exact, as the reader noted. That should be checked carefully before publication.

Other soft spots are minor. The laboratory effect is tiny, around 10^-16 eV for protons, and the astrophysical coherence assumptions are optimistic, but the paper is honest about these limitations.

This is a paper that deserves a serious referee. The central physical conjecture is plausible but not established by the calculation presented. The fix might be straightforward—either prove the limit in a way consistent with the boundary condition, or relegate the perturbation-theory result and lean on the harmonic oscillator method as the primary derivation. I would send it to peer review and ask for a revision that addresses the large-l inconsistency and re-checks the relativistic transformation.

Referee Report

3 major / 5 minor

Summary. The paper studies a charged particle moving in a uniform magnetic field on a Schwarzschild background. It derives an approximate Newtonian radial equation (Eq. (15)) and computes first-order gravitational corrections to Landau levels by time-independent perturbation theory with a Dirichlet boundary at a finite sphere radius ρ0, obtaining Eqs. (22)–(24). A harmonic-oscillator expansion (Section 3.2) is presented as an independent confirmation. The paper also criticizes a biconfluent-Heun polynomial truncation method, proposes tests of Yukawa and power-law deviations from the inverse-square law (Section 4), and gives a relativistic treatment with a tortoise-coordinate reduction (Section 5). The central claim is that the gravitational field of a spherical mass removes the infinite Landau degeneracy and, for the first level and large orbital quantum number 𝓁, produces a correction proportional to GMm√B/√𝓁 (Eq. (24)).

Significance. If the central result is established, the gravitational splitting of Landau levels with the specific 1/√𝓁 scaling and the GMm√B product is an interesting and potentially testable effect, relevant both for tabletop gravity experiments and for magnetized astrophysical objects. The paper has notable strengths: two independent approximation schemes agree qualitatively, the appendix supplies explicit integral evaluations for the matrix elements, and the discussion of the biconfluent-Heun approach usefully clarifies a limitation of that method. However, the primary perturbative derivation of Eq. (24) contains a technical inconsistency between the boundary condition and the large-𝓁 asymptotic limit, so the central formula is not established as presented. The qualitative conclusion may survive, but the derivation needs repair.

major comments (3)

[§3.1 and Appendix A, Eqs. (19), (24), (A15), (A19)] The boundary condition (19) is incompatible with the large-𝓁 asymptotic used to derive the headline formula (24). For n=1, Eq. (19) reduces to 1F1(-1;𝓁+1;βρ0²/2)=1-βρ0²/[2(𝓁+1)]=0, so βρ0²/2=𝓁+1 and hence 𝓁=βρ0²/2−1. The large-𝓁 evaluation of M1𝓁 and P1𝓁 in Eqs. (A15) and (A19) discards the boundary-dependent incomplete-gamma terms; the text states this is valid only for 𝓁>eβρ0²/2 (see the paragraph following Eq. (A15)). No n=1 state satisfying Eq. (19) can meet this condition, since 𝓁=βρ0²/2−1<eβρ0²/2. The neglected series terms are of order (βρ0²/2)^{𝓁+1}/(𝓁+1)!∼e^{𝓁}/√𝓁, which grow rather than vanish. Consequently Eq. (24) is not the large-𝓁 limit of the matrix elements for the boundary-adapted states, and the stated 1/√𝓁 scaling is not proven by this calculation. The harmonic-oscillator method independently suggests the same qualitative behavior, so the conclusion may survive, but the perturbative derivation must be repaired, for example by using full-space Landau states (for which ⟨n,𝓁|1/ρ|n,𝓁⟩ is finite) or by analyzing the boundary-adapted states with the correct relation between 𝓁 and βρ0².
[§3.1, Eqs. (18)–(24)] The interpretation of Eq. (24) as the gravitational splitting of the first Landau level is not supported by the boundary-adapted calculation. For n=1, Eq. (19) has a unique solution 𝓁=βρ0²/2−1, so the unperturbed first Landau level already has only one allowed orbital in this basis. The gravitational correction (23)–(24) is therefore a shift of a single boundary-selected orbital, not a splitting of a degenerate level. The claim in the abstract and Section 6 that gravity removes the infinite degeneracy is thus conflated with the effect of the impenetrable-sphere boundary condition, which itself restricts 𝓁. The authors should either compute the 1/ρ perturbation in the full plane with ρ0=0, where the Landau degeneracy is genuinely infinite and the matrix element is finite, or explicitly separate the boundary-induced reduction of degeneracy from the gravitational splitting.
[§5, Eqs. (58)–(64)] The relativistic derivation replaces the tortoise coordinate ρ* by ρ to zeroth order in GM/c² while retaining first-order GM/c² terms in the potential. This is not a controlled first-order truncation of Eq. (60) unless the first-order correction to the kinetic term is also taken into account. Specifically, from Eq. (58) one has dρ/dρ*=(1+2GM/(c²ρ))^{-1}, so the transformation of d²/dρ*² introduces a first-order correction of order (GM/c²ρ) times the kinetic operator. When acting on the unperturbed states, this omitted term is of order (GM/c²ρ)ħ eB/m, which is comparable to retained relativistic terms such as −GMm/ρ(1−ħ eB𝓁/(m²c²)) for 𝓁 of order one. The derivation of Eq. (64) requires either a consistent first-order expansion in GM/c²ρ including the kinetic factor or an explicit estimate showing that the omitted term is higher order in the small parameters uniformly in 𝓁.

minor comments (5)

[§3.1] Typo: 'Iimportantly' should be 'Importantly' in the paragraph introducing the finite-radius boundary condition.
[Appendix A, before Eq. (A1)] Typo: 'Polchhammer symbol' should be 'Pochhammer symbol'.
[§3.1, after Eq. (26)] Typo: 'cannot be used to find be energy levels' should read 'cannot be used to find the energy levels'.
[Abstract and §4] The phrase 'deviations from the inverse-square law' is used in the abstract and Section 4, while Section 4 itself sometimes says 'square-law'; the terminology should be unified.
[§4.2, Eq. (56)] The polar-cap result Eq. (56) is imported from Ref. [23] without derivation; a brief derivation or a more explicit citation of the relevant equation in that reference would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the gravitational Landau splitting is derived from the Schwarzschild metric by standard perturbation theory with no fitted parameters and no load-bearing self-citation.

full rationale

The central derivation chain is self-contained. The paper begins from the Klein-Gordon equation in the Schwarzschild metric (Eqs. (1), (4), (6)), expands to leading Newtonian order (Eq. (15)), converts to Schrodinger form (Eq. (16)), and computes the gravitational correction by first-order perturbation theory in the standard Landau basis (Eqs. (17)-(23)), with no parameter fitted to the target result. The M=0 limit reproduces the textbook Landau levels (Eq. (13)), serving as an internal consistency check. The large-l formula (24) is obtained from the explicit gamma-function asymptotics of the computed matrix elements M and P in Appendix A, not by assuming the answer. The harmonic-oscillator method of Section 3.2 independently reaches the same scaling, so the central claim does not reduce to an input or a fit. The only use of the authors' prior work is the polar-cap splitting in Eq. (56), imported from Ref. [23] as an auxiliary application; it is not used to derive the main claim and does not constrain the central derivation. The paper's admitted assumption that combinations of B, rho0, n, and l satisfying the Dirichlet condition (19) exist is a gap in support rather than circularity, and the possible inconsistency between Eq. (19) and the large-l limit is a correctness risk, not a circular reduction. Accordingly, no circular step is present.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central derivation introduces no new particles, forces, or fitted constants. It relies on standard quantum mechanics in curved spacetime plus several approximation and idealization assumptions; the most fragile is the existence of the boundary-adapted Landau states required by Eq. (19).

assumptions (8)

domain assumption Klein-Gordon equation with minimal electromagnetic coupling in curved spacetime describes the charged test particle.
Used at the start of Section 2 to build Eq. (1); spin is neglected by design.
domain assumption The magnetic field's backreaction on spacetime is negligible, so the Schwarzschild-Melvin factor Lambda is set to 1.
Invoked after Eq. (3) to replace the metric with pure Schwarzschild; valid for tabletop fields and for fields up to about 10^12 G used in the paper.
domain assumption The non-relativistic and weak-field approximations E much less than mc^2 and GM/(c^2 rho) much less than 1 hold.
These approximations convert Eq. (14) into Eq. (15) and justify treating the Newtonian potential as a small perturbation.
domain assumption The motion is restricted to the equatorial plane theta = pi/2, and the wavefunction depends only on rho = r sin theta.
Used immediately after Eq. (5); the paper notes that other planes require the latitude angle to be included.
ad hoc to paper The spherical mass is impenetrable and Landau states satisfying Eq. (19), 1F1(-n; l+1; beta rho_0^2/2)=0, exist.
The perturbation calculation in Section 3.1 uses only these boundary-adapted states and assumes a suitable combination of B, rho_0, n, and l rather than proving existence.
ad hoc to paper In Section 5, the tortoise coordinate rho* is approximated by rho to zeroth order in GM/c^2 while GM terms are kept in the potential.
Needed to pass from Eq. (60) to the perturbative Eq. (62); this mixing of orders is an approximation and not exact.
domain assumption The effective potential in the oscillator approach is truncated at quadratic order around its minimum, and the parameter x in Eq. (32) is small.
This is the harmonic oscillator approximation of Section 3.2, appropriate for weak gravity and laboratory-level fields.
standard math Standard properties of confluent hypergeometric functions, Laguerre polynomials, incomplete gamma functions, and biconfluent Heun functions are used without proof.
The special-function identities are taken from standard references and are used throughout Appendix A.

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Cite this review

Pith. "Pith review of Landau levels in a gravitational field: The Schwarzschild spacetime case." pith.science (2026). https://pith.science/paper/M5IKXXDC

@misc{pith2026190901827,
  author       = {Pith},
  title        = {Pith review of: Landau levels in a gravitational field: The Schwarzschild spacetime case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M5IKXXDC}},
  note         = {Machine review of arXiv:1909.01827}
}

read the original abstract

We investigate the gravitational effect on Landau levels. We show that the familiar infinite Landau degeneracy of the energy levels of a quantum particle moving inside a uniform and constant magnetic field is removed by the interaction of the particle with a gravitational field. Two independent approaches are used to solve the relevant Schr\"odinger equation within the Newtonian approximation. It is found that both approaches yield qualitatively similar results within their respective approximations. With the goal of clarifying some results found in the literature concerning the use of a third independent approach for extracting the quantization condition based on a similar differential equation, we show that such an approach cannot yield a general and yet consistent result. We point out to the more accurate, but impractical, way to use such an approach; a way which does in principle yield a consistent quantization condition. We discuss how our results could be used to contribute in a novel way to the existing methods for testing gravity at the tabletop experiments level as well as at the astrophysical observational level by deriving the corrections brought by Yukawa-like and power-law deviations from the inverse-square law. The full relativistic regime is also examined in detail.

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Works this paper leans on

99 extracted references · 63 canonical work pages

[1]

Quantum states of neutrons in the Earth’s gravitational ﬁeld

Nesvizhevsky, V .V .; Börner, H.G.; Petukhov A.K.; Abele H.; Baeßler, S; Rueß, F.J.; Stöferle T.; Westphal A.; Gagarski, A.M.; Petrov, G.A.; Strelkov, A.V . Quantum states of neutrons in the Earth’s gravitational ﬁeld. Nature415, 297 (2002)

2002
[2]

Measurement of quantum states of neutrons in the Earth's gravitational field

Nesvizhevsky, V .V .; Börner, H.G.; Gagarski, A.M.; Petoukhov, A.K.; Petrov, G.A.; Abele, H.; Baeßler, S.; Divkovic, G.; Rueß, F.J.; Stöferle, Th.; Westphal, A.; Strelkov, A.V .; Protasov, K.V .; Voronin, A.Yu. Measurement of quantum states of neutrons in the Earth’s gravitational ﬁeld. Phys. Rev. D 67, 102002 (2003) [arXiv:hep-ph/0306198]

work page Pith review arXiv 2003
[3]

Study of the neutron quantum states in the gravity field

Nesvizhevsky, V .V .; Petukhov, A.K.; Börner, H.G.; Baranova, T.A.; Gagarski, A.M.; Petrov, G.A.; Protasov, K.V .; Voronin, A.Yu.; Baeßler, S.; Abele, H.; Westphal, A.; Lucovac, L. Study of the neutron quantum states in the gravity ﬁeld. Eur. Phys. J. C 40, 479 (2005) [arXiv:hep-ph/0502081]

work page Pith review arXiv 2005
[4]

Rauch, H.H.; Werner, S.A.Neutron Interferometry, Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglment, 2nd ed.; Oxford University Press: Oxford, UK, 2015

2015
[5]

Spectrometer for new gravitational experiment with UCN

Kulin, G.V .; Frank, A.I.; Goryunov, S.V .; Kustov, D.V .; Geltenbort, P .; Jentschel, M.; Strepetov, A.N.; Bushuev, V .A. Spectrometer for new gravitational experiment with UCN. In Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 792, 38 (2015) [arXiv:1502.03243]

work page Pith review arXiv 2015
[6]

Precision experiments with cold and ultra-cold neutrons

Abele, H. Precision experiments with cold and ultra-cold neutrons. Hyperﬁne Interact 237, 155 (2016)

2016
[7]

Witnessing the quantumness of a system by observing only its classical features

Marletto, C.; Vedral, V . Witnessing the quantumness of a system by observing only its classical features. NPJ. Quantum Information3, 43 (2017) [arXiv:1704.00120]

arXiv 2017
[8]

Gravitationally Induced Entanglement between Two Massive Particles is Sufﬁcient Evidence of Quantum Effects in Gravity

Marletto, C.; Vedral, V . Gravitationally Induced Entanglement between Two Massive Particles is Sufﬁcient Evidence of Quantum Effects in Gravity. Phys. Rev. Lett. 119, 240402 (2017) [arXiv:1707.06036]

arXiv 2017

Show all 99 references

[9]

A Spin Entanglement Witness for Quantum Gravity

Bose, S.; Mazumdar, A.; Morley, G.W.; Ulbricht, H.; Toroš, M.; Paternostro, A.; Geraci, A.A.; Barker, P .F; Kim, A.S.; Milburn, G. A Spin Entanglement Witness for Quantum Gravity. Phys. Rev. Lett. 119, 240401 (2017) [arXiv:1707.06050]

2017 arXiv
[10]

Prospects for testing the inverse-square law and gravitomagnetism using quantum interference

Hammad, F.; Landry, A.; Mathieu, K. Prospects for testing the inverse-square law and gravitomagnetism using quantum interference. Int. J. Mod. Phys. D 30, 2150004 (2021) [arXiv:1910.13814]

2021 arXiv
[11]

A simple superconductor quantum interference device for testing gravity

Hammad, F.; Landry, A. A simple superconductor quantum interference device for testing gravity. Mod. Phys. Lett. A35, 2050171 (2020) [arXiv:2005.05798]

2020 arXiv
[12]

Possible Daily and Seasonal Variations in Quantum Interference Induced by Chern-Simons Gravity

Okawara, H.; Yamada, K.; Asada, H. Possible Daily and Seasonal Variations in Quantum Interference Induced by Chern-Simons Gravity. Phys. Rev. Lett. 109, 231101 (2012) [arXiv:1210.4628]

2012 arXiv
[13]

Possible latitude effects of Chern-Simons gravity on quantum interference

Okawara, H.; Yamada, K.; Asada, H. Possible latitude effects of Chern-Simons gravity on quantum interference. Phys. Rev. D 87, 084038 (2013) [arXiv:1302.0002]

2013 arXiv
[14]

Possible altitudinal, latitudinal and directional dependence of relativistic Sagnac effect in Chern-Simons modiﬁed gravity

Kikuchi, D.; Omoto, N.; Yamada, K.; Asada, H. Possible altitudinal, latitudinal and directional dependence of relativistic Sagnac effect in Chern-Simons modiﬁed gravity. Phys. Rev. D 90, 064036 (2014) [arXiv:1405.7472]

2014 arXiv
[15]

Gravitationally induced quantum transitions

Landry, A.; Paranjape, M.B. Gravitationally induced quantum transitions. Phys. Rev. D 93, 122006 (2016) [arXiv:1601.06132]

2016 arXiv
[16]

Graviton Laser

Landry, A.; Paranjape, M.B. Graviton Laser. Int. J. Mod. Phys. D 25, 1644016 (2016) [arXiv:1604.02762]

2016 arXiv
[17]

Quantum Mechanics: Non-Relativistic Theory, 2nd ed.; Pergamon Press: Oxford, UK, 1965

Landau, L.D.; Lifshitz, E.M. Quantum Mechanics: Non-Relativistic Theory, 2nd ed.; Pergamon Press: Oxford, UK, 1965

1965
[18]

New degeneracies and modiﬁcation of Landau levels in the presence of a parallel linear electric ﬁeld

Edery, A.; Audin, Y. New degeneracies and modiﬁcation of Landau levels in the presence of a parallel linear electric ﬁeld. J. Phys. Commun. 3, 025013 (2019) [arXiv:1808.00369]

2019 arXiv
[19]

(Eds.) The Quantum Hall Effect; Springer: Heidelberg, Germany, 1992

Prange, R.E.; Girvin, S.M. (Eds.) The Quantum Hall Effect; Springer: Heidelberg, Germany, 1992

1992
[20]

Introduction to the Theory of the Integer Quantum Hall Effect ; VCH: Weinheim, Germany, 1994

Janssen, M.; Viehweger, O.; Fastenrath, U.; Hajdu, J. Introduction to the Theory of the Integer Quantum Hall Effect ; VCH: Weinheim, Germany, 1994. 35 of 37

1994
[21]

Is the Quantum Hall Effect Inﬂuenced by the Gravitational Field? Phys

Hehl, F.W.; Obukhov, Y.N.; Rosenow, B. Is the Quantum Hall Effect Inﬂuenced by the Gravitational Field? Phys. Rev. Lett.93, 096804 (2004) [arXiv:cond-mat/0310281]

2004 arXiv
[22]

Splitting of Landau levels in the presence of external potentials

Grosse, H.; Stubbe, J. Splitting of Landau levels in the presence of external potentials. Lett. Math. Phys. 34 59, (1995)

1995
[23]

A fresh look at the inﬂuence of gravity on the quantum Hall effect

Hammad, F.; Landry, A.; Mathieu, K. A fresh look at the inﬂuence of gravity on the quantum Hall effect. Eur. Phys. J. Plus 135, 449 (2020) [arXiv:2005.10631]

2020 arXiv
[24]

Parker, L.E.; Toms, D.J.Quantum Field Theory in Curved Spacetime, Quantized Fields and Gravity; Cambridge University Press: Cambridge, UK, 2009

2009
[25]

Exact Space-Times in Einstein’s General Relativity; Cambridge University Press: New York, NY, USA, 2009

Grifﬁths, J.B.; Podolský, J. Exact Space-Times in Einstein’s General Relativity; Cambridge University Press: New York, NY, USA, 2009

2009
[26]

Black holes in a magnetic universe

Ernst, F.J. Black holes in a magnetic universe. J. Math. Phys. 17, 54 (1976)

1976
[27]

Removal of the nodal singularity of the C−metric

Ernst, F.J. Removal of the nodal singularity of the C−metric. J. Math. Phys. 17, 515 (1976)

1976
[28]

black hole in an external magnetic ﬁeld

Gal’tsov, D.V .; Petukhov, V .I. black hole in an external magnetic ﬁeld. Sov. Phys. JETP47, 419 (1978). (Russian original: Zh. Eksp. Teor. Fiz. 74, 801 (1978))

1978
[29]

Chaotic motion of test particles in the Ernst space-time

Karas, V .; Vokrouhlicky, D. Chaotic motion of test particles in the Ernst space-time. Gen. Relat. Gravit. 24, 729 (1992)

1992
[30]

Static Magnetic Fields in General Relativity

Bonnor, W.B. Static Magnetic Fields in General Relativity. Proc. Phys. Soc. A 67 225 (1954)

1954
[31]

Pure magnetic and electric geons

Melvin, M.A. Pure magnetic and electric geons. Phys. Lett. 8, 65 (1964)

1964
[32]

Dirac equation and the Melvin metric

Santos, L.C.N.; Barros, C.C., Jr. Dirac equation and the Melvin metric. Eur. Phys. J. C 76, 560 (2016) [arXiv:1508.07307]

2016 arXiv
[33]

Neutron Stars 1: Equation of State and Structure; Springer: Berlin, Germany, 2007

Haensel, P .; Potekhin, A.Y.; Yakovlev, D.G. Neutron Stars 1: Equation of State and Structure; Springer: Berlin, Germany, 2007

2007
[34]

General Relativity; University of Chicago Press: Chicago, IL, USA, 1984

Wald, R. General Relativity; University of Chicago Press: Chicago, IL, USA, 1984

1984
[35]

Magnus, W.; Oberhettinger, F.; Soni, R.P .Formulas and Theorems for the Special Functions of Mathematical Physics , 3rd ed.; Springer: Berlin, Germany, 1966

1966
[36]

The Conﬂuent Hypergeometric Function, with Special Emphasis on its Applications; Springer: Berlin, Germany, 1969

Buchholz, H. The Conﬂuent Hypergeometric Function, with Special Emphasis on its Applications; Springer: Berlin, Germany, 1969

1969
[37]

Magnetic White Dwarfs

Ferrario, L.; de Martino, D.; Gaensicke, B. Magnetic White Dwarfs. Space Sci. Rev. 191, 111 (2015) [arXiv:1504.08072]

2015 arXiv
[38]

The Hill determinant: An application to a class of conﬁnement potentials

Chaudhuri, R.N. The Hill determinant: An application to a class of conﬁnement potentials. J. Phys. A: Math. Gen. 16, 209 (1983)

1983
[39]

Two electrons in an external oscillator potential: Particular analytic solutions of a Coulomb correlation problem

Taut, M. Two electrons in an external oscillator potential: Particular analytic solutions of a Coulomb correlation problem. Phys. Rev. A 48, 3561 (1993)

1993
[40]

Exact low-lying states of two interacting equally charged particles in a magnetic ﬁeld

Truong, T.T.; Bazzali, D. Exact low-lying states of two interacting equally charged particles in a magnetic ﬁeld. Phys. Lett. A 269, 186 (2000)

2000
[41]

Harmonium

Karwowski, J.; Cyrnek, L. Harmonium. Ann. Phys. (Leipzig) 13. 181 (2004)

2004
[42]

Few-particle systems: Quasi-exactly solvable models

Karwowski, J. Few-particle systems: Quasi-exactly solvable models. J. Phys.: Conf. Ser. 104, 012033 (2008)

2008
[43]

Separable N-particle Hookean models

Karwowski, J.; Szewc, K. Separable N-particle Hookean models. J. Phys.: Conf. Ser. 213 012016 (2010)

2010
[44]

Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconﬂuent Heun Equation

Caruso, F.; Martins, J.; Oguri, V . Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconﬂuent Heun Equation. Ann. Phys. 347, 130 (2014) [arXiv:1308.0815]

2014 arXiv
[45]

Biconﬂuent Heun equation in quantum chemistry: Harmonium and related systems

Karwowski, J.; Witek, H.A. Biconﬂuent Heun equation in quantum chemistry: Harmonium and related systems. Theor. Chem. Acc. 133 (2014)

2014
[46]

Energy spectra of Hartmann and ring-shaped oscillator potentials using the quantum Hamilton-Jacobi formalism

Gharbi, A.; Bouda, A. Energy spectra of Hartmann and ring-shaped oscillator potentials using the quantum Hamilton-Jacobi formalism. Phys. Scr. 88, 045007 (2013) [arXiv:1312.0087]

2013 arXiv
[47]

Exact Solutions of the Schrodinger Equation with Inverse-Power Potential in Two Dimensions

Dong, S.-H.; Ma, Z.-Q.; Esposito, G. Exact Solutions of the Schrodinger Equation with Inverse-Power Potential in Two Dimensions. Found. Phys. Lett. 12, 465 (1999) [arXiv:quant-ph/9902081]

1999 arXiv
[48]

Exact Solutions of the Two-Dimensional Schrödinger Equation with Certain Central Potentials

Dong, S.-H. Exact Solutions of the Two-Dimensional Schrödinger Equation with Certain Central Potentials. Int. J. Theor. Phys. 39, 1119 (2000) [arXiv:quant-ph/0003100]

2000 arXiv
[49]

Exact solutions of the radial Schrödinger equation for some physical potentials

Ikhdair, S.; Sever, R. Exact solutions of the radial Schrödinger equation for some physical potentials. C. Eur. Phys. J. 5, 516 (2007)

2007
[50]

Variational and perturbative schemes for a spiked harmonic oscillator

Aguilera-Navarro, V .C.; Estévez, G.A.; Guardiola, R. Variational and perturbative schemes for a spiked harmonic oscillator. J. Math. Phys. 31, 99 (1990)

1990
[51]

Matrix elements for a generalized spiked harmonic oscillator

Hall, R.L.; Saad, N.; von Keviczky, A.B. Matrix elements for a generalized spiked harmonic oscillator. J. Math. Phys. 39, 6345 (1998) [arXiv:quant-ph/9812048]

1998 arXiv
[52]

Variational analysis for a generalized spiked harmonic oscillator

Hall, R.L.; Saad, N. Variational analysis for a generalized spiked harmonic oscillator. J. Phys. A 33, 569 (2000) [arXiv:quant- ph/9911118]

2000
[53]

Perturbation expansions for the spiked harmonic oscillator and related series involving the gamma function

Hall, R.L.; Saad, N. Perturbation expansions for the spiked harmonic oscillator and related series involving the gamma function. J. Phys. A 33, 5531 (2000) [arXiv:math-ph/0006024]

2000 arXiv
[54]

Generalized spiked harmonic oscillator

Hall, R.L.; Saad, N.; von Keviczky, A.B. Generalized spiked harmonic oscillator. J. Phys. A 34, 1169 (2001) [arXiv:math-ph/0101006]

2001 arXiv
[55]

Integrals containing conﬂuent hypergeometric functions with applications to perturbed singular potentials

Saad, N.; Hall, R.L. Integrals containing conﬂuent hypergeometric functions with applications to perturbed singular potentials. J. Phys. A 36, 7771 (2003)[arXiv:math-ph/0306043]

2003 arXiv
[56]

Quantum Mechanics, 1st ed.; McGraw-Hill Book Company, Inc.: New York, NY, USA, 1949

Schiff, J.I. Quantum Mechanics, 1st ed.; McGraw-Hill Book Company, Inc.: New York, NY, USA, 1949

1949
[57]

Atoms in high magnetic ﬁelds (white dwarfs)

Garstang, R.H. Atoms in high magnetic ﬁelds (white dwarfs). Rep. Prog. Phys. 40, 105 (1977). 36 of 37

1977
[58]

Atoms in Strong Magnetic Fields: Quantum Mechanical Treatment and Applications in Astrophysics and Quantum Chaos; Springer: Berlin, Germany, 1994

Ruder, H.; Wunner, G.; Herold, H.; Geyer, F. Atoms in Strong Magnetic Fields: Quantum Mechanical Treatment and Applications in Astrophysics and Quantum Chaos; Springer: Berlin, Germany, 1994

1994
[59]

(Eds.) Solutions of Quartic Equations

Abramowitz, M.; Stegun, I.A. (Eds.) Solutions of Quartic Equations. In Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover: New York, NY, USA, 1972

1972
[60]

Arscott, F.M.; Slavyanov, S.Yu.; Schmidt, D.; Wolf, G.; Maroni, P .; Duval; A.Heun’s Differential Equation; Ronveaux, A., Ed.; Oxford University Press: Oxford, UK, 1995

1995
[61]

Heun Functions and Some of Their Applications in Physics

Hortacsu, M. Heun Functions and Some of Their Applications in Physics. Adv. High Energy Phys. 2018, 8621573 (2018) [arXiv:1101.0471]

2018 arXiv
[62]

Spectral properties of the biconﬂuent Heun differential equation

Arriola, E.R.; Zarzo, A.; Dehesa, J.S. Spectral properties of the biconﬂuent Heun differential equation. J. Com. Appl. Math. 37, 161 (1991)

1991
[63]

Global solutions of the biconﬂuent Heun equation

Ferreira, E.M.; Sesma, J. Global solutions of the biconﬂuent Heun equation. Numer. Algor. 71, 797 (2016)

2016
[64]

Quantum Newtonian cosmology and the biconﬂuent Heun functions

Vieira, H.S.; Bezerra, V .B. Quantum Newtonian cosmology and the biconﬂuent Heun functions. J. Math. Phys. 56, 092501 (2015) [arXiv:1502.03071]

2015 arXiv
[65]

Relativistic Landau levels in the rotating cosmic string spacetime

Cunha, M.S.; Muniz, C.R.; Christiansen, H.R.; Bezerra, V .B. Relativistic Landau levels in the rotating cosmic string spacetime. Eur. Phys. J. C. 76, 512 (2016) [arXiv:1606.04481]

2016 arXiv
[66]

Asymptotic Methods for Integrals; Series in Analysis Volume 6; World Scientiﬁc: Hackensack, NJ, USA, 2015

Temme, N.M. Asymptotic Methods for Integrals; Series in Analysis Volume 6; World Scientiﬁc: Hackensack, NJ, USA, 2015

2015
[67]

Quantum states of neutrons in the gravitational ﬁeld and limits for non-Newtonian interaction in the range between 1 micron and 10 microns

Abele, H.; Baessler, S.; Westphal, A. Quantum states of neutrons in the gravitational ﬁeld and limits for non-Newtonian interaction in the range between 1 micron and 10 microns. Lect. Notes Phys. 631, 355 (2003) [arXiv:hep-ph/0301145]

2003 arXiv
[68]

Testing gravity with cold-atom interferometers

Biedermann, G.W.; Wu, X.; Deslauriers, L.; Roy, S.; Mahadeswaraswamy, C.; Kasevich, M.A. Testing gravity with cold-atom interferometers. Phys. Rev. A 91, 033629 (2015) [arXiv:1412.3210]

2015 arXiv
[69]

Constraints on New Gravitylike Forces in the Nanometer Range

Kamiya, Y.; Itagami, K.; Tani, M.; Kim, G.N.; Komamiya, S. Constraints on New Gravitylike Forces in the Nanometer Range. Phys. Rev. Lett. 114, 161101 (2015) [arXiv:1504.02181]

2015 arXiv
[70]

Constraining the range of Yukawa gravity interaction from S2 star orbits

Borka, D.; Jovanovi´ c, P .; Jovanovi´ c, V .B.; Zakharov, A.F. Constraining the range of Yukawa gravity interaction from S2 star orbits. JCAP .11, 050 (2013) [arXiv:1311.1404]

2013 arXiv
[71]

Tests of the Gravitational Inverse-Square Law

Adelberger, E.G.; Heckel, B.R.; Nelson, A.E. Tests of the Gravitational Inverse-Square Law. Ann. Rev. Nucl. Part. Sci. 53, 77 (2003) [arXiv:hep-ph/0307284]

2003 arXiv
[72]

Torsion balance experiments: A low-energy frontier of particle physics

Adelberger, E.G.; Gundlach, J.H.; Heckel, B.R.; Hoedl, S.; Schlamminger, S. Torsion balance experiments: A low-energy frontier of particle physics. Prog. Part. Nucl. Phys. 62, 102 (2009)

2009
[73]

Review of short-range gravity experiments in the LHC era

Murata, J.; Tanaka, S. Review of short-range gravity experiments in the LHC era. Class. Quantum Grav. 32, 033001 (2015) [arXiv:1408.3588]

2015 arXiv
[74]

Physics of white dwarf stars

Koester, D.; Chanmugam, G. Physics of white dwarf stars. Rep. Prog. Phys. 53, 837 (1990)

1990
[75]

The Properties of Matter in White Dwarfs and Neutron Stars

Balberg, S.; Shapiro, S.L. The Properties of Matter in White Dwarfs and Neutron Stars. In Handbook of Elastic Properties of Solids, Liquids, and Gases, Four-Volume Set, 1st ed.; Levy, M., Bass, H., Stern, R., Eds.; Academic Press: Cambridge, MA, USA, 2000

2000
[76]

Mass-Radius Relation for Magnetic White Dwarfs

Suh, I.-S.; Mathews, G.J. Mass-Radius Relation for Magnetic White Dwarfs. ApJ 530 949 (2000) [arXiv:astro-ph/9906239]

2000 arXiv
[77]

Strongly magnetized cold electron degenerate gas: Mass-radius relation of the magnetized white dwarf

Das, U.; Mukhopadhyay, B. Strongly magnetized cold electron degenerate gas: Mass-radius relation of the magnetized white dwarf. Phys. Rev. D 86, 042001 (2012) [arXiv:1204.1262]

2012 arXiv
[78]

The Equation of State of Neutron Star Matter in Strong Magnetic Fields

Broderick, A.; Prakash, M.; Lattimer, J.M. The Equation of State of Neutron Star Matter in Strong Magnetic Fields. Astrophysical J. 537, 351 (2000)

2000
[79]

Role of Landau quantization on the neutron-drip transition in magnetar crusts

Chamel, N.; Stoyanov, Z.K.; Mihailov, L.M.; Mutafchieva, Y.D.; Pavlov, R.L.; Velchev, C.J. Role of Landau quantization on the neutron-drip transition in magnetar crusts. Phys. Rev. C 91, 065801 (2015)

2015
[80]

Landau quantization and neutron emissions by nuclei in the crust of a magnetar

Chamel, N.; Mutafchieva, Y.D.; Stoyanov, Z.K.; Mihailov, L.M.; Pavlov, R.L. Landau quantization and neutron emissions by nuclei in the crust of a magnetar. J. Phys.: Conf. Ser. 724, 012034 (2016) [arXiv:1607.05934]

2016 arXiv
[81]

Neutron star crusts with magnetic ﬁelds

Yakovlev, D.G.; Kaminker, A.D. Neutron star crusts with magnetic ﬁelds. In The Equation of State in Astrophysics ; Chabrier, G., Schatzman, E., Eds.; Cambridge University Press: Cambridge, UK, 1994; pp. 214–238

1994
[82]

Landau levels in a gravitational ﬁeld: The Levi-Civita and Kerr spacetimes case

Hammad, F.; Landry, A. Landau levels in a gravitational ﬁeld: The Levi-Civita and Kerr spacetimes case. Eur. Phys. J. Plus 135, 90 (2020) [arXiv:1910.01899]

2020 arXiv
[83]

A new interior Schwarzschild solution

Florides, P .S. A new interior Schwarzschild solution. Proc. R. Soc. Lond. A 1974, 337, 529

1974
[84]

Superﬂuidity and Superconductivity in Neutron Stars

Haskell, B.; Sedrakian, A. Superﬂuidity and Superconductivity in Neutron Stars. In The Physics and Astrophysics of Neutron Stars. Astrophysics and Space Science Library; Rezzolla, L., Pizzochero, P ., Jones, D., Rea, N., Vidaña, I., Eds.; Springer: Cham, Switzerland, 2018; Volume 457

2018
[85]

and Tüxenb, J

Eibenberger, S.; Gerlich, S.; Arndt, M.; Mayor, M. and Tüxenb, J. Matter-wave interference with particles selected from a molecular library with masses exceeding 10,000 amu. Phys. Chem. Chem. Phys. 15, 14696 (2013) [arXiv:1310.8343]

2013 arXiv
[86]

and Bouwmeester, D

Marshall, W.; Simon, C; Penrose, R. and Bouwmeester, D. Towards quantum superpositions of a mirror. Phys. Rev. Lett.91, 130401 (2003), Erratum: Phys. Rev. Lett. 91, 159903 (2003) [arXiv:quant-ph/0210001]. 37 of 37

2003 arXiv
[87]

and Cirac, J.I

Romero-Isart, O.; Juan, M.L.; Quidant, R. and Cirac, J.I. Toward Quantum Superposition of Living Organisms. New J. Phys. 12, 033015 (2010), [arXiv:0909.1469]

2010 arXiv
[88]

and Zoller, P

Chang, D.E.; Regal, C.A.; Papp, S.B.; Wilson, D.J.; Ye, J.; Painter, O.; Kimble, H.J. and Zoller, P . Cavity opto-mechanics using an optically levitated nanosphere. Proc. Nat. Acad. Sci. U. S. A. 107, 1005 (2010) [arXiv:0909.1548]

2010 arXiv
[89]

Cavity cooling of an optically trapped nanoparticle

Barker, P .F.; Shneider, M.N. Cavity cooling of an optically trapped nanoparticle. Phys. Rev. A 81, 023826 (2010) [arXiv:0910.1221]

2010 arXiv
[90]

and Cirac, J.I

Romero-Isart, O.; Pﬂanzer, A.C.; Blaser, F.; Kaltenbaek, R.; Kiesel, N.; Aspelmeyer, M. and Cirac, J.I. Large Quantum Superpositions and Interference of Massive Nanometer-Sized Objects. Phys. Rev. Lett. 107, 020405 (2011) [arXiv:1103.4081]

2011 arXiv
[91]

and Novotny, L

Gieseler, J.; Deutsch, B.; Quidant, R. and Novotny, L. Sub-kelvin Parametric Feedback Cooling of a Laser-Trapped Nanoparticle. Phys. Rev. Lett. 109, 103603 (2012) [arXiv:1202.6435]

2012 arXiv
[92]

and Aspelmeyer, M

Kiesel, N.; Blaser, F.; Deli´ c, U.; Grass, D.; Kaltenbaek, R. and Aspelmeyer, M. Cavity cooling of an optically levitated submicron particle. Proc. Nat. Acad. Sci. U. S. A. 110, 14180 (2013) [arXiv:1304.6679]

2013 arXiv
[93]

Cavity cooling of free silicon nanoparticles in high vacuum

Asenbaum, P .; Kuhn, S.; Nimmrichter, S.; Sezer, U.; Arndt, M. Cavity cooling of free silicon nanoparticles in high vacuum. Nat. Commun. 4, 2743 (2013) [arXiv:1306.4617]

2013 arXiv
[94]

Near-ﬁeld interferometry of a free-falling nanoparticle from a point-like source

Bateman, J.; Nimmrichter, S.; Hornberger, K.; Ulbricht, H. Near-ﬁeld interferometry of a free-falling nanoparticle from a point-like source. Nat. Commun. 5, 4788 (2014) [arXiv:1312.0500]

2014 arXiv
[95]

Cavity Cooling a Single Charged Levitated Nanosphere

Millen, J.; Fonseca, P .Z.G.; Mavrogordatos, T.; Monteiro, T.S.; Barker, P .F. Cavity Cooling a Single Charged Levitated Nanosphere. Phys. Rev. Lett. 114, 123602 (2015) [arXiv:1407.3595]

2015 arXiv
[96]

On-chip quantum interference of a superconducting microsphere

Pino, H.; Prat-Camps, J.; Sinha, K.; Venkatesh, B.P .; Romero-Isart, O. On-chip quantum interference of a superconducting microsphere. Quantum Sci. Technol. 3, 25001 (2018) [arXiv:1603.01553]

2018 arXiv
[97]

Some Integrals of the products of Laguerre polynomials

Poh-aun, L.; Ong, S.-H.; Srivastava, H.M. Some Integrals of the products of Laguerre polynomials. Int. J. Computer Math. 78, 303 (2000)

2000
[98]

Prudnikov, A.P .; Brychkov, Y.A.; Marichev, O.I.Integrals and Series; Special Functions; Gordon and Breach: New York, NY, USA, 1986; Volume 2

1986
[99]

Generalized Hypergeometric Function; Cambridge University Press: Cambridge, UK, 1966

Slater, L.J. Generalized Hypergeometric Function; Cambridge University Press: Cambridge, UK, 1966

1966

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Quantum states of neutrons in the Earth’s gravitational ﬁeld

Nesvizhevsky, V .V .; Börner, H.G.; Petukhov A.K.; Abele H.; Baeßler, S; Rueß, F.J.; Stöferle T.; Westphal A.; Gagarski, A.M.; Petrov, G.A.; Strelkov, A.V . Quantum states of neutrons in the Earth’s gravitational ﬁeld. Nature415, 297 (2002)

2002

[2] [2]

Measurement of quantum states of neutrons in the Earth's gravitational field

Nesvizhevsky, V .V .; Börner, H.G.; Gagarski, A.M.; Petoukhov, A.K.; Petrov, G.A.; Abele, H.; Baeßler, S.; Divkovic, G.; Rueß, F.J.; Stöferle, Th.; Westphal, A.; Strelkov, A.V .; Protasov, K.V .; Voronin, A.Yu. Measurement of quantum states of neutrons in the Earth’s gravitational ﬁeld. Phys. Rev. D 67, 102002 (2003) [arXiv:hep-ph/0306198]

work page Pith review arXiv 2003

[3] [3]

Study of the neutron quantum states in the gravity field

Nesvizhevsky, V .V .; Petukhov, A.K.; Börner, H.G.; Baranova, T.A.; Gagarski, A.M.; Petrov, G.A.; Protasov, K.V .; Voronin, A.Yu.; Baeßler, S.; Abele, H.; Westphal, A.; Lucovac, L. Study of the neutron quantum states in the gravity ﬁeld. Eur. Phys. J. C 40, 479 (2005) [arXiv:hep-ph/0502081]

work page Pith review arXiv 2005

[4] [4]

Rauch, H.H.; Werner, S.A.Neutron Interferometry, Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglment, 2nd ed.; Oxford University Press: Oxford, UK, 2015

2015

[5] [5]

Spectrometer for new gravitational experiment with UCN

Kulin, G.V .; Frank, A.I.; Goryunov, S.V .; Kustov, D.V .; Geltenbort, P .; Jentschel, M.; Strepetov, A.N.; Bushuev, V .A. Spectrometer for new gravitational experiment with UCN. In Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 792, 38 (2015) [arXiv:1502.03243]

work page Pith review arXiv 2015

[6] [6]

Precision experiments with cold and ultra-cold neutrons

Abele, H. Precision experiments with cold and ultra-cold neutrons. Hyperﬁne Interact 237, 155 (2016)

2016

[7] [7]

Witnessing the quantumness of a system by observing only its classical features

Marletto, C.; Vedral, V . Witnessing the quantumness of a system by observing only its classical features. NPJ. Quantum Information3, 43 (2017) [arXiv:1704.00120]

arXiv 2017

[8] [8]

Gravitationally Induced Entanglement between Two Massive Particles is Sufﬁcient Evidence of Quantum Effects in Gravity

Marletto, C.; Vedral, V . Gravitationally Induced Entanglement between Two Massive Particles is Sufﬁcient Evidence of Quantum Effects in Gravity. Phys. Rev. Lett. 119, 240402 (2017) [arXiv:1707.06036]

arXiv 2017