From Short-Range to Contact Interactions in the 1d Bose Gas

Marcel Griesemer; Michael Hofacker; Ulrich Linden

REVIEW 1 major objections 4 minor 21 references

From Short-Range to Contact Interactions in the 1d Bose Gas

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any number of bosons in one dimension, delta-interaction Hamiltonians are norm-resolvent limits of rescaled short-range potentials, with explicit rate O(epsilon^s).

desk verdict Solid new result on norm resolvent convergence for the 1D Bose gas with delta interactions; the main proof has a repairable logical gap in the limit identification. read the letter →

arxiv 1908.05705 v1 pith:DP7QXZYA submitted 2019-08-15 math-ph math.MP

Marcel Griesemer , Michael Hofacker , Ulrich Linden This is my paper · ORCID

classification math-phmath.MP MSC 81Q1081Q1581V70

keywords one-dimensionalBosegasLieb-LinigermodeldeltainteractionscontactnormresolventconvergenceGamma-convergenceKreinformulashort-rangepotentials

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 proves that the usual Hamiltonian for N bosons in one dimension with two-body $\delta$-interactions—infinitely short-range potential spikes—is a norm-resolvent limit of genuine Schrödinger operators with rescaled short-range potentials. For an even potential $V\in L^1\cap L^2(\mathbb{R})$, it defines $H_\varepsilon=-\Delta - g_\varepsilon \sum_{i

What carries the argument

The load-bearing object is a Krein-type resolvent identity (Appendix B), a formula expressing $(H_\varepsilon+z)^{-1}$ as the free resolvent $(H_0+z)^{-1}$ plus one additional term built from $\varphi_\varepsilon(z)=B_\varepsilon(H_0+z)^{-1}A_\varepsilon^*$, where $A_\varepsilon$ and $B_\varepsilon$ absorb the scaled potential through multiplication and scaling operators. The proof then establishes operator-norm convergence of $A_\varepsilon(H_0+z)^{-1}$ to $S(z)$ and of each pair contribution $\varphi_{ij,\varepsilon}(z)$ to $\varphi_{ij,0}(z)$ using explicit Green's function kernels in dimensions 1, 3, and 4 in relative-and-center-of-mass coordinates. A $\Gamma$-convergence argument on the quadratic forms identifies the strong-resolvent limit with $H$, and the rate $O(\varepsilon^s)$ comes from Hölder-type estimates on translates of Green's functions (Lemma A.3).

What would settle it

For $N=2$ and a Gaussian potential $V(r)=e^{-r^2}$, the resolvent difference can be evaluated numerically at fixed $z$ for several $\varepsilon$; since $\int |r|^{2s}|V|\,dr<\infty$ for every $s<1$, the theorem predicts decay no slower than $O(\varepsilon^{0.9})$. Observably slower decay, or none, would refute the rate claim. For the identification claim itself, one can check directly that the operator $R(z)$ in (1.13) satisfies the $\delta$-interaction boundary condition (the jump condition) on a dense set; failure to satisfy it would mean the limit is a different Hamiltonian.

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Extended reading notes

Core claim

The central claim, Theorem 1.1, is that $H_\varepsilon\to H$ in the norm resolvent sense for every $N\geq 2$, with the explicit resolvent identity $(H+z)^{-1}=(H_0+z)^{-1}+gS(z)^*(1-g\varphi(z))^{-1}JS(z)$ for $z\in\rho(H_0)\cap\rho(H)$. Here $S(z)$ and $\varphi(z)$ are limits of operators built from the scaled potential, and after integrating out the potential the formula depends on $V$ only through $\alpha=g\int V\,dr$. Under the moment condition $\int |r|^{2s}|V(r)|\,dr<\infty$ and the coupling-rate condition $|g_\varepsilon-g|=O(\varepsilon^s)$, the paper proves $\|(H+z)^{-1}-(H_\varepsilon+z)^{-1}\|=O(\varepsilon^s)$. For $N=2$ and $N=3$ such convergence was known; the contribution here is the general $N$ statement and the explicit rate.

Load-bearing premise

The load-bearing premise is that the energy cost of a wavefunction whose particles coincide is controlled by its kinetic energy; the Gamma-convergence proof uses this control to pin down the limit. If this control failed, the limit of the approximating operators could be a different self-adjoint operator, and the explicit resolvent formula would not describe the $\delta$-interaction Hamiltonian.

Editorial extensions

If this is right

Norm resolvent convergence implies convergence of spectra, so eigenvalues and spectral gaps of the short-range models approach those of the $\delta$-interaction Hamiltonian as $\varepsilon\to 0$.
The unitary time evolutions converge in a weighted operator norm: $\|(e^{-iH_\varepsilon t}-e^{-iHt})(H+i)^{-1}\|\to 0$ uniformly on compact time intervals, and at rate $O(\varepsilon^s)$ on growing intervals when $s>0$.
The rate estimate needs only an $L^1\cap L^2$ potential with finite $2s$-th moment; no further smoothness of $V$ is required.
The limiting Hamiltonian does not remember the shape of $V$: any potential with the same integrated strength $\alpha=g\int V\,dr$ produces the same resolvent formula (1.13).
The result holds for all $N\geq 2$, going beyond the previously settled two- and three-particle cases.

Reading between the lines

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

Since the limiting resolvent depends on $V$ only through $\alpha$, the same $O(\varepsilon^s)$ rate should be universal across different choices of $V$ with equal integrated strength and identical moment decay; this could be checked numerically for Gaussian, exponential, and compactly supported potentials.
The abstract pieces of the proof—the Krein identity and the $\Gamma$-convergence identification—do not use one-dimensional geometry, so a plausible extension is a two-dimensional analogue for distinguishable particles with a renormalized coupling, where $\varphi_\varepsilon(z)$ diverges but its singular part is expected to cancel.
The trace inequalities (C.1)–(C.2) are the only place where collision-plane regularity enters; testing whether the rate $O(\varepsilon^s)$ can fail when $\int |r|^{2s}|V|=\infty$ would clarify whether the moment condition is sharp.

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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

Dear colleague,

You should know this: the paper proves the first norm resolvent convergence result for the 1D Bose gas Hamiltonian with delta interactions, for arbitrary N, with explicit rates. Previous work covered N=2 and N=3; the N=3 proof used Faddeev equations that do not generalize. The pair-decomposed Krein formula approach is new, and the convergence estimates in Sections 3 and 4 are detailed and explicit. This is a genuine advance.

The proof is self-contained and mostly well organized. The weak spot is in Section 5, where the limit operator R(z) is identified with the actual Hamiltonian H. The proof defines R(z) as the limit of the resolvents, then says 'To see that R(z) = (H+z)^{-1} we use Corollary 5.1, below.' But Corollary 5.1 is stated as an identity for (H+z)^{-1}, not for R(z), and its derivation appears after the theorem and presupposes that (5.4) is the resolvent of H. As written, that is a formal circularity. It is not fatal: the algebra in (5.7)–(5.12) is independent of the identification and can be reorganized as a lemma showing R(z) equals the V-independent expression. Then a compactly supported V' with the same integral and Corollary C.4 complete the proof. The authors just need to move that calculation before the theorem and state it for R(z). The gap is repairable and does not affect the main convergence estimates.

Other concerns are minor. Lemma A.1 is left to the reader, which is fine. The Gamma-convergence appendix requires only ∫|V||r|^{1/2} < ∞, weaker than the theorem's decay assumption, but it is used only for the strong-resolvent identification, so no problem. The citation pattern is clean; the one self-citation, [13], is unrelated to the central argument.

Overall, the reader's ACCEPT verdict is fair, provided the authors reorder the proof to remove the circularity. This paper is for mathematical physicists working on zero-range limits, Lieb-Liniger-type models, and resolvent convergence; it deserves a serious referee. I would cite it once the logic is cleaned up.

Referee Report

1 major / 4 minor

Summary. This paper proves that the N-boson Schrödinger operator H_ε with two-body rescaled potentials V_ε(x_j−x_i)=ε^{−1}V((x_j−x_i)/ε) converges in norm resolvent sense, as ε→0, to the δ-interaction Hamiltonian H with strength α=g∫V dr, for any even V∈L^1∩L^2 and g=lim g_ε. The proof is based on a generalized Krein formula, explicit norm convergence of the auxiliary operators A_ε(H0+z)^{−1} and φ_ε(z), and Γ-convergence of the associated quadratic forms. Under the additional decay condition ∫|r|^{2s}|V(r)|dr<∞ and |g_ε−g|=O(ε^s), the resolvent error is O(ε^s). A factorization step shows the limiting resolvent depends only on α, yielding a closed-form resolvent identity (1.13)/(5.13).

Significance. The result is a significant advance over the known N=2 and N=3 cases: it establishes norm resolvent convergence for arbitrary N bosons in one dimension, with explicit rates and a V-independent formula for the limit, without recourse to Faddeev equations. The method (Krein formula plus Γ-convergence) is robust and the lemmas are stated with precise estimates. If the proof's one structural gap is repaired, the paper will be a useful reference for singular perturbation theory and the Lieb-Liniger model.

major comments (1)

[Section 5, Eq. (5.4) and Corollary 5.1] The proof of Theorem 1.1 identifies the norm limit R(z) with (H+z)^{-1} by invoking Corollary 5.1, which is stated below. Taken literally, the corollary is an identity for (H+z)^{-1}, and its derivation in the closing paragraph of Section 5 uses the phrase "the expression (5.4) for the resolvent of H", i.e., it presupposes the very identification R(z)=(H+z)^{-1} that Theorem 1.1 is proving. This is a formal circularity in a load-bearing step. The gap is repairable without changing the results: the factorization and the algebra leading to (5.12) can be reorganized as an independent lemma showing R(z)=R0(z)+α \tilde S(z)^*(1−α \tilde φ(z))^{-1}\tilde S(z) for the operator R(z) defined in (5.4); then, since the right side depends on V only through α, choosing a compactly supported V' with the same integral and applying Corollary C.4 gives R(z)=(H+z)^{-1} by uniqueness of strong limits. Until this reorganization is made, the proof of the central identification is not fully self-contained.

minor comments (4)

[Abstract and title page] There are typographical errors ("semi-bounde d" and "S tuttgart"); please correct them.
[Appendix A, Lemma A.1] The proof of Lemma A.1 is left as an exercise to the reader. Since properties (i)-(vi) are used repeatedly in the main text, a sketch or a reference would improve the self-containedness of the appendix.
[Section 1, Remark 2] The phrase "the left hand side of (1.13) seems to depend on V" is only resolved later by Corollary 5.1; consider adding a forward reference at that point.
[Equation (5.12)] The identity (5.12) is stated without derivation; a one-line verification using (1−A)^{-1}=1+A(1−A)^{-1} with A=g|u⟩⟨v|⊗\tilde φ would help the reader.

Circularity Check

1 steps flagged · score 6.0 of 10

Formal circularity in the resolvent-limit identification: Theorem 1.1's proof appeals to Corollary 5.1, whose derivation presupposes the identification it is used to prove.

self definitional [Section 5, proof of Theorem 1.1, Eq. (5.4) and Corollary 5.1 (after proof of Theorem 1.1)]
""To see that R(z) = (H +z)−1 we use Corollary 5.1, below. By this corollary, the operator R(z) depends on α = g∫ V (r) dr only, but not on the particular choice of V . Finally, we show that the potential V may be integrated out in the expression (5.4) for the resolvent of H.""

Corollary 5.1 is stated as an identity for (H+z)^{-1}. Its derivation substitutes the factorization into the expression (5.4) and calls (5.4) 'the resolvent of H', i.e. it assumes R(z)=(H+z)^{-1}. The theorem proof then invokes this corollary to prove R(z)=(H+z)^{-1}. The corollary's conclusion is therefore the same as the missing identification, obtained from the same expression, so the step is circular by construction. The gap is repairable: the factorization (5.7)-(5.12) itself shows R(z) is V-independent, and with Corollary C.4 for a compactly supported comparison potential the identification follows without assuming the theorem. But as written the forward reference does not supply an independent proof.

full rationale

The central numerical content of the paper—norm resolvent convergence with rate O(epsilon^s)—rests on the Krein-formula convergence estimates of Sections 3 and 4 and the Gamma-convergence result Appendix C; these are independent external benchmarks and contain no fitted parameters. The only circularity I can exhibit is the proof-order loop in Section 5: R(z) is defined as the norm limit of (H_epsilon+z)^-1, and the identification R(z)=(H+z)^-1 is justified by Corollary 5.1, whose statement and proof already treat (5.4) as the resolvent of H. Because the needed factorization and the strong-resolvent comparison argument are present in the paper, the loop is formal rather than substantive; nevertheless, as written the derivation of the main theorem contains a self-referential step. Hence a partial-circularity score of 6, not 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the coupling g and the exponent s are prescribed assumptions. No new physical or mathematical entities are introduced; all operators are constructed from the given potential and the free Hamiltonian.

assumptions (4)

domain assumption The 1D delta-interaction Hamiltonian H is defined as the self-adjoint operator associated with the closed semibounded quadratic form q on H^1(R^N) with contact term -alpha sum ||gamma_ij psi||^2.
Used in Section 1 and throughout Appendix C; standard for Lieb-Liniger and cited to [3,14], but assumed without rederivation.
domain assumption The potential family is V_epsilon(r) = epsilon^{-1} V(r/epsilon) with V in L1 intersect L2, even, and g_epsilon times the integral of V tends to alpha.
Theorem 1.1's standing hypothesis; the theorem does not cover other rescaling shapes or pair-dependent scalings.
standard math The abstract Krein formula Theorem B.1 gives the resolvent representation (1.9) for H0 - g A* B under relative H0-boundedness of A* A and A* B.
Proved in Appendix B from Kato-Rellich and resolvent identities; used in Section 5 to take the epsilon to 0 limit.
standard math Gamma-convergence of quadratic forms is equivalent to strong resolvent convergence, and Theorem 2.19 of [8] extends invertibility of 1 - g phi(z) from large z to all z in rho(H0) intersect rho(H).
Used in Corollary C.4 and in the final paragraph of the proof of Theorem 1.1; these are external results from [9] and [8].

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Cite this review

Pith. "Pith review of From Short-Range to Contact Interactions in the 1d Bose Gas." pith.science (2026). https://pith.science/paper/DP7QXZYA

@misc{pith2026190805705,
  author       = {Pith},
  title        = {Pith review of: From Short-Range to Contact Interactions in the 1d Bose Gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DP7QXZYA}},
  note         = {Machine review of arXiv:1908.05705}
}

abstract

For a system of $N$ bosons in one space dimension with two-body $\delta$-interactions the Hamiltonian can be defined in terms of the usual closed semi-bounded quadratic form. We approximate this Hamiltonian in norm resolvent sense by Schr\"odinger operators with rescaled two-body potentials, and we estimate the rate of this convergence.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [13]

Griesemer and U

M. Griesemer and U. Linden. Spectral Theory of the Fermi Polaron. Ann. Henri Poincaré, 20(6):1931–1967, 2019

work page 1931

[2] [1]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden . Solvable models in quantum me- chanics. AMS Chelsea Publishing, Providence, RI, second edition, 2 005. With an appendix by Pavel Exner

work page

[3] [2]

Albeverio and P

S. Albeverio and P. Kurasov. Singular perturbations of diﬀerential operators , volume 271 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge,

work page

[4] [3]

The three- body problem in dimension one: from short-range to contact i nteractions

Giulia Basti, Claudio Cacciapuoti, Domenico Finco, and Alessandro Teta. The three- body problem in dimension one: from short-range to contact i nteractions. J. Math. Phys. , 59(7):072104, 18, 2018

work page 2018

[5] [4]

On the quantum mechani cal three-body problem with zero-range interactions

Giulia Basti and Alessandro Teta. On the quantum mechani cal three-body problem with zero-range interactions. In Functional analysis and operator theory for quantum physic s, EMS Ser. Congr. Rep., pages 71–93. Eur. Math. Soc., Zürich, 2 017

work page

[6] [5]

de Oliveira, and Alessandra A

Roxana Bedoya, César R. de Oliveira, and Alessandra A. Ve rri. Complex Γ-convergence and magnetic Dirichlet Laplacian in bounded thin tubes. J. Spectr. Theory , 4(3):621–642, 2014

work page 2014

[7] [6]

Universality in few-body sy stems with large scattering length

Eric Braaten and H-W Hammer. Universality in few-body sy stems with large scattering length. Physics Reports, 428(5-6):259–390, 2006

work page 2006

[8] [7]

On contact interactions as limits of short-range potentials

Gerhard Bräunlich, Christian Hainzl, and Robert Seirin ger. On contact interactions as limits of short-range potentials. Methods Funct. Anal. Topology, 19(4):364–375, 2013

work page 2013