Antiadiabatic Phonons and Superconductivity in Eliashberg-McMillan Theory

M.V. Sadovskii

REVIEW 3 major objections 5 minor 24 references

Antiadiabatic Phonons and Superconductivity in Eliashberg-McMillan Theory

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

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Pith's one-line read The paper claims that when phonon frequencies exceed the Fermi energy, mass renormalization vanishes but the superconducting transition temperature is still set by the Eliashberg–McMillan coupling λ.

desk verdict A useful, partially novel note on antiadiabatic phonons, with a definitional ambiguity in λ that should be fixed before publication. read the letter →

arxiv 1908.00718 v1 pith:OG4PNFQ5 submitted 2019-08-02 cond-mat.supr-con

M.V. Sadovskii This is my paper · ORCID

classification cond-mat.supr-con

keywords Eliashberg–McMillantheoryelectron–phononinteractionantiadiabaticphononsnonadiabaticsuperconductivitymassrenormalizationCoulombpseudopotentialTolmachevlogarithmcriticaltemperature

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

The paper claims that Eliashberg–McMillan theory of superconductivity remains valid even when phonon frequencies exceed the Fermi energy, a regime usually thought to break its adiabatic foundations. It shows that two different coupling constants control normal and superconducting properties: mass renormalization is set by a bandwidth-corrected coupling $\tilde{\lambda}$ that shrinks to $\lambda D/\Omega_0$ in the antiadiabatic limit, while pairing and $T_c$ remain set by the standard coupling $\lambda$. In that limit the new perturbation parameter is $\lambda D/\Omega_0 \ll 1$, so vertex corrections are suppressed and the theory acquires an anti-Migdal character. The paper derives a general $T_c$ formula for discrete optical phonons and shows that antiadiabatic phonons drop out of Tolmachev's logarithm in the Coulomb pseudopotential, leaving adiabatic phonons to screen the repulsion.

What carries the argument

The central objects are the two frequency-dependent coupling constants derived from the finite-bandwidth self-energy: the pairing constant $\lambda = 2\int d\omega\,\alpha^2(\omega)F(\omega)/\omega$ and the mass-renormalization constant $\tilde{\lambda} = 2\int d\omega\,\alpha^2(\omega)F(\omega)\,D/[\omega(\omega+D)]$. In the antiadiabatic limit they are related by $\lambda_D = \lambda D/\Omega_0$, which is the small parameter that suppresses vertex corrections. The argument is carried by Eq. (42) for $T_c$ and Eq. (57) for $\mu^\star$, both built from the same weighted logarithmic frequency $\langle\Omega\rangle = \prod_i (D/(1+D/\Omega_i))^{\lambda_i/\lambda}$ that replaces the phonon cutoff by the Fermi energy in the Cooper channel.

What would settle it

Compute the normal-state quasiparticle residue and $T_c$ from the full Eliashberg equations on a realistic band structure with $\Omega_0 > E_F$ (for example a non-half-filled or strongly dispersive density of states); if the mass renormalization stays of order $\lambda$ rather than $\lambda D/\Omega_0$, or if the $T_c$ prefactor tracks $\Omega_0$ instead of $D$, the paper's central claim collapses.

Watch

Extended reading notes

Core claim

Within a half-filled two-dimensional band of half-width $D = E_F$ and constant density of states, the paper establishes that a high-frequency optical phonon ($\Omega_0 \gg E_F$) does not destroy the Eliashberg–McMillan description but reorders it. The electron mass renormalization is governed by $\tilde{\lambda} = \sum_i \lambda_i D/(\Omega_i + D)$, which in the antiadiabatic limit reduces to the small constant $\lambda_D = \lambda D/\Omega_0$, so the quasiparticle residue stays close to unity. The superconducting transition temperature, however, is still controlled by the ordinary Eliashberg–McMillan constant $\lambda$, with $T_c \sim \prod_i (D/(1+D/\Omega_i))^{\lambda_i/\lambda} \exp(-(1+\tilde{\lambda})/\lambda)$. The same split appears in the Coulomb repulsion: $\mu^\star = \mu/(1 + \mu \ln \prod_i (1 + D/\Omega_i)^{\lambda_i/\lambda})$, so only phonons with $\Omega_i \lesssim D$ contribute to Tolmachev's logarithm. The result is an anti-Migdal regime in which the electron–phonon coupling looks weak for self-energy and vertex corrections but remains effective for pairing.

Load-bearing premise

The cancellation that produces the small parameter $\lambda D/\Omega_0$ and the Fermi-energy cutoff relies on a half-filled two-dimensional band of half-width $D=E_F$ with a constant density of states; away from this toy band the quantitative conclusions are not established.

Editorial extensions

If this is right

In the antiadiabatic limit the Cooper-channel cutoff is the Fermi energy (band half-width) rather than the phonon frequency, so high-frequency phonons set the prefactor of $T_c$ only through $D$.
Mass renormalization in this limit is determined by $\lambda_D = \lambda D/\Omega_0 \ll \lambda$, so the quasiparticle residue remains near one even when the pairing coupling is not small.
Vertex corrections are suppressed by the same small parameter, extending the validity of Eliashberg–McMillan theory into the strongly nonadiabatic regime (an anti-Migdal theorem).
Antiadiabatic phonons do not contribute to Tolmachev's logarithm; the value of $\mu^\star$ is fixed by adiabatic phonons alone, which weakens the Coulomb suppression less than a single antiadiabatic phonon model would suggest.
In a mixed spectrum, Eq. (42) interpolates smoothly between adiabatic and antiadiabatic behavior, providing a single formula for $T_c$ across the crossover.

Reading between the lines

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

If the anti-Migdal picture holds, materials with very large bare electron–phonon couplings should show small normal-state mass enhancement but substantial pairing, a laboratory signature that could distinguish this regime from polaronic physics.
The same bandwidth-corrected coupling $\tilde{\lambda}$ could be tested in systems where phonon frequency is tuned across the Fermi energy, such as pressurized hydrides or interface superconductors, by comparing normal-state specific heat with $T_c$.
Nothing in the derivation fixes the number of phonon branches, so the discrete-mode formula should extend to a continuous $\alpha^2 F(\omega)$; evaluating Eq. (47) on realistic spectra would be a quantitative check.

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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

Colleague, here's my take on 1908.00718. The core message—that antiadiabatic phonons do not simply vanish from pairing but contribute with a Fermi-energy cutoff, while mass renormalization is controlled by a smaller λ̃—is physically sensible, and the algebra in Sections 4–6 checks out. The genuinely new items are the multi-phonon Tc formula (42) and the product form of the Tolmachev logarithm (57); the rest is a clear restatement of the author's earlier single-phonon results.

What it does well: the distinction between the pairing constant λ and the mass-renormalization constant λ̃ is made concrete, and the finite-band cutoff is handled honestly in the Eliashberg equations. The μ* analysis is self-consistent and yields a testable structure.

The soft spots are real. First, the stress-test concern is legitimate: Eq. (13) is called the most general pairing λ, but in the finite-band model with Ω0>D that expression is identically zero, which would contradict the later use of λ=2α²(Ω0)/Ω0 in Eq. (27) and Eq. (42). The resolution is that Eq. (13) describes a real scattering process with a final state at energy Ω0, not the virtual pairing kernel; but the text does not say so. This is a confusing overgeneralization and should be fixed.

Second, the anti-Migdal vertex-correction suppression is asserted via Ref. [22] rather than demonstrated in this model. That is acceptable in a short paper, but it should be flagged as an assumption or cited more carefully.

Third, the constant-DOS, half-filled band is a toy. The authors acknowledge it, but the conclusions about the antiadiabatic small parameter λD/Ω0 are explicitly derived in that toy, and the paper does not discuss how DOS variations change the cancellation. That limits the scope, though not fatally.

The reader's report gives this a conditional, which matches my view. The central claim holds up within the stated model. The paper deserves a serious referee, but the referee should ask for a reconciliation of Eq. (13) with the finite-band definition and a clearer separation of new content from the review of Refs. [14,15]. I would accept it for peer review after revision.

Referee Report

3 major / 5 minor

Summary. The paper studies electron-phonon superconductivity in the antiadiabatic regime, where the characteristic phonon frequency Ω0 is comparable to or larger than the Fermi energy EF, within Eliashberg–McMillan theory. It introduces a generalized mass-renormalization coupling \tildeλ that accounts for the finite conduction bandwidth and reduces to the standard λ in the adiabatic limit and to λ_D ≈ λ D/Ω0 in the strongly antiadiabatic limit. The paper's main claims are that in the antiadiabatic limit mass renormalization becomes irrelevant because \tildeλ is small, while the superconducting transition temperature is still controlled by the standard Eliashberg–McMillan constant λ, with the band half-width D serving as the Cooper-channel cutoff. For a discrete set of Einstein phonon modes it derives a multi-phonon Tc formula, Eq. (42), and a Coulomb pseudopotential μ* formula, Eq. (57), and concludes that antiadiabatic phonons do not contribute to Tolmachev's logarithm.

Significance. If correct, the paper would establish an 'anti-Migdal' regime in which Eliashberg–McMillan theory remains applicable despite Ω0/EF being large, because the effective coupling for vertex corrections and mass renormalization is λD/Ω0 while the pairing interaction still scales with λ. The paper is self-contained in its finite-band Eliashberg treatment: the derivative leading to \tildeλ in Eq. (22) is correctly derived, and the multi-phonon formulas (42) and (57) are explicit and testable. The significance is tempered by the restrictive model assumptions (constant density of states, half-filled band, D = EF) and by a serious internal inconsistency in the definition of the pairing constant λ, which is described below.

major comments (3)

[Sec. 3, Eq. (13); Sec. 4, Eq. (27); Sec. 5, Eq. (42)] The manuscript uses two mutually inconsistent definitions of the pairing constant λ. Equation (13), which follows from the expression called 'the most general' in Eq. (11), contains δ(ε_{p+q} − Ω0). In the finite-band, constant-DOS, half-filled model of Sections 4 and 5 (D = EF), ε_{p+q} is bounded by D, so for an antiadiabatic phonon with Ω0 > D this delta has no support and Eq. (13) yields λ = 0. If that were the pairing constant, antiadiabatic phonons would not contribute to Tc at all, contradicting the abstract and Eq. (42). In contrast, Section 4 defines λ = 2α2(Ω0)/Ω0 (Eq. (27)), and Section 5 uses this λ in the gap equation (36) and in the final Tc formula (42), obtaining a cutoff D. The paper never reconciles these two usages. The central claim that Tc in the antiadiabatic limit is still determined by the Eliashberg–McMillan coupling constant therefore rests on an unstated choice of definition. The authors should either drop the claim that Eq. (11)/(13) is the general pairing constant, or justify why the on-shell delta in Eq. (13) does not apply to the virtual intermediate states in the pairing equation.
[Sec. 5, Eqs. (36)–(42)] The derivation of the central Tc formula is not demonstrated. The text passes from Eq. (36), or equivalently Eq. (41), to Eq. (42) with the statement 'This equation is easily solved', but the solution requires approximating the integral ∫_0^D dε′/[ε′(ε′+Ω_i)] th(ε′/2T) and, more importantly, Eq. (36) itself is obtained from Eq. (33) by assuming a constant gap Δ(ε′) = Δ in the integrand. For the kernel 1/(ε′(ε′+Ω_i)), the actual eigenfunction of the linearized Eliashberg equation is generally not constant, and the error introduced by this ansatz is not quantified. Since Eq. (42) is the main quantitative result and is also used to define ⟨Ω⟩ and μ* in Section 6, the authors should provide the intermediate steps and justify the constant-gap assumption, or explicitly state that Eq. (42) is a variational estimate.
[Sec. 4, 'anti-Migdal theorem'] The 'anti-Migdal theorem' claim is asserted rather than derived. The paper states that in the antiadiabatic limit all vertex corrections are small because λ_D = λ D/Ω0 ≪ 1, citing Ref. [22]. However, the standard Migdal parameter λΩ0/EF is large in this limit, and the paper does not show, within the finite-band model used here, how the vertex corrections are suppressed by λD/Ω0 rather than enhanced by λΩ0/D. Since the validity of the Eliashberg approach in the antiadiabatic regime is a central claim, this point requires at least a clear argument, or a more explicit reference to a direct calculation in the same model, before the 'anti-Migdal' conclusion is accepted.

minor comments (5)

[Title and Section 1] The title contains a typo ('Eliashber g – McMillan'), and the first paragraph repeats 'is is'.
[Eqs. (26)–(28)] The notation for the antiadiabatic coupling constant λ_D is easily confused with the product λ·D; a distinct symbol or an explicit definition would improve readability.
[Section 5, first paragraph] 'In difference with the standard approach' should be 'In contrast to the standard approach'.
[References] Reference [21] appears to merge two separate publications (a JETP article and a Phys. Rev. B article); please split and correct.
[Section 5, Eq. (42) discussion] The paper does not specify the limits of validity of the approximations leading to the product formula (42) in the intermediate regime where Ω_i ∼ D; a brief statement would help the reader assess the interpolation claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central Tc and μ* expressions are derived in-paper from Eliashberg equations; self-citations are not load-bearing.

full rationale

The derivation chain is self-contained. The generalized coupling constant λ̃ (Eq. 23), the antiadiabatic coupling λ_D (Eq. 24), the linearized Eliashberg equations with finite bandwidth (Eqs. 32-33), the Tc formula (Eq. 42), the average logarithmic frequency (Eqs. 46-47), and the Coulomb pseudopotential (Eq. 57) are all obtained from the stated model assumptions (constant DOS, half-filled band, D=EF, discrete Einstein phonons) by explicit calculation. No quantity is fitted to Tc or to μ*, and no output is used as an input under another name. The references to the author's own Refs. [14,15] are historical pointers to previously published estimates; the present paper rederives the relevant results rather than importing them as unverified premises, so the self-citations are not load-bearing. The reviewer-identified tension between the delta-function pairing constant of Eq. (13) and the λ of Eq. (27) is a definitional inconsistency (whether λ should be zero for Ω0>D in a finite band), not a circular reduction: the paper does not define its prediction in terms of itself, it simply uses two λ conventions without explicitly reconciling them. Under the stated convention, Eq. (42) follows from Eq. (36) by standard BCS-type solution. Thus there is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results rest on the standard Eliashberg-McMillan equations plus a set of modeling assumptions: a constant-DOS half-filled band, a generalized momentum average in the Eliashberg function, smallness of vertex corrections taken from Ref. [22], and two-step models for the gap and Coulomb kernel. No free parameters are fitted to data.

assumptions (5)

domain assumption Standard Eliashberg-McMillan theory and Migdal's theorem are valid in the adiabatic regime; the paper restricts to λ < 1 to avoid polaronic vertex corrections.
Section 1 states the adiabatic basis and cites Ref. [8] for the restriction to weak coupling.
domain assumption In the non-adiabatic case the Eliashberg function is obtained by replacing δ(ε_p') with δ(ε_p' − Ω_{p−p'}) in the momentum average (Eq. (4)).
Section 2 presents Eq. (4) as the direct generalization for Ω0 ~ EF, without deriving it from a controlled microscopic expansion.
ad hoc to paper The electronic band has constant density of states over width 2D and is half-filled, so D = EF.
Section 4 introduces 'for simplicity we assume the case of half-filled band'; Section 5 repeats it. The explicit small parameter λD/Ω0 and the cutoff at D follow from this density of states.
ad hoc to paper Vertex corrections are negligible in the antiadiabatic limit because λD/Ω0 is small.
Section 4 asserts all vertex corrections are small and cites Ref. [22], but no vertex diagram is evaluated in this paper.
ad hoc to paper The linearized gap equation can be analyzed with a constant gap Δ(ε') and, for the Coulomb part, a two-step kernel with cutoff ⟨Ω⟩.
Section 5 passes from Eq. (33) to Eq. (36) by taking ε → 0 and effectively assuming Δ(ε') = Δ; Section 6 models the kernel as step functions in Eq. (49).

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Cite this review

Pith. "Pith review of Antiadiabatic Phonons and Superconductivity in Eliashberg-McMillan Theory." pith.science (2026). https://pith.science/paper/OG4PNFQ5

@misc{pith2026190800718,
  author       = {Pith},
  title        = {Pith review of: Antiadiabatic Phonons and Superconductivity in Eliashberg-McMillan Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OG4PNFQ5}},
  note         = {Machine review of arXiv:1908.00718}
}

abstract

The standard Eliashberg - McMillan theory of superconductivity is essentially based on the adiabatic approximation. Here we present some simple estimates of electron - phonon interaction within Eliashberg - McMillan approach in non - adiabatic and even antiadiabatic situation, when characteristic phonon frequency $\Omega_0$ becomes large enough, i.e. comparable or exceeding the Fermi energy $E_F$. We discuss the general definition of Eliashberg - McMillan (pairing) electron - phonon coupling constant $\lambda$, taking into account the finite value of phonon frequencies. We show that the mass renormalization of electrons is in general determined by different coupling constant $\tilde\lambda$, which takes into account the finite width of conduction band, and describes the smooth transition from the adiabatic regime to the region of strong nonadiabaticity. In antiadiabatic limit, when $\Omega_0\gg E_F$, the new small parameter of perturbation theory is $\lambda\frac{E_F}{\Omega_0}\sim\lambda\frac{D}{\Omega_0}\ll 1$ ($D$ is conduction band half -- width), and corrections to electronic spectrum (mass renormalization) become irrelevant. However, the temperature of superconducting transition $T_c$ in antiadiabatic limit is still determined by Eliashberg - McMillan coupling constant $\lambda$. We consider in detail the model with discrete set of (optical) phonon frequencies. A general expression for superconducting transition temperature $T_c$ is derived, which is valid in situation, when one (or several) of such phonons becomes antiadiabatic. We also analyze the contribution of such phonons into the Coulomb pseudopotential $\mu^{\star}$ and show, that antiadiabatic phonons do not contribute to Tolmachev's logarithm and its value is determined by partial contributions from adiabatic phonons only.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[22]

Ikeda, A

M.A. Ikeda, A. Ogasawara, M. Sugihara. Phys. Lett. A 170, 319 (1992)

work page 1992
[1]

Schrieﬀer

J.R. Schrieﬀer. Theory of Superconductivity, W A Ben- jamin, NY, 1964

work page 1964
[2]

Superconductivity

D.J. Scalapino. In “Superconductivity”, p. 449, Ed. by R.D. Parks, Marcel Dekker, NY, 1969

work page 1969
[3]

White, T.H

R.M. White, T.H. Geballe. Long Range Order in Solids. Academic Press, New York, San Francisco, London, 1979

work page 1979
[4]

Vonsovsky, Yu.A

S.V. Vonsovsky, Yu.A. Izyumov, E.Z. Kurmaev. Super- conductivity of Transition metals, Their Alloys and Com- pounds, Springer, Berlin – Heidelberg, 1982

work page 1982
[5]

Allen, B

P.B. Allen, B. Mitrovi´ c. Solid State Physics, Vol. Vol. 3 7 (Eds. F. Seitz, D. Turnbull, H. Ehrenreich), Academic Press, NY, 1982, p. 1

work page 1982
[6]

A.B. Migdal. Zh. Eksp. Teor. Fiz. 34, 1438 (1958) [Sov. Phys. JETP 7, 996 (1958)]

work page 1958
[7]

Alexandrov

A.S. Alexandrov. A.B. Krebs. Usp. Fiz. Nauk 162, 1 (1992) [Physics Uspekhi 35, 345 (1992)]

work page 1992

Show all 24 references

[8]

Esterlis, B

I. Esterlis, B. Nosarzewski, E.W. Huang, D. Moritz, T.P. Devereux, D.J. Scalapino, S.A. Kivelson. Phys. Rev. B 97, 140501(R) (2018)

2018
[9]

Sadovskii

M.V. Sadovskii. Usp. Fiz. Nauk 178, 1243 (2008) [Physics Uspekhi 51, 1243 (2008)]

2008
[10]

Gor’kov, V.Z

L.P. Gor’kov, V.Z. Kresin. Rev. Mod. Phys. 90, 011001 (2018)

2018
[11]

L.P. Gor’kov. PNAS 113, 4646 (2018)

2018
[12]

L.P. Gor’kov. Phys. Rev. B 93, 054517 (2016)

2016
[13]

L.P. Gor’kov. Phys. Rev. B 93, 060507 (2016)

2016
[14]

Sadovskii.Zh

M.V. Sadovskii.Zh. Eksp. Teor. Fiz 155, 527 (2019) [JETP 128, 455 (2019)]

2019
[15]

Sadovskii

M.V. Sadovskii. Pis’ma Zh. Eksp. Teor. Fiz. 109, 165 (2010) [JETP Letters 109, 166 (2019)]

2010
[16]

Sadovskii

M.V. Sadovskii. Diagrammatics. World Scientiﬁc, Singa - pore, 2006

2006
[17]

P.B. Allen. Phys. Rev. B 6, 2577 (1972)

1972
[18]

Lee, F.T

J.J. Lee, F.T. Schmitt, R.G. Moore, S. Johnston, Y.T. Cui,W. Li, Z.K. Liu, M. Hashimoto, Y. Zhang, D.H. Lu, T.P. Devereaux, D.H. Lee, Z.X. Shen. Nature 515, 245 (2014)

2014
[19]

Kuli´ c, O.V

M.L. Kuli´ c, O.V. Dolgov. New J. Phys. 19, 013020 (2017)

2017
[20]

Nekrasov, N.S

I.A. Nekrasov, N.S. Pavlov, V.V. Sadovskii. Pis’ma Zh. Eksp. Teor. Fiz. 105, 354 (2017) [JETP Letters 105, 370 (2017)]

2017
[21]

Nekrasov, N.S

I.A. Nekrasov, N.S. Pavlov, M.V. Sadovskii. Zh. Eksp. Teor. Fiz. 153, 590 (2018) [JETP 126, 485 (2018)] Phys. Rev. B 93, 134513 (2016)

2018
[23]

Karakozov, E.G

A.E. Karakozov, E.G. Maksimov, S.A. Mashkov. Zh. Eksp. Teor. Fiz. 68, 1937 (1975) [JETP 41, 971 (1975)]

1975
[24]

Kirzhnits, E.G

D.A. Kirzhnits, E.G. Maksimov, D.I. Khomskii. J. Low. Temp. Phys. 10, 79 (1973)

1973

Pith tools

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

[1] [22]

Ikeda, A

M.A. Ikeda, A. Ogasawara, M. Sugihara. Phys. Lett. A 170, 319 (1992)

work page 1992

[2] [1]

Schrieﬀer

J.R. Schrieﬀer. Theory of Superconductivity, W A Ben- jamin, NY, 1964

work page 1964

[3] [2]

Superconductivity

D.J. Scalapino. In “Superconductivity”, p. 449, Ed. by R.D. Parks, Marcel Dekker, NY, 1969

work page 1969

[4] [3]

White, T.H

R.M. White, T.H. Geballe. Long Range Order in Solids. Academic Press, New York, San Francisco, London, 1979

work page 1979

[5] [4]

Vonsovsky, Yu.A

S.V. Vonsovsky, Yu.A. Izyumov, E.Z. Kurmaev. Super- conductivity of Transition metals, Their Alloys and Com- pounds, Springer, Berlin – Heidelberg, 1982

work page 1982

[6] [5]

Allen, B

P.B. Allen, B. Mitrovi´ c. Solid State Physics, Vol. Vol. 3 7 (Eds. F. Seitz, D. Turnbull, H. Ehrenreich), Academic Press, NY, 1982, p. 1

work page 1982

[7] [6]

A.B. Migdal. Zh. Eksp. Teor. Fiz. 34, 1438 (1958) [Sov. Phys. JETP 7, 996 (1958)]

work page 1958

[8] [7]

Alexandrov

A.S. Alexandrov. A.B. Krebs. Usp. Fiz. Nauk 162, 1 (1992) [Physics Uspekhi 35, 345 (1992)]

work page 1992