Pith. sign in

REVIEW 4 major objections 4 minor 33 references

A spatially varying effective mass in a Klein–Gordon field produces an exactly solvable spectrum whose vacuum energy splits into a Landau-like sector that reproduces the standard plate Casimir energy and an additional sector that diverges a

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 00:15 UTC pith:HWY6YQRN

load-bearing objection A careful Casimir calculation in a quadratic mass profile, undone by an unproven and likely incorrect replacement of the transverse density of states. the 4 major comments →

arxiv 2607.15070 v1 pith:HWY6YQRN submitted 2026-07-16 hep-th gr-qcmath-phmath.MPquant-ph

Casimir effect for a massive scalar field confined between parallel plates with a spatially varying effective mass

R. L. Ara\'ujo Xavier , M. H. B. Chaves , E. R. Bezerra de Mello , Herondy Mota This is my paper
classification hep-th gr-qcmath-phmath.MPquant-ph
keywords Casimir effectposition-dependent effective massKlein-Gordon equationLandau levelsvacuum energyzeta-function regularizationparallel platesscalar field
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks what happens to the Casimir energy of a massive real scalar field between two perfectly reflecting parallel plates when the effective mass grows quadratically with the radial distance from the plate axis, m_eff^2 = m^2 + α^2 ρ^2. Solving the Klein–Gordon equation exactly, the authors find that the transverse motion is quantized into equally spaced levels, with the same counting as Landau levels in a magnetic field even though no magnetic field is present. They then regularize the vacuum energy with a generalized zeta function and renormalize it. Their central result is that the renormalized vacuum energy per unit area separates into a Landau-like contribution that smoothly reproduces the standard massive-scalar Casimir energy as α→0, and an additional contribution that diverges as 1/α in that same limit. The paper argues this singular limit is a direct consequence of the quantization condition being valid only for α>0, and that away from this singular point the Landau-like sector dominates.

Core claim

The central result is an exact spectrum ω_σ = √(k_j² + m² + α(2n+|l|+1)) with k_j = jπ/L, giving a renormalized vacuum energy per unit area E_ren/A = −(1/(8π²L³)) ∑_j [I_L(j,m₀,α₀) + I_c(j,m₀,α₀)]. The paper shows that as α₀→0 the Landau-like term I_L reduces exactly to the standard massive-scalar Casimir integral, while I_c diverges as (2/α₀) times that same integral, so the total energy has no smooth α→0 limit. For large α₀ both terms decay exponentially, quenching the Casimir force. The authors trace the singular small-α behavior to the quantization condition holding only for α>0, and numerical results show the Landau-like term dominates everywhere except very close to that singularity.

What carries the argument

The carrying mechanism is the exact normal-mode spectrum of a Klein–Gordon field with quadratic position-dependent mass: α²ρ². The transverse quantum numbers (n, |l|) organize into equally spaced 'Landau-like' ladders with spacing 2α, analogous to Landau levels in a magnetic field but with no magnetic field present. To compute the vacuum energy, the continuum integral over transverse momenta is replaced by a discrete sum over these ladders with a density factor A α/(2π) (Eq. 25). The vacuum energy is evaluated with generalized zeta-function regularization, and the renormalization subtracts the bulk and plate-separation-linear contributions, leaving the boundary-induced integrals I_L and I_c.

Load-bearing premise

The load-bearing premise is the mode-counting prescription in Eq. (25), which assigns a constant α/(2π) density per unit area to every transverse mode even though the spectrum depends on |l|; if that counting is wrong, the singular α→0 limit of the extra Casimir term is an artifact rather than a physical prediction.

What would settle it

Recompute the vacuum energy with the true density of states of the two-dimensional harmonic oscillator spectrum (one state per (j,n,l), not α/(2π) per unit area) and check whether the α→0 divergence of I_c persists. If it vanishes or changes, the paper's singular-limit claim is an artifact of the chosen mode-counting measure; a direct numerical evaluation of the mode sum with the correct multiplicity would settle the question.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • For large values of the dimensionless coupling α₀ = αL², both sectors of the Casimir energy decay exponentially, so the vacuum force between the plates is strongly suppressed once the mass varies rapidly across the gap.
  • In the limit α₀→0, the Landau-like sector reproduces exactly the standard Casimir energy of a massive scalar between parallel plates, so the model contains the known result as its weak-coupling limit.
  • The additional, mass-gradient-induced sector diverges as 1/α₀ as α₀→0, meaning the full spectrum has no smooth α=0 limit; the paper interprets this as a genuine consequence of the quantization condition's restricted domain.
  • Except in a narrow neighborhood of this singular limit, the ratio of total to Landau-like energy approaches unity, so the Landau-like sector dominates the vacuum energy for essentially all nonzero couplings.
  • The model provides a new exactly solvable framework for studying boundary-induced vacuum phenomena in spatially inhomogeneous relativistic scalar systems.

Where Pith is reading between the lines

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

  • The Landau-like structure suggests the same zeta-regularization machinery extends to other position-dependent mass profiles — linear, quartic, or discontinuous — yielding a family of exact Casimir energies with the same two-sector decomposition.
  • The analogy between α and the magnetic scale eB implies the plate force could be tuned continuously from a standard vacuum force to an exponentially suppressed one; engineered effective-mass gradients in semiconductor heterostructures or ultracold-atom platforms might provide a test bed.
  • The singular α→0 limit could be regularized by including the α=0 sector explicitly and matching the two spectra, converting the divergence into a finite matching condition and a well-defined interpolation between homogeneous and inhomogeneous cases.
  • If the mode-counting is correct, the method gives a template for boundary-induced vacuum energies in any confined system whose transverse spectrum is an equally spaced ladder, independent of the physical source of the ladder.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper considers a massive real scalar field in Minkowski spacetime with a position-dependent effective mass m_eff^2 = m^2 + α^2ρ^2, confined between two parallel Dirichlet plates separated by L. The Klein-Gordon equation is solved in cylindrical coordinates, giving transverse modes labeled by (n,l) with energies ω = sqrt(k_j^2 + m^2 + α(2n+|l|+1)), where k_j = jπ/L. The field is quantized and the vacuum energy is computed using generalized zeta-function regularization and a subtraction scheme that removes bulk and L-linear contributions. The renormalized energy splits into a 'Landau-like' term E_L and an additional term E_c. The paper claims that for α0=αL^2→0, E_L reduces to the standard massive-scalar Casimir energy while E_c diverges as 1/α0, and that for large α0 both terms are exponentially suppressed with E_L dominating. The central derivation, however, relies on an unproved replacement of the transverse continuum integral by a discrete sum with constant weight Aα/(2π), which is not justified for the spectrum obtained.

Significance. If the calculation were correct, the paper would provide a new exactly solvable model connecting position-dependent effective masses with boundary-induced vacuum phenomena, and the Landau-like spectrum would be a noteworthy structural analogy. The exact solution of the Klein-Gordon equation in Sec. II and the nonrelativistic harmonic-oscillator limit are standard and appear sound. However, the main physical results—the separation into E_L and E_c, the exponential suppression, and especially the α0→0 singularity—all depend on the density-of-states replacement in Eq. (25), which is never derived and is inconsistent with the spectrum in Eq. (17). Because this is load-bearing for the paper's central claim, the significance of the results is not established by the manuscript as written.

major comments (4)
  1. [§III.A, Eq. (25)] The replacement V/(2π)^3∫d^3k → Aα/(2π)Σ_jΣ_nΣ_l is the load-bearing step of the paper and is never derived. For the spectrum (17) the energy depends explicitly on |l|, so l is not a degeneracy label; the shell q=2n+|l| has multiplicity q+1 (including both signs of l). In the Landau problem the constant degeneracy Aα/(2π) is valid only because the summed quantum number does not enter the energy. Here the assumed constant weight contradicts the spectrum itself, and the correct 2D-oscillator density of states is not a constant per mode. All subsequent results—Eqs. (34), (37), (38), and the α0→0 divergence in Eq. (43)—are built on this measure. An independent derivation of the transverse mode density is required.
  2. [Eqs. (27) and (29)] After separating l=0 and l≥1, the second sum is written with n=1,2,... . This omits the modes n=0, |l|≥1 that are present in the original sum (20). Moreover, the later evaluation in Eq. (36) effectively uses a sum over n≥0 and l≥1 (the factor (e^{ατ^2}−1) corresponds to summing l≥1 with n≥0), so the chain (27)→(29)→(36) is internally inconsistent. Correcting the lower limit changes E_c and hence the claimed singular behavior.
  3. [Eq. (43)] The stated limit I_c ≈ (2/α0) m0^2K2(2jm0)/j^2 does not follow from Eq. (37). For w=α0τ^2, 1/[sinh w (e^w−1)] ∼ w^{−2}, so I_c ∼ (2/α0)∫ dτ τ^{−7} e^{−m0^2τ^2−j^2/τ^2}. The leading term is proportional to (m0^2/j^2)^{3/2}K3(2jm0), not (2/α0) times the I_L limit. The two expressions differ even in the massless limit (∼1/(α0 j^6) vs ∼1/(α0 j^4)). Thus the claimed divergence structure and the ratio shown in Fig. 3 are not reliable even if Eq. (25) is accepted.
  4. [Eq. (39)] The large-α0 asymptotic uses K_a(x) ∼ √(π/2)e^{−x}; the correct leading asymptotics is K_a(x) ∼ √(π/(2x))e^{−x}. Consequently the exponent in the final line should be e^{−2j√(fα0)}, not e^{−2jfα0}. The qualitative conclusion of exponential suppression survives, but the quantitative expression in Eq. (39) is incorrect.
minor comments (4)
  1. [Eq. (37)] After the change of variables τ→Lτ, the denominator is written as sinh(ατ^2); for consistency with Eq. (34) it should be sinh(α0τ^2).
  2. [Eqs. (40)–(41)] The notation is confusing: g(w) is said to be defined by the denominators but is then listed as sinh(w) and sinh(w)(e^w−1), while the substitution into Eq. (40) requires the reciprocal. The parameter f is also overloaded with the f used in Eq. (39).
  3. [Introduction and Sec. III.A] The sentence that the Landau-like term differs from the conventional Landau spectrum 'only by an overall factor of two associated with the two degrees of freedom of a complex scalar field' is unclear for a real scalar field and should be justified or rephrased.
  4. [Sec. II] The statement that the α→0 limit cannot be obtained directly from the exact spectrum is correct, but the paper later attempts expansions around α0=0; the relation between these two points should be clarified.

Circularity Check

0 steps flagged

No significant circularity: the central vacuum-energy calculation is a forward derivation from the model action; the main caveat (Eq. 25) is an unproven density-of-states assumption, not a circular input.

full rationale

The derivation chain is self-contained: the spectrum (11)/(17) follows from solving the Klein-Gordon equation with m_eff^2 = m^2 + α^2ρ^2; the vacuum energy (26) is the direct half-sum of these mode frequencies; zeta regularization and the renormalization subtractions in Eqs. (32)-(37) are standard prescriptions imposed by requiring the boundary-induced energy to vanish as L→∞ and not grow with α. No parameter is fitted to the predicted Casimir energy, and the α0→0 recovery of the standard massive parallel-plate result (Eq. (42), compared with independent Refs. [32] and [33]) is an external consistency check, not an input. The self-citations ([31], [33]) are used for routine geometric-series/Hurwitz-zeta manipulations and for quoting the known Casimir result; they are not load-bearing uniqueness claims and do not import the present paper's conclusions. The serious caveat is Eq. (25): the replacement V/(2π)^3 ∫ d^3k → Aα/(2π) Σ_j Σ_n Σ_l is asserted without proof and conflicts with the l-dependence of (17), so the claimed 1/α0 divergence of E_c (Eq. (43)) is an artifact of that measure rather than an inevitable consequence of the exact spectrum. That is a derivation gap/correctness risk, not circular reasoning, because the target result is not fed back as a premise or fitted parameter. Hence score 2 (minor self-citation, no substantive circularity).

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 0 invented entities

The model postulates a position-dependent effective mass m_eff^2 = m^2 + alpha^2 rho^2, which is a background profile rather than a new particle or force. The calculation rests on standard special functions and regularization methods, but it also relies on two ad hoc choices: the mode-sum replacement (25) and the subtraction scheme. The former is the most consequential, as it determines the structure of the vacuum energy and the alpha->0 divergence.

free parameters (2)
  • alpha
    Coupling constant in m_eff^2 = m^2 + alpha^2 rho^2. It is a model parameter, not fitted to data, but the central results depend on it; the alpha->0 limit is singular by construction of the mode sum.
  • m
    Mass of the scalar field. Enters results through the dimensionless combination m0 = mL. Not fitted; a standard model parameter.
axioms (7)
  • standard math The Kummer/confluent hypergeometric equation provides the complete set of normal modes when the first argument is a non-positive integer.
    Used in Section II to derive the spectrum (11). Relies on textbook properties of the confluent hypergeometric function.
  • standard math Generalized zeta-function regularization and analytic continuation to s = -1/2 give a finite vacuum energy after subtraction of bulk and L-proportional terms.
    Introduced in Section III.A and used throughout; a standard QFT regularization technique.
  • standard math Poisson resummation formula (31) is valid for the sum over j.
    Applied in Eq. (31) to the sum over the plate-mode index j. Standard identity.
  • domain assumption The spacetime is Minkowski in cylindrical coordinates, ds^2 = dt^2 - d rho^2 - rho^2 d phi^2 - dz^2.
    Background geometry stated in Eq. (3). Natural for the axially-symmetric mass profile.
  • domain assumption Two perfectly reflecting parallel plates impose Dirichlet boundary conditions phi(0) = phi(L) = 0.
    Boundary conditions given in Eq. (16); defines the confinement between plates.
  • ad hoc to paper The replacement (25), V/(2 pi)^3 integral d^3k -> A alpha/(2 pi) sum_j sum_n sum_l, converts the continuum transverse integral into a discrete sum with constant weight alpha/(2 pi).
    Stated without derivation in Section III.A. This is the load-bearing assumption that produces the 'Landau-like' form. It mimics the Landau-level degeneracy but the spectrum (17) depends on |l|, so the analogy is not justified.
  • ad hoc to paper Renormalization subtraction scheme (33): remove 'bulk' contributions and terms growing linearly in L from the zeta-regulated energy.
    The subtraction is designed to enforce E_ren -> 0 as L -> infinity and to avoid unbounded growth in alpha. It is physically plausible but not uniquely fixed by a first-principles condition.

pith-pipeline@v1.3.0-alltime-deepseek · 11554 in / 22765 out tokens · 230054 ms · 2026-08-02T00:15:20.193077+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Casimir effect for a massive scalar field confined between parallel plates with a spatially varying effective mass." pith.science (2026). https://pith.science/paper/HWY6YQRN

@misc{pith2026260715070,
  author       = {Pith},
  title        = {Pith review of: Casimir effect for a massive scalar field confined between parallel plates with a spatially varying effective mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HWY6YQRN}},
  note         = {Machine review of arXiv:2607.15070}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

We investigate the Casimir effect for a massive real scalar field confined between two perfectly reflecting parallel plates in the presence of a position-dependent effective mass, a mechanism for coupling a scalar background to a scalar field. Exact normal modes are obtained by solving the corresponding Klein-Gordon equation, leading to a transverse energy spectrum that exhibits a characteristic Landau-like structure despite the absence of an external magnetic field. Upon quantization of the field, the vacuum energy is evaluated by means of generalized zeta-function regularization together with an appropriate renormalization procedure. The renormalized vacuum energy naturally separates into a Landau-like contribution and an additional term induced by the spatial dependence of the effective mass. We show analytically and numerically that both contributions are exponentially suppressed in the strong-coupling regime. In the opposite limit, the Landau-like contribution smoothly reproduces the standard vacuum energy for a massive scalar field confined between parallel plates, whereas the additional contribution becomes singular owing to the restricted domain of validity of the exact spectrum. Except in the vicinity of this singular limit, the vacuum energy is shown to be dominated by the Landau-like sector. Our results establish a direct connection between position-dependent effective masses and boundary-induced quantum vacuum phenomena, providing a new exactly solvable framework for the investigation of Casimir effects in spatially inhomogeneous relativistic systems.

Figures

Figures reproduced from arXiv: 2607.15070 by E. R. Bezerra de Mello, Herondy Mota, M. H. B. Chaves, R. L. Ara\'ujo Xavier.

Figure 1
Figure 1. Figure 1: Schematic illustration of two perfectly reflecting parallel plates of area [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Renormalized vacuum energy contributions as functions of the dimensionless parameter [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ratio between the total renormalized vacuum energy, [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 1 canonical work pages

  1. [1]

    H. B. Casimir,On the attraction between two perfectly conducting plates, inProc. Kon. Ned. Akad. Wet., vol. 51, p. 793, 1948

    1948

  2. [2]

    M. J. Sparnaay,Measurements of attractive forces between flat plates,Physica24(1958) 751–764

    1958

  3. [3]

    S. K. Lamoreaux,Demonstration of the casimir force in the 0.6 to6µmrange,Phys. Rev. Lett.78(Jan, 1997) 5–8

    1997

  4. [4]

    Lamoreaux,Erratum: Demonstration of the casimir force in the 0.6 to 6µm range [phys

    S. Lamoreaux,Erratum: Demonstration of the casimir force in the 0.6 to 6µm range [phys. rev. lett. 78, 5 (1997)],Physical Review Letters81(1998) 5475

    1997

  5. [5]

    Mohideen and A

    U. Mohideen and A. Roy,Precision measurement of the Casimir force from 0.1 to 0.9 micrometers,Phys. Rev. Lett.81(1998) 4549–4552, [physics/9805038]

    Pith/arXiv arXiv 1998

  6. [6]

    Roy and U

    A. Roy and U. Mohideen,Demonstration of the nontrivial boundary dependence of the casimir force,Phys. Rev. Lett.82(May, 1999) 4380–4383

    1999

  7. [7]

    Bressi, G

    G. Bressi, G. Carugno, A. Galvani, R. Onofrio, G. Ruoso and F. Veronese,Experimental studies of macroscopic forces in the micrometre range,Classical and Quantum Gravity18(2001) 3943–3961

    2001

  8. [8]

    Bressi, G

    G. Bressi, G. Carugno, R. Onofrio and G. Ruoso,Measurement of the Casimir force between parallel metallic surfaces,Phys. Rev. Lett.88(2002) 041804, [quant-ph/0203002]

    Pith/arXiv arXiv 2002

  9. [9]

    Bordag, G

    M. Bordag, G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko,Advances in the Casimir effect, vol. 145. OUP Oxford, 2009

    2009

  10. [10]

    K. A. Milton,The Casimir effect: physical manifestations of zero-point energy. World Scientific, 2001

    2001

  11. [11]

    Mostepanenko, N

    V. Mostepanenko, N. Trunov and R. Znajek,The Casimir Effect and Its Applications. Oxford science publica- tions. Clarendon Press, 1997

    1997

  12. [12]

    H. G. Dosch, J. H. D. Jensen and V. F. M¨ uller,Einige bemerkungen zum klein ’schen paradoxon,Physica Norvegica 5(1971) 151–164

    1971

  13. [13]

    G. Soff, B. M¨ uller, J. Rafelski and W. Greiner,Solution of the dirac equation for scalar potentials and its implications in atomic physics,Zeitschrift f¨ ur Naturforschung A28(1973) 1389–1396

    1973

  14. [14]

    Bordag and N

    M. Bordag and N. Khusnutdinov,A remark on bound states in conical spacetime,Classical and Quantum Gravity 13(1996) L41–L44

    1996

  15. [15]

    A. L. Cavalcanti de Oliveira and E. R. Bezerra de Mello,Exact solutions of the Klein-Gordon equation in the presence of a dyon, magnetic flux and scalar potential in the specetime of gravitational defects,Class. Quant. Grav.23(2006) 5249–5264, [hep-th/0603036]

    Pith/arXiv arXiv 2006

  16. [16]

    Bakke and C

    K. Bakke and C. Furtado,On the Klein–Gordon oscillator subject to a Coulomb-type potential,Annals Phys. 355(2015) 48–54, [1411.6988]

    Pith/arXiv arXiv 2015

  17. [17]

    R. L. L. Vit´ oria, C. Furtado and K. Bakke,On a relativistic particle and a relativistic position-dependent mass particle subject to the Klein–Gordon oscillator and the Coulomb potential,Annals Phys.370(2016) 128–136, [1511.05072]

    Pith/arXiv arXiv 2016

  18. [18]

    Vit´ oria and K

    R. Vit´ oria and K. Bakke,Relativistic quantum effects of confining potentials on the klein-gordon oscillator,The European Physical Journal Plus131(2016) 36

    2016

  19. [19]

    R. L. L. Vit´ oria and K. Bakke,Torsion effects on a relativistic position-dependent...,Gen. Rel. Grav.48(2016) 161, [1611.06111]

    Pith/arXiv arXiv 2016

  20. [20]

    von Roos,Position-dependent effective masses in semiconductor theory,Phys

    O. von Roos,Position-dependent effective masses in semiconductor theory,Phys. Rev. B27(1983) 7547–7552

    1983

  21. [21]

    L´ evy-Leblond,Position-dependent effective mass and galilean invariance,Phys

    J.-M. L´ evy-Leblond,Position-dependent effective mass and galilean invariance,Phys. Rev. A52(Sep, 1995) 1845–1849

    1995

  22. [22]

    Bastard,Wave Mechanics Applied to Semiconductor Heterostructures

    G. Bastard,Wave Mechanics Applied to Semiconductor Heterostructures. John Wiley & Sons, New York, 1988

    1988

  23. [23]

    B. D. Serot and J. D. Walecka,Recent progress in quantum hadrodynamics,Int. J. Mod. Phys. E6(1997) 515–631, [nucl-th/9701058]

    Pith/arXiv arXiv 1997

  24. [24]

    B. D. Serot and J. D. Walecka,The Relativistic Nuclear Many Body Problem,Adv. Nucl. Phys.16(1986) 1–327

    1986

  25. [25]

    M. G. Silveirinha and N. Engheta,Transformation electronics: Tailoring the effective mass of electrons,Phys. Rev. B86(Oct, 2012) 161104(R)

    2012

  26. [26]

    Greiner,Relativistic Quantum Mechanics: Wave Equations

    W. Greiner,Relativistic Quantum Mechanics: Wave Equations. Springer, Berlin, Heidelberg, 3 ed., 2000, 10.1007/978-3-662-04275-5

    work page doi:10.1007/978-3-662-04275-5 2000

  27. [27]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55 ofNational Bureau of Standards Applied Mathematics Series. Dover Publications, New York, 1965

    1965

  28. [28]

    Elizalde, S

    E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko and S. Zerbini,Zeta regularization techniques with applications. World Scientific Publishing, Singapore, 1994, 10.1142/2065

    doi:10.1142/2065 1994

  29. [29]

    Elizalde,Ten Physical Applications of Spectral Zeta Functions, vol

    E. Elizalde,Ten Physical Applications of Spectral Zeta Functions, vol. 35 ofLecture Notes in Physics Monographs. Springer-Verlag, Berlin, Heidelberg, 1995, 10.1007/978-3-540-44757-3

    doi:10.1007/978-3-540-44757-3 1995

  30. [30]

    Kirsten,Basic zeta functions and some applications in physics,MSRI Publ.57(2010) 101–143, [1005.2389]

    K. Kirsten,Basic zeta functions and some applications in physics,MSRI Publ.57(2010) 101–143, [1005.2389]. 13

    Pith/arXiv arXiv 2010

  31. [31]

    A. J. D. F. Junior, A. Erdas and H. F. Santana Mota,Vacuum energy and topological mass from a constant mag- netic field and boundary conditions in coupled scalar field theories,Phys. Rev. D112(2025) 125026, [2508.15121]

    arXiv 2025

  32. [32]

    M. V. Cougo-Pinto, C. Farina, M. R. Negrao and A. C. Tort,Bosonic Casimir effect in an external magnetic field,J. Phys. A32(1999) 4457–4462, [hep-th/9809214]

    Pith/arXiv arXiv 1999

  33. [33]

    M. B. Cruz, E. R. Bezerra de Mello and H. F. Santana Mota,Casimir energy and topological mass for a massive scalar field with Lorentz violation,Phys. Rev. D102(2020) 045006, [2005.09513]

    Pith/arXiv arXiv 2020