Bounds on the heat kernel of the Schroedinger operator in a random electromagnetic field

Z. Haba

REVIEW

Bounds on the heat kernel of the Schroedinger operator in a random electromagnetic field

Z. Haba

Quantum Physics · Submitted 10 Apr 2006 · 42 references

Pith's verdict: New heat-kernel bounds for growing random vector potentials, but the advertised Green-function decay is unsupported and likely wrong—don't cite it without a fix.

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arXiv IDquant-ph/0604068 (version 2)

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CitationZ. Haba. "Bounds on the heat kernel of the Schroedinger operator in a random electromagnetic field." arXiv:quant-ph/0604068 (2006). https://pith.science/paper/TQUSQFSX

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Editor's note

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Private take: the heat-kernel lower bounds are new and plausible; the Green-function claim (38) is not justified and likely false, and the paper's main application rides on it.

The paper derives averaged heat-kernel bounds for Schrödinger operators in a random Gaussian vector potential with growing, scale-invariant covariance. The route is standard—Feynman–Kac, Gaussian integration, Jensen—but the lower bound for γ>0 in Theorem 3 appears to be genuinely new. The explicit estimates for three of the six stochastic-integral terms in (41) look right, and the scaling form (29) gives a coherent sanity check. The paper is also candid about several technical limitations, and the citation pattern is clean.

The soft spot is serious. Equation (38) is supposed to follow from (28) by integrating over τ. But the τ-independent term in (28) already gives exp(-c |x-x'|^{2+2γ}) for large separation, not the exp(-b|x-x'|) claimed in (38). The τ integral only adds more decay, so the lower bound from (28) is super-exponential, far below the exponential claim. The paper's own remark after (38)—that it is unclear whether the decay is intrinsic or an artifact of the bound—is an admission that the step is unproven. In fact, the exact Gaussian variance of the line integral grows like |x-x'|^{2+2γ}, so the annealed heat kernel is exp(-c R^{2+2γ}); after τ integration the Green function cannot be bounded below by C R^{-D+2} exp(-aR^2-bR) at large R. The advertised Ginzburg-Landau correlation decay (72) rests on this unsupported bound.

So: the heat-kernel part is worth keeping, but the Green-function section needs major revision or a clear retraction. I would send this to a serious referee, because the heat-kernel result is novel and the proof is mostly checkable, but the referee should insist on removing or fixing (38). A reader looking for a rigorous localization statement from random magnetic fields should not rely on this paper's Green-function claim.

Referee report

Significance

If the lower bound (28) is valid, it is a substantive rigorous result: it shows that growing random electromagnetic correlations can force a much faster than Gaussian decay of the averaged heat kernel, in contrast to the diamagnetic upper bound, and it provides a tool for studying localization and the integrated density of states. The method is standard and transparent, with no fitted parameters and no apparent circularity: the constants a_j are existential and the estimates follow from the stated covariance assumptions. The paper also supplies a concrete model (Section 5) where a growing vector potential provably improves localization, and it is candid about the main gap in the Green-function argument. However, the Green-function lower bound (38) is not actually established, and the proof of Theorem 3 leaves three of the six terms in eq. (41) unestimated; both issues must be addressed before the results can be regarded as complete.

Major objections (3)

[Sec. 3, eqs. (28) and (38)] The lower bound (38) for the Green function does not follow from the heat-kernel bound (28). The τ-independent term in (28) is exp(-a1 ℏ^{-2}(|x|^{2γ}+|x'|^{2γ})(x-x')^2); taking x'=0 and |x|=R gives exp(-a1 ℏ^{-2} R^{2+2γ}). Substituting (28) into the τ-integral (37) and integrating would therefore produce a large-R decay of order exp(-c R^{2+2γ}) rather than the linear exponential exp(-(b/ℏ+m)R) claimed in (38). A direct Gaussian calculation of the variance in (19) along an almost straight Brownian bridge from 0 to R under covariance (23) also gives a variance of order R^{2+2γ}, consistent with this faster decay. The text after (38) explicitly concedes the gap: "It is not clear whether this exponential decay comes solely from the unprecise lower bound or if it is an intrinsic property of growing vector potentials." Since the abstract and introduction advertise exponentially decaying Green functions, this is a load-bearing point: (38) should either be proved with a detailed τ-integral estimate or be explicitly labeled as a heuristic conjecture.
[Sec. 4, eq. (41) and following] The proof of Theorem 3 bounds only three of the six stochastic-integral terms in eq. (41) (eqs. (48), (50), and (56)) and states that the remaining three "can be estimated in a similar way." This is a gap: the omitted terms include those with different powers of (x-x') and of the Brownian bridge (the second, fifth, and sixth lines of (41)), and their contributions could in principle modify the exponents a_1,...,a_8 in (28). Please provide the missing estimates or a domination argument showing that each omitted term is bounded by the expressions already estimated. Without this, Theorem 3 is not proven as stated.
[Sec. 3, paragraph after eq. (23)] The passage from a smooth regularized covariance G_κ to the distributional covariance (23) is asserted but not justified. The equality (19) and the estimates (48)-(56) require evaluating the covariance at points q(s), q(s'), while the covariance (23) is defined only in the distributional sense for γ<1. The statement "The inequalities discussed in this paper are preserved under such limits" needs a proof (for example, via convergence of the relevant expectation values or a monotone/fatou argument). Relatedly, the use of the stochastic representation (7) for merely Hölder-continuous A is delegated to reference [26]; the precise regularity assumptions and the functional setting for the Gaussian expectation should be stated explicitly.

Minor objections (6)

[Sec. 4, eq. (41)] Eq. (41) contains malformed parentheses in several terms, e.g., "E[G(q(s)), q(s′))]" and "E[b(s/τ−s)G(q(s)), q(s′))b(s′/τ−s′)]"; these should read E[G(q(s), q(s′))] and E[b(s/τ−s) G(q(s), q(s′)) b(s′/τ−s′)]. Eq. (48) also has an extra bracket and a misplaced comma in "Gjj(q(s)), q(s))".
[Theorems 2 and 3] The symbol a is used both as a generic constant in the bounds (e.g., "-a ℏ^{-1} τ" in (25)) and as the assumed growth constant in the hypothesis (27) "for certain a>0 and B>0" with V(x) ≤ B|x|^{2β}+a; this dual use is confusing and should be disambiguated.
[Sec. 3, eq. (26)] In the final expression of (26), the variable y is written as a scalar in what should be a D-dimensional Gaussian integral; it should be (2π)^{-D/2} exp(-|y|²/2) with y ∈ R^D.
[Sec. 5, eq. (68)] In eq. (68), the exponential reads "exp(-τV2(x,y))", but the preceding equation (67) has "exp(-τ/ℏ V2(x,y))"; the factor 1/ℏ appears to be missing in (68).
[References] Reference [27] bundles two distinct works by Dudley and by Fernique; these should be separated. The arXiv identifiers in references [10] and [12] are also cited inconsistently (as "arxiv" in lowercase), and the Introduction has the typo "Cwickel-Lieb-Rosenbljum" for "Cwikel-Lieb-Rosenbljum".
[Sec. 3, eq. (40)] The bound (40) for the diagonal difference is asserted after eq. (39) with only a vague reference to eqs. (43)-(46) and no derivational details; a concise argument or a reference to a lemma would make this estimate reproducible.

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The paper

The paper claims

Rising random magnetic correlations force exponential heat-kernel decay

The reading

The paper aims to establish that when a random electromagnetic vector potential has correlations that grow with distance, the averaged heat kernel of the Schrödinger operator decays exponentially in both time and spatial separation. The relevant regime is a transverse Gaussian vector potential with scale-invariant covariance and rough, Hölder-continuous samples. Using a stochastic Feynman-Kac representation, a diamagnetic upper bound, and Jensen's inequality, the paper derives lower bounds containing exponential factors such as exp(−a5 $ℏ^{{−1+γ}}$$τ^{{1+γ}}$ − a6 $ℏ^{{−2+γ}}$|x−x'|²τ^γ). If these bounds are correct, growing random fields suppress long-range propagation and enhance localization, a conclusion that matters for random magnetic field models and for correlation decay in scalar field models of superconductivity.

What carries the argument

The carrying mechanism is the stochastic Feynman-Kac representation, which writes the heat kernel as a Gaussian smearing over Brownian bridges times the phase factor exp(i/ℏ ∫ A(q(s))∘dq_s). Averaging over the Gaussian field converts this phase factor into exp(−(1/2ℏ²)⟨(∫ A dq)²⟩), and Jensen's inequality applied to the Brownian average turns the averaged kernel into an exponential of an explicit second-moment estimate. The estimates are made with the scale-invariant covariance (23) and the Ito calculus identities (43)–(46), which bound each of the six terms in eq. (41). This machinery is what converts the growth condition on the covariance into concrete exponential factors in the lower bound (28).

Extended reading notes

Core claim

The central claim is Theorem 3: for a transverse Gaussian vector potential whose covariance is the scale-invariant growing law with 0<γ<1, the averaged heat kernel obeys the lower bound (28), which contains exponential factors such as exp(−a5 $ℏ^{{−1+γ}}$$τ^{{1+γ}}$ − a6 $ℏ^{{−2+γ}}$|x−x'|²τ^γ − ...). Integrating this bound over proper time then gives the averaged Green function the exponential decay exp(−(b/ℏ + m)|x−x'|) in eq. (38). The paper presents the growing vector potential as acting like a growing scalar potential: it suppresses propagation and improves localization, and the same mechanism is mirrored by the deterministic constant magnetic field example. The diagonal and trace of the heat kernel also acquire exponential lower bounds, which the paper interprets as evidence for a discrete spectrum.

Load-bearing premise

The results rest on the assumption that the Feynman-Kac representation and the Jensen/diamagnetic inequalities remain valid when the random vector potential is only Hölder continuous rather than differentiable, after a regularization limit that is asserted to preserve the inequalities.

Editorial extensions

If this is right

At large separation and time, the averaged heat kernel is exponentially small, so propagation of a quantum particle through a strongly correlated random field is strongly suppressed.
The averaged Green function decays as exp(−(b/ℏ + m)|x−x'|), so two-point correlation functions of a scalar field in the corresponding Euclidean model lose long-range order.
The trace of the heat kernel has an exponential lower bound with a τ^{−ν} prefactor, which the paper reads as supporting a discrete spectrum for growing random electromagnetic fields.
In the model with a vector potential depending on two coordinates and a scalar potential |x|^α|y|^α, any growth index γ>0 yields a finite trace, whereas γ=0 would not; this is a concrete situation where growing correlations provably enhance localization.

Reading between the lines

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

The same Jensen-plus-second-moment route would plausibly extend to non-Gaussian random vector potentials, since only the covariance enters the lower bound; verifying this would require additional large-deviation control on higher cumulants.
Because the paper's upper bound is the field-free diamagnetic one and the lower bound is field-suppressed, the true averaged kernel is pinned between two exponentials; this suggests an intermediate asymptotics that might be testable with direct numerical path-integral simulations.
The gauge-dependence of the exponential decay means only gauge-invariant quantities like the diagonal and the trace should be compared with physical signatures; a gauge-invariant formulation of the bound would be a natural next step.
The method should extend to multi-point scalar-field correlations along multiple Brownian paths, and the paper leaves open whether the exponential decay survives in all separations; deriving the analogue of eq. (28) for such correlations would be a direct test.

Original abstract

We obtain lower and upper bounds on the heat kernel and Green functions of the Schroedinger operator in a random Gaussian magnetic field and a fixed scalar potential. We apply stochastic Feynman-Kac representation, diamagnetic upper bounds and the Jensen inequality for the lower bound. We show that if the covariance of the electromagnetic (vector) potential is increasing at large distances then the lower bound is decreasing exponentially fast for large distances and a large time.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We obtain lower and upper bounds on the heat kernel and Green functions of the Schroedinger operator in a random Gaussian magnetic field and a fixed scalar potential. We apply stochastic Feynman-Kac representation, diamagnetic upper bounds and the Jensen inequality for the lower bound. We show that if the covariance of the electromagnetic (vector) potential is increasing at large distances then the lower bound is decreasing exponentially fast for large distances and a large time.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We obtain lower and upper bounds on the heat kernel and Green functions of the Schroedinger operator in a random Gaussian magnetic field and a fixed scalar potential.	contribution 0.58	0.96	0.72	asserted 1.00
s1	We apply stochastic Feynman-Kac representation, diamagnetic upper bounds and the Jensen inequality for the lower bound.	method 1.00	0.92	0.04	asserted 0.99
s2	We show that if the covariance of the electromagnetic (vector) potential is increasing at large distances then the lower bound is decreasing exponentially fast for large distances and a large time.	result 1.00	0.94	0.50	asserted 1.00

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References

Works cited (42)

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [26]

K. Broderix, D. Hundertmark and H. Leschke, Rev. Math.Phys. 12,181(2000)

View on Pith 2000

[2] [1]

P.W. Milonni, The Quantum Vacuum, Academic, New York,1994

View on Pith 1994

[3] [2]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics,Ca mbridge Univ.Press,1995

View on Pith 1995

[4] [3]

J.W. Goodman, Statistical Optics,Wiley,New York,1985

View on Pith 1985

[5] [4]

L.B. Ioﬀe and P.B. Wiegmann, Phys.Rev.Lett. 65,653(1990)

View on Pith 1990

[6] [5]

G. Murthy and R.Shankar,Rev.Mod.Phys. 75,1101(2003)

View on Pith 2003

[7] [6]

A.L.Fetter and J.D. Walecka, Quantum Theory of Many-Particle Sy stems, McGraw-Hill,1971

View on Pith 1971

[8] [7]

B.I. Halperin, T.C. Lubensky and S.-K. Ma, Phys.Rev.Lett. 32,292(1974)

View on Pith 1974

[9] [8]

P.W. Anderson, Phys.Rev. 109,1492(1958)

View on Pith 1958

[10] [9]

L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Oper- ators,Springer, Berlin,1992

View on Pith 1992

[11] [10]

L. Erd¨ os, arxiv:math-ph/0510055

View on Pith Pith review arXiv

[12] [11]

N. Ueki,Ann.Inst.Henri Poincare, 1,473(2000)

View on Pith 2000

[13] [12]

H. Leschke,S. Warzel and A. Weichlein,arxiv:math-ph/0507035

View on Pith Pith review arXiv

[14] [13]

J.M Combes, R. Schrader and R.Seiler, Ann.Phys. 111,1(1978) 19

View on Pith 1978

[15] [14]

B. Simon, Functional Integration and Quantum Physics, Acade mic Press, 1979

View on Pith 1979

[16] [15]

P. Malliavin,C.R.Acad.Sci.Paris Ser.I.Math. 302,481(1986)

View on Pith 1986

[17] [16]

L. Erd¨ os, Duke Math.Journ.76,541(1994)

View on Pith 1994

[18] [17]

M. Loss and B. Thaller, Commun.Math.Phys. 186,95(1997)

View on Pith 1997

[19] [18]

S. Nakamura,Commun.Math.Phys. 214,565(2000)

View on Pith 2000

[20] [19]

J.E. Avron, I.Herbst and B.Simon, Duke Math.Journ. 45,847(1978)

View on Pith 1978

[21] [20]

B. Helﬀer and A. Mohamed, Ann.l’inst. Fourier, 38,95(1988)

View on Pith 1988

[22] [21]

Z. Shen,Trans.Amer.Math.Soc. 348,4465(1996)

View on Pith 1996

[23] [22]

J.Schwinger, Phys.Rev. 82,664(1951)

View on Pith 1951

[24] [23]

A.V. Skorohod, Random Linear Operators, Reidel, Dordrecht,1 984

View on Pith

[25] [24]

D.K. Elworthy, A. Truman and K. Watling, J.Math.Phys. 26,984(1985)

View on Pith 1985

[26] [25]

K.Broderix, H. Leschke and P. M¨ uller, Journal Funct.Anal. 212,287(2004)

View on Pith 2004

[27] [27]

R.M. Dudley, Ann.Probab. 1,66(1973) X. Fernique,in Lect.Notes.in Math. 480,Berlin,Springer,1975

View on Pith 1973

[28] [28]

H. Hess,R. Schrader and D. Uhlenbrock, J.Diﬀ.Geom. 15,27(1980)

View on Pith 1980

[29] [29]

N. Ikeda and S. Watanabe, Stochastic Diﬀerential Equations a nd Diﬀusion Processes, North Holland,1981

View on Pith 1981

[30] [30]

M.A. Berger and V.J.Mizel, Trans.Amer.Math.Soc. 252,249(1979)

View on Pith 1979

[31] [31]

P. Levy, Processus Stochastiques et Mouvement Brownien, d euxieme edi- tion, Revue et Augmentee,Paris,1965

View on Pith 1965

[32] [32]

J.L.W. Jensen, Acta Math. 30,175(1906)

View on Pith 1906

[33] [33]

A. W. Marshall and I. Olkin, Inequalities:Theory of Majorization a nd Its Applications, Academic Press,1979

View on Pith 1979

[34] [34]

D.V.Vassilevich, Phys.Rep. 388,279(2003)

View on Pith 2003

[35] [35]

K. Maurin, Methods of Hilbert Spaces,PWN,Warszawa,1972

View on Pith 1972

[36] [36]

J.Br¨ uning, V.Geyler and K.Pankrashkin, J.Math.Phys. 46,113508(2005) 20

View on Pith 2005

[37] [37]

Z.Haba, Phys.Rev. D26,3506(1982)

View on Pith 1982

[38] [38]

S. Golden, Phys.Rev. 137,B1127(1965)

View on Pith 1965

[39] [39]

C.J. Thompson, J.Math.Phys. 6,1812(1965)

View on Pith 1965

[40] [40]

M.B. Ruskai,Commun.Math.Phys. 26,280(1972)

View on Pith 1972

[41] [41]

T.W.B.Kibble, Phys.Rev. 150,1060(1966)

View on Pith 1966

[42] [42]

B. Simon, Ann.Phys. 146,209(1983) 21

View on Pith 1983