Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations

Arup Chattopadhyay; Chandan Pradhan; Teun D.H. van Nuland

arxiv: 2404.18422 · v3 · submitted 2024-04-29 · 🧮 math.FA

Differentiation, Taylor series, and all order spectral shift functions, for relatively bounded perturbations

Arup Chattopadhyay , Teun D.H. van Nuland , Chandan Pradhan This is my paper

Pith reviewed 2026-05-24 02:24 UTC · model grok-4.3

classification 🧮 math.FA

keywords spectral shift functionsmultiple operator integralsrelatively bounded perturbationsGateaux derivativeTaylor seriesSchatten classesKrein-Koplienko functionsoperator perturbation

0 comments

The pith

Under relative boundedness and Schatten conditions, Krein-Koplienko spectral shift functions exist for all orders.

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

This work proves the existence of the Gateaux derivative of f(H + tV) at t=0 in the operator norm for every order n, along with an explicit formula using multiple operator integrals. It also establishes conditions for the absolute convergence of the corresponding Taylor series in norm when the relative bound of V is less than 1. The key advance is showing that the Krein-Koplienko spectral shift functions η_k exist for arbitrarily large k when V satisfies V(H-i)^{-p} in S^{s/p} for p up to s, and this holds independently of s and even for bounded V. These results rely on combining multiple operator integral techniques with prior perturbation results and have implications for trace formulas in operator theory.

Core claim

Given self-adjoint H and symmetric relatively H-bounded V, the nth Gateaux derivative of f(H+tV) at t=0 exists in operator norm for all natural n and mild f, with an explicit multiple operator integral formula. When the H-bound of V is less than 1, the Taylor series converges absolutely under suitable conditions on f. Under the Schatten assumption V(H-i)^{-p} ∈ S^{s/p} for p=1 to s, the spectral shift functions η_{k,H,V} exist for every k and satisfy the stated trace identity with the kth derivative of f.

What carries the argument

Multiple operator integrals used to express the derivatives, combined with Schatten class membership to ensure the existence of all-order Krein-Koplienko spectral shift functions η_{k,H,V}.

If this is right

The perturbation formulas for multiple operator integrals hold under relatively bounded perturbations.
The Taylor expansion converges absolutely in operator norm when the H-bound is less than 1.
The spectral shift functions exist independently of the parameter s in the Schatten condition.
Results apply to quantum physics and noncommutative geometry as discussed.

Where Pith is reading between the lines

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

The independence from s may allow uniform treatment across different manifold dimensions in geometric applications.
This could enable higher-order corrections in spectral invariants for unbounded operators in quantum field theory.
Extensions to non-self-adjoint cases or time-dependent perturbations might be possible using similar techniques.

Load-bearing premise

V must satisfy the Schatten class conditions V(H-i)^{-p} ∈ S^{s/p} for p=1 to s to guarantee existence of the higher order spectral shift functions.

What would settle it

A concrete operator H and perturbation V that is relatively bounded with the Schatten conditions violated, for which the trace identity fails to hold for some smooth f with compact support and some k, would falsify the existence claim.

read the original abstract

Given $H$ self-adjoint, $V$ symmetric and relatively $H$-bounded, and $f:\mathbb{R}\to\mathbb{C}$ satisfying mild conditions, we show that the Gateaux derivative $$\frac{d^n}{dt^n}f(H+tV)|_{t=0}$$ exists in the operator norm topology, for every natural $n$, give a new explicit formula for this derivative in terms of multiple operator integrals, and establish useful perturbation formulas for multiple operator integrals under relatively bounded perturbations. Moreover, if the $H$-bound of $V$ is less than 1, we obtain sufficient conditions on $f$ which ensure that the Taylor expansion $$f(H+V)=\sum_{n=0}^\infty\frac{1}{n!}\frac{d^n}{dt^n} f(H+tV)\big|_{t=0}$$ exists and converges absolutely in operator norm. Finally, assuming that $V(H-i)^{-p}\in\mathcal{S}^{s/p}$ for $p=1,\ldots,s$ for some $s\in\mathbb{N}$ (for instance, when $H$ is an order 1 differential operator on an $s-1$ dimensional space), we show that the Krein--Koplienko spectral shift functions $\eta_{k,H,V}$, satisfying $${Tr}\left(f(H+V)-\sum_{m=0}^{k-1}\frac{1}{m!}\frac{d^m}{dt^m} f(H+tV)\big|_{t=0}\right)=\int_{\mathbb{R}} f^{(k)}(x)\eta_{k,H,V}(x)dx,$$ exist for every $k=1,2,3,\ldots$, independently of $s$. The latter result (which is significantly stronger than \cite{vNS22}) is completely new also in the case that $V$ is bounded. The proof is based on \cite{PSS}, combined with a generalisation of the multiple operator integral compatible with \cite{HMvN}. We discuss applications of our results to quantum physics and noncommutative geometry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They prove all-order Krein-Koplienko spectral shift functions exist independently of s under the stated Schatten conditions, including for bounded V, with explicit multiple-integral derivative formulas.

read the letter

The main takeaway is that the paper establishes existence of the Krein-Koplienko functions η_k for every k, independently of s, when V(H-i)^{-p} sits in S^{s/p} for p=1 to s. This holds even if V is bounded, which goes beyond the cited vNS22 result. They also give explicit formulas for the Gateaux derivatives of f(H+tV) via multiple operator integrals and show norm convergence of the Taylor series when the relative bound of V is less than 1, under suitable conditions on f. Perturbation formulas for the integrals themselves are included as well. The argument combines the PSS framework with a generalized multiple-operator-integral calculus that fits HMvN, reducing the trace identities to properties of those integrals. The stress-test found no internal gaps in the reduction steps or domain handling beyond the stated relative boundedness and Schatten membership. The assumptions are laid out clearly up front, so the scope is not overstated. One limitation is that the Schatten conditions are required for the all-order result; without them the proof does not go through, though the authors note this is satisfied for standard examples like order-1 differential operators on lower-dimensional spaces. The relative bound less than 1 for the series part is a standard restriction in this setting. The work is aimed at functional analysts working on perturbation theory for unbounded self-adjoint operators, with possible use in quantum physics and noncommutative geometry. A reader looking for higher-order tools or explicit integral expressions would find concrete value here. The result is new, the citations are used to mark the advance rather than prop up the core claim, and the logic appears coherent on its own terms. It deserves a serious referee.

Referee Report

0 major / 0 minor

Summary. The manuscript shows that for self-adjoint H and symmetric relatively H-bounded V, the Gateaux derivative d^n/dt^n f(H+tV) at t=0 exists in the operator-norm topology for every natural number n; it supplies an explicit formula for this derivative in terms of multiple operator integrals and derives perturbation formulas for those integrals under relative boundedness. When the H-bound of V is less than 1, it gives conditions on f ensuring absolute norm-convergence of the Taylor series. Under the additional assumption that V(H-i)^{-p} belongs to the Schatten class S^{s/p} for p=1 to s (for some fixed s), the Krein-Koplienko spectral shift functions η_{k,H,V} exist for every k=1,2,… independently of s; this last result is new even when V is bounded. The proofs combine the framework of PSS with a generalization of multiple operator integrals compatible with HMvN, and applications to quantum physics and noncommutative geometry are indicated.

Significance. If the stated reductions and domain arguments hold, the work supplies the first existence proof for all-order Krein-Koplienko functions under relative boundedness with the given Schatten-class hypotheses, and the independence of s (including the bounded-V case) is a genuine strengthening of vNS22. The explicit multiple-operator-integral formulas and the perturbation results for those integrals constitute concrete, usable tools that extend the reach of higher-order perturbation theory in functional analysis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. The recommendation for minor revision is noted. No specific major comments were provided in the report, so we have no points requiring detailed rebuttal or revision at this stage. We are pleased that the significance of the all-order results and the independence from s (including the bounded case) were recognized.

Circularity Check

0 steps flagged

No significant circularity; derivations rest on external cited frameworks

full rationale

The paper's central results on existence of all-order Krein-Koplienko functions η_k under the stated Schatten-class assumptions V(H-i)^{-p} ∈ S^{s/p} proceed by reducing trace identities to properties of multiple operator integrals, using the external framework of [PSS] together with a generalization compatible with [HMvN]. These are treated as independent prior results; no step equates a claimed prediction or first-principles output to a quantity defined or fitted inside the paper itself. The relative boundedness and Schatten conditions are explicit hypotheses, not outputs, and the Taylor expansion and Gateaux derivative formulas are derived without self-referential closure or renaming of known patterns as new unifications. The comparison to [vNS22] is merely a strength claim and does not bear the proof load.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Central claims rest on standard domain assumptions for self-adjoint and symmetric operators plus generalizations of multiple operator integrals drawn from the cited papers PSS and HMvN; no free parameters or invented entities are introduced.

axioms (3)

domain assumption H is self-adjoint on a Hilbert space
Stated as given at the opening of the abstract.
domain assumption V is symmetric and relatively H-bounded
Stated as given; required for all subsequent statements.
domain assumption f satisfies mild conditions (unspecified in abstract)
Invoked for existence of derivatives and Taylor series.

pith-pipeline@v0.9.0 · 5933 in / 1591 out tokens · 23595 ms · 2026-05-24T02:24:51.700856+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

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[1] [1]

B., & Peller, V

Aleksandrov A. B., & Peller, V . V . (2022). Functions of Pa irs of Unbounded Noncommuting Self-Adjoint Operators under Perturbation. Dokl. Math. 106, 407–411. 7

work page 2022

[2] [2]

B., & Peller, V

Aleksandrov A. B., & Peller, V . V . (2024). Functions of se lf-adjoint operators under relatively bounded and relatively trace class perturbations. Relativ ely operator Lipschitz functions. Preprint, arXiv:2411.01901 [math.FA] 4

work page arXiv 2024

[3] [3]

Al-Gwaiz, M. A. (1992). Theory of distributions . Monographs and Textbooks in Pure and Ap- plied Mathematics, 159. Marcel Dekker, Inc., New York, XII+25 7 pp. 26, 30

work page 1992

[4] [4]

A., Carey , A

Azamov , N. A., Carey , A. L., Dodds, P . G., & Sukochev , F. A.(2009). Operator integrals, spec- tral shift, and spectral ﬂow. Canad. J. Math. 61, no. 2, 241–263. 2, 7, 11, 32 39

work page 2009

[5] [5]

Birman, M. Sh. & Solomyak, M. Z. (1966). Double Stieltjes operator integrals. Problems of mathe- matical physics, No. I: Spectral theory and wave processes ( Russian). Izdat. Leningrad. Univ ., Leningrad, pp. 33–67. 2

work page 1966

[6] [6]

Clark, C. (1968). On relatively bounded perturbations o f ordinary differential operators. Pac. J. Math. 25, no. 1, 59–70. 7

work page 1968

[7] [7]

Cwikel, M. (1977). Weak type estimates for singular value s and the number of bound states of Schr ¨ odinger operators. Ann. Math.106, 93–100. 31

work page 1977

[8] [8]

Chattopadhyay , A., & Skripka, A. (2018). Trace formulas for relative Schatten class perturba- tions. J. Funct. Anal. 274, 3377–3410. 9, 35

work page 2018

[9] [9]

Daletskii, Y . L. (1990). A noncommutative taylor formul a and functions of triangle operators. Funct. Anal. Its Appl. 24, 64–66. 3

work page 1990

[10] [10]

Daletski˘ ı, Y . L. & Kre˘ ın, S. G. (1965). Integration and differentiation of functions of Hermi- tian operators and applications to the theory of perturbati ons. Vorone˘ z. Gos. Univ. T rudy Sem. Funkcional. Anal. 1956.1, pp. 81–105. 2

work page 1965

[11] [11]

Dunford, N., & Schwartz, Jacob T. (1988). Linear Operators. I. General Theory. John Wiley & Sons, Inc., Hoboken, New Jersey . 14

work page 1988

[12] [12]

Feynman, R. P . (1951). An operator calculus having applications in quantum electrodynamics. Physical Review, 84(1), 108. 3

work page 1951

[13] [13]

& Miehe, J

Batu G ¨ uneysu, B. & Miehe, J. (2024). Fermionic Dyson exp ansions and stochastic Duistermaat-Heckman localization on loop spaces. Preprin t, arXiv:2410.14034v1 [math.DG]. 3

work page arXiv 2024

[14] [14]

Hansen, F. (2006). Trace functions as Laplace transfor ms. J. Math. Phys. 47 (4), 043504. 3

work page 2006

[15] [15]

Hekkelman, E., McDonald E., & van Nuland, T. D. H. (2024). Multiple operator inte- grals, pseudodifferential calculus, and asymptotic expan sions. Preprint, arXiv:2404.16338 [math.FA]. 1, 2, 3, 7

work page arXiv 2024

[16] [16]

Iochum, B., Levy , C., & Vassilevich, D. (2012). Spectra l action beyond the weak-ﬁeld approx- imation. Commun. Math. Phys. 316, 595–613. 32

work page 2012

[17] [17]

Iochum B., Masson T. (2018). Heat asymptotics for nonmin imal Laplace type operators and application to noncommutative tori. J. Geom. Phys. 129, 1–24. 3

work page 2018

[18] [18]

Kato, T. (1951). Fundamental properties of Hamiltonian operators of Schr ¨ odinger type.T rans. Am. Math. Soc. 70, no. 2, 195–211. 7, 34

work page 1951

[19] [19]

Kato, T. (1995). Perturbation Theory for Linear Operators (Reprint of the 1980 edition). Springer-V erlag, Berlin.35

work page 1995

[20] [20]

Koplienko, L. S. (1984). The trace formula for perturbat ions of nonnuclear type. Sibirsk. Mat. Zh. 25, no. 5, 62–71; English translation, Siberian Math. J. 25, 735–743. 4 40

work page 1984

[21] [21]

Krein, M. G. (1953). On the trace formula in perturbation t heory. Mat. Sbornik N.S. 33, no. 75, 597–626. 3, 23

work page 1953

[22] [22]

Lesch M. (2017). Divided differences in noncommutative geometry: rearrangement lemma, functional calculus and expansional formula. J. Noncommut. Geom. 11, no. 1, 193–223. 3

work page 2017

[23] [23]

Levitina, G., Sukochev , F., & Zanin, D. (2020). Cwikel e stimates revisited. P . Lond. Math. Soc. 120, no. 2, 265–304. 31, 33

work page 2020

[24] [24]

McDonald, E., & Ponge, R. (2022). Cwikel estimates and ne gative eigenvalues of Schr ¨ odinger operators on noncommutative tori. J. Math. Phys. 63, no. 4. 31

work page 2022

[25] [25]

van Nuland, T. D. H., & Skripka, A. (2022). Spectral shif t for relative Schatten class perturba- tions. J. Spectr. Theor. 12, 1347–1382. 1, 4, 9, 12, 19, 30, 31, 35

work page 2022

[26] [26]

van Nuland, T. D. H., & Skripka, A. (2022). Higher-order spectral shift function for resolvent comparable perturbations. J. Operat. Theor. , to appear, arXiv:2211.03330 [math.FA]. 4, 9, 19, 28, 35

work page arXiv 2022

[27] [27]

van Nuland, T. D. H., & van Suijlekom, W. D. (2021). Cycli c cocycles in the spectral action. J. Noncommut. Geom. 16, 1103–1135. 3, 9, 17, 27, 35

work page 2021

[28] [28]

van Nuland, T. D. H., & van Suijlekom, W. D. (2022). One-l oop corrections to the spectral action. J. High Energy Phys. , Paper No. 078, 14 pp. 3, 32

work page 2022

[29] [29]

Pavlov , B. (1971). On multidimensional integral operators. Linear Operators and Operator Equa- tions, pp. 81–97. 2

work page 1971

[30] [30]

Peller, V . V . (2006). Multiple operator integrals and higher operator derivatives. J. Funct. Anal. 233, 515–544. 2, 7

work page 2006

[31] [31]

Potapov , D., & Sukochev , F. (2009). Unbounded Fredholm modules and double operator in- tegrals. J. Reine Angew. Math. 626, 159–185. 6, 17

work page 2009

[32] [32]

(2013) Spectra l shift function of higher order

Potapov , D., Skripka, A., & Sukochev , F. (2013) Spectra l shift function of higher order. Invent. Math. 193, no. 3, 501–538. 1, 4, 8, 22

work page 2013

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(1972) Methods of modern mathematical physics

Reed, M., & Simon, B. (1972) Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London. 23, 35

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Rennie, A. C. (2004). Summability for nonunital spectr al triples. K-Theory 31, no. 1, 71–100. 4, 31, 32, 33

work page 2004

[35] [35]

Simon, B. (2005). Trace ideals and their applications. Mathematical surveys and monographs, volume 120, American mathematical society , Providence, RI. 14, 31, 34, 35

work page 2005

[36] [36]

Skripka, A. (2014). Asymptotic expansions for trace fu nctionals. J. Funct. Anal. 266, 2845–2866. 3

work page 2014

[37] [37]

Skripka, A. (2021). Lipschitz estimates for functions of Dirac and Schr ¨ odinger operators, J. Math. Phys. 62, no. 1, Paper No. 013506, 28 pp. 4, 31, 32 41

work page 2021

[38] [38]

Skripka, A., & Tomskova, A. (2019). Multilinear Operator Integrals: Theory and Applications. Lecture Notes in Math. 2250, Springer International Publish ing, XI+192 pp. 7, 22, 36

work page 2019

[39] [39]

& Sten’kin, V

Solomyak, M. & Sten’kin, V . (1971). On one class of Stieltjes multiple-integral operators. Linear Operators and Operator Equations, pp. 99–108. 2

work page 1971

[40] [40]

Sten’kin, V . (1977). Multiple operator integrals. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 4, pp. 102–115. 2

work page 1977

[41] [41]

van Suijlekom, W. D. (2011). Perturbations and operato r trace functions. J. Funct. Anal. 260, 2483–2496. 3

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