{"id":"78037c59-1564-46e9-8c66-62e8e601eb8c","arxiv_id":"2509.02938","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"A Lippmann-Schwinger based scheme with Hermite recursion efficiently computes converged spectra for three/four 1D bosons and three 1D fermions with zero-range interactions, validated by Bose-Fermi mapping and free-space binding energies.","lead":"This paper develops a new numerical method to compute energy spectra of three and four identical 1D bosons and three identical 1D fermions in a harmonic trap. It reports the first direct calculation for three fermions without the Bose-Fermi mapping and the first full four-boson spectrum, validated against exact limits.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"I agree with the reader that empirical cutoff convergence is the weakest part of the paper, but I do not regard it as a load-bearing defect. The paper's strongest claims are independently corroborated: the FFF spectrum is validated by the Bose-Fermi mapping without using that mapping in the derivation, and the BBBB lowest branch is validated against the McGuire free-space tetramer energy. These checks make a systematic error in the derivation or implementation unlikely. The absence of a rigorous error bound and the lack of an explicit BBBB basis-size convergence study are limitations, but they are standard in numerical few-body work and do not rise to the level of rejecting or conditioning the central claim. The proposed larger-cutoff rerun would settle the only residual uncertainty about the four-boson spectrum.","tokens_in":23485,"tokens_out":13049,"duration_ms":155345,"concrete_test":"Recompute the BBBB spectrum of Fig. 4 with smax=tmax=kmax=ellmax=48 and pmax at least 256 over the same g+ range, then compare the 20 lowest eigenenergies; if any curve shifts by more than about 0.01 hbar omega, the four-boson convergence claim would need revision. As a secondary check, recompute the FFF spectrum using Boost precision for the full matrix without replacing k>ell elements and confirm that quantitative residuals against the mapped BBB spectrum stay below the 0.0005 hbar omega grid spacing used in Fig. 3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the Lippmann-Schwinger truncations (basis size and pmax) are adequate for the 20+ reported eigenstates and that the FFF derivation handles the derivative-delta regularization correctly. I found no internally inconsistent step and no demonstrated flaw. The BBB spectrum is checked against five basis sizes; the FFF spectrum is benchmarked against the Bose-Fermi-mapped BBB spectrum (Fig. 3); low-lying levels are checked against perturbation theory and degeneracy counting; and the BBBB tetramer branch tracks the McGuire free-space energy (Fig. 4). The C++, Python, and Mathematica implementations were cross-checked. The weakest spot is that the BBBB spectrum in Fig. 4 is shown at a single cutoff (smax=tmax=kmax=ellmax=32, pmax=192) with no basis-size comparison, and none of the spectra carry rigorous error bounds. This is a standard numerical-convergence limitation rather than a load-bearing correctness objection, and it does not undermine the central claim as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives and solves Lippmann-Schwinger eigenvalue equations for three identical bosons (BBB), three identical fermions (FFF), and four identical bosons (BBBB) in a one-dimensional harmonic trap with zero-range two-body interactions: an even-parity delta potential for bosons and an odd-parity derivative-delta-derivative potential for fermions. The eigenvalue equations are given in truncated harmonic-oscillator bases in Eqs. (21), (27), and (32), and the numerical scheme is built on a recursive evaluation of overlapped Hermite-polynomial integrals (Eq. (37)). The main results are energy spectra as functions of the coupling strength: the FFF spectrum is computed directly without invoking the Bose-Fermi mapping and is benchmarked against the mapped BBB spectrum (Fig. 3); the BBBB spectrum is presented as the first such Lippmann-Schwinger calculation, and its tetramer branch is compared with the McGuire free-space binding energy (Fig. 4). Additional checks include degeneracy counting and first-order perturbative slopes for weak coupling.","tokens_in":23750,"tokens_out":9319,"duration_ms":112544,"significance":"If correct, the paper establishes a stable and efficient Lippmann-Schwinger treatment of a derivative-regularized zero-range potential for more than two identical fermions and provides the first full four-boson trapped energy spectrum by this method. There are no fitted parameters, and the validation is genuinely independent: C++, Python, and Mathematica implementations are cross-checked; the FFF spectrum coincides with the Bose-Fermi-mapped BBB spectrum; low-lying levels follow perturbative slopes; and the BBBB tetramer branch tracks the free-space McGuire energy. These are substantive strengths. The principal caveat is numerical evidence: convergence is demonstrated in detail for the BBB matrix elements and for selected FFF matrix elements, but the BBBB spectrum itself is shown at only one cutoff. This is a missing-support issue rather than a demonstrated internal inconsistency, and it is fixable within the scope of the manuscript.","major_comments":[{"comment":"The BBBB spectrum is computed at a single truncation, smax=tmax=kmax=ellmax=32 with pmax=192, and no basis-size or pmax variation is shown for the four-boson eigenenergies. Section III.B asserts that the BBB and BBBB implementations are 'numerically stable and converged', but for BBBB the displayed evidence is one curve. The agreement with the McGuire free-space tetramer energy validates only the lowest bound branch; it does not validate the roughly 20 other eigenenergies claimed in the abstract. Because the BBBB spectrum is a central new result, please add a convergence study (e.g., overlay spectra for two additional cutoffs or provide a table of selected eigenenergies versus cutoff). This is a missing-support issue, not a demonstrated error.","section":"§IV, Fig. 4"}],"minor_comments":[{"comment":"A summation over j is missing on the right-hand side after the integration by parts; as written the expression depends on an undefined j. Later equations restore the sum, so this is an intermediate typo, but it should be corrected.","section":"Appendix A, Eq. (A21)"},{"comment":"The title and abstract promise determination of eigenstates, but Section IV explicitly shows only energy spectra. Either add representative eigenstate densities (e.g., for a few BBB/FFF/BBBB states) or soften the wording so the claim matches the presented results.","section":"Title/Abstract and §IV"},{"comment":"Please state precisely where the rule 'replace A(3F)_{ell,k} for k>ell by the converged A(3F)_{k,ell}' is applied (all off-diagonal pairs, all energies?) and give the tolerance used to judge the matrix elements converged in Fig. 1(b).","section":"§III.B, Fig. 1"},{"comment":"The recursion for W_{ijkℓ} is not accompanied by initial conditions or a statement of how negative subscripts are handled. Adding these details would make the implementation reproducible.","section":"Eq. (37)"},{"comment":"The caption says 'ℓmax/2 eigenenergies are shown'; briefly explain that this is because only even ℓ values are retained in the basis.","section":"Fig. 2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid numerical-methods contribution and the central derivation appears sound. My main reservation is the lack of convergence evidence for the four-boson spectrum, which is the paper's flagship new result. Once that is supplied, I would support acceptance. The FFF replacement procedure is partially mitigated by the Bose-Fermi mapping benchmark, though an explicit tolerance statement would help."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a solid, careful numerical-methods paper that delivers what it promises. The recursion-based W-integral scheme makes the Lippmann-Schwinger approach efficient enough to get converged spectra for 3 and 4 particles, and the direct FFF and BBBB spectra are genuinely new.\n\nThe strongest parts are the checks. The BBB spectrum reproduces D'Amico and Rontani; the FFF spectrum matches the Bose-Fermi-mapped BBB spectrum essentially perfectly (a non-trivial check, since the FFF calculation treats the derivative-delta regularization from scratch); the BBBB tetramer branch tracks the McGuire free-space energy; low-lying levels pass perturbation theory and degeneracy counting; and the C++, Python, and Mathematica implementations agree. For a paper whose main currency is numerical reliability, that is the right kind of evidence. The derivation of the FFF eigenequation in Appendix A is also careful about the discontinuities introduced by the odd-parity pseudopotential, which is exactly where a less careful treatment would go wrong.\n\nSoft spots, in proportion: The convergence analysis is empirical. The rule pmax+1 > 2(k+ell) is plausible and tested on selected matrix elements, but there are no rigorous error bounds, and the BBBB spectrum is shown at a single cutoff. That is not a load-bearing flaw, but a reader who wants to use a specific eigenvalue near the edge of the plotted range will have to trust the heuristic. The FFF implementation also explicitly replaces unstable A^(3F)_{ell,k} elements with their converged A^(3F)_{k,ell} counterparts; the paper shows both converge to the same number, so this is reasonable, but it is an ad hoc workaround. No code is released, which slows replication but is common in this subfield.\n\nI also do not share the worry about circularity in using the Bose-Fermi mapping as a benchmark. The mapping is a known theorem and is not what the paper is trying to prove; using it to check a direct FFF calculation is validation, not circular reasoning.\n\nVerdict: this is a paper for the few-body cold-atom crowd, and it deserves a real referee. I'd send it to peer review and expect acceptance after a revision that adds a bit more convergence documentation, especially for BBBB and for the highest-energy states claimed. The work is honest and the technical contribution is real.","headline":"Solid, well-benchmarked numerical method with first direct FFF and full BBBB spectra; minor convergence-documentation gaps do not undermine it.","tokens_in":24223,"tokens_out":2871,"would_cite":true,"duration_ms":32975,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that a Lippmann-Schwinger eigenvalue scheme, applied to derivative-delta interactions, yields stable and converged energy spectra for three identical fermions and four identical bosons in a 1D harmonic trap, with the four-bo","keywords":["Lippmann-Schwinger equation","one-dimensional quantum gases","harmonic confinement","odd-parity pseudopotential","identical fermions","identical bosons","Bose-Fermi mapping","few-body spectra"],"falsifier":"Take a low-lying FFF eigenvalue near E = 2.5 hbar omega, such as the one used in the convergence tests, and recompute it with basis ellmax = kmax = 100 and pmax much larger than 2(k+ell) using 50-digit precision; if the eigenvalue moves by more than the line thickness in Fig. 3, the claimed convergence is not real. Alternatively, compare the predicted BBBB tetramer energy at g+ = -10 hbar omega aho with an independent exact-diagonalization or Monte Carlo calculation; agreement to quoted precision would settle it.","tokens_in":23392,"feed_emoji":"⚛️","tokens_out":5572,"duration_ms":57916,"temperature":0.7,"pith_summary":"This paper claims that the Lippmann-Schwinger equation, applied to few atoms with zero-range interactions, can be made into an efficient and numerically stable scheme for three and four identical particles in a one-dimensional harmonic trap. For three identical fermions the interaction is a derivative delta function, and the paper shows that this odd-parity potential leads to well-defined eigenvalue equations without invoking the Bose-Fermi mapping. It also presents what it says are the first full energy spectra of this kind for four identical bosons. If correct, the fermionic spectrum independently reproduces the Bose-Fermi mapping relation g+ = -2 hbar^4/(m^2 g-), and the four-boson bound tetramer matches the free-space McGuire energy. The wider point is that a computationally cheap route now exists to few-body spectra over a broad interaction range, not just at the Tonks-Girardeau limits.","feed_headline":"Direct solver produces stable 1D three-fermion spectra","feed_subtitle":"Lippmann-Schwinger equations with derivative-delta interactions now converge, matching Bose-Fermi mapping predictions.","key_machinery":"The central objects are the Lippmann-Schwinger eigenvalue equations (21), (27), and (32), in which 1/g is treated as an eigenvalue of an energy-dependent matrix. The matrix elements combine the two-body Green's function at zero separation with integrals over harmonic-oscillator states (I^(3B), J^(3F), K^(3F), I^(4B)). These integrals are rewritten, using the Hermite argument-scaling identity (44) and the product-separation identity (52), as linear combinations of the four-index recursion W (37), which is evaluated once and reused for every energy. For the fermion case, an integration-by-parts step (Appendix A) removes the discontinuity of the derivative Green's function so that a smooth inte","core_discovery":"Starting from the Lippmann-Schwinger equation with the harmonic-oscillator Green's function, the authors reduce the three- and four-body problems to finite matrix eigenvalue problems whose energy-dependent matrix elements involve two-body Green's functions and integrals of products of Hermite polynomials. For three identical fermions, the derivative operators in the interaction are handled through an integration-by-parts resolution of the discontinuity at zero separation, giving the eigenvalue equation (27). The paper's central numerical claim is that these equations converge: with cutoffs such as ellmax=kmax=60 and pmax=240 for three fermions, and smax=tmax=kmax=ellmax=32 with pmax=192 for","pith_inferences":["A natural extension the paper does not develop is time-dependent dynamics: since the scheme returns eigenstates as well as eigenenergies, evolving an initial state in this basis would give few-body quench dynamics with the same computational machinery.","The observed numerical instability in A^(3F) for k>ell, remedied by symmetry replacement, suggests that a more symmetric formulation of the fermion matrix element might eliminate the need for high-precision arithmetic entirely.","If the convergence heuristic pmax+1 > 2(k+ell) is as universal as the tests suggest, it could be converted into an adaptive cutoff rule for larger N, where full convergence checks become expensive.","The method's explicit two-body correlations plus full harmonic-oscillator basis appear to capture three-body correlations effectively, as the tetramer match shows; testing this against a dedicated three-body-correlation measure for the trimer branch would sharpen what the basis is actually learning."],"forward_implications":["Three identical fermions can now be treated directly, without relying on the Bose-Fermi mapping, so the odd-parity pseudopotential is confirmed to give well-defined few-body matrix elements.","The four-boson spectrum provides the first such calculation for this system, including a clear tetramer branch that matches the free-space McGuire binding energy on the strongly attractive side.","Because the energy-independent sums are precomputed, spectra over broad coupling ranges cost minutes to hours on a laptop, making extensions to N=5 plausible.","The same eigenvalue-equation structure can be reused for other parity sectors, Bose-Bose and Fermi-Fermi mixtures, and anyonic systems via the bosonic-fermionic anyon mappings."],"supporting_citations":[{"why":"Supplies the Lippmann-Schwinger equation and Green's function formalism on which all the eigenvalue equations are built.","marker":"[17]"},{"why":"Supplies the odd-parity derivative pseudopotential and the relation between the two-fermion and two-boson Green's functions used in the FFF matrix elements.","marker":"[18]"},{"why":"Provides the Bose-Fermi mapping relation used to benchmark the FFF spectrum against the BBB spectrum.","marker":"[32]"},{"why":"Earlier Lippmann-Schwinger treatment of trapped three-body systems whose alternative integral evaluation is used as a benchmark.","marker":"[53]"},{"why":"Provides the two-component FFF' and FFF'F' eigenvalue equations that the authors adapt to benchmark their numerical approach.","marker":"[54]"},{"why":"Previously derived three-boson eigenvalue equation that the BBB calculation extends and benchmarks against.","marker":"[57]"},{"why":"Recursion formulas for integrals of Hermite polynomial products, the key computational tool that makes the scheme efficient.","marker":"[58]"},{"why":"Free-space three- and four-boson binding energies used to benchmark the trapped trimer and tetramer branches.","marker":"[63]"},{"why":"Existing variational treatment of one-dimensional few-fermion systems that this direct calculation complements.","marker":"[40]"}],"fun_headline_variants":["Efficient finite-matrix solver for 1D few-body spectra (N=3-4)","Convergent finite-matrix method solves 1D few-body spectra","Efficient eigenenergy solver for 1D bosons and fermions (N=3-4)","1D few-body spectra from a convergent finite-matrix approach","Stable 1D few-body spectra from finite-matrix equations"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The spectra are as converged as claimed; the finite basis size and pmax cutoffs, plus the rule that replaces numerically unstable FFF matrix elements with their symmetric partners, are enough to capture every eigenstate shown, and no rigorous error bound is given.","fun_headline_variants_meta":{"raw":{"variants":["Efficient finite-matrix solver for 1D few-body spectra (N=3-4)","Convergent finite-matrix method solves 1D few-body spectra","Efficient eigenenergy solver for 1D bosons and fermions (N=3-4)","1D few-body spectra from a convergent finite-matrix approach","Stable 1D few-body spectra from finite-matrix equations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001023,"raw_usage":{"total_tokens":4095,"prompt_tokens":630,"completion_tokens":3465,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":374,"completion_tokens_details":{"reasoning_tokens":3361}},"tokens_in":374,"tokens_out":3465,"duration_ms":28316,"temperature":1.0,"reasoning_tokens":3361,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:15:45.205026+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a low-lying FFF eigenvalue near E = 2.5 hbar omega, such as the one used in the convergence tests, and recompute it with basis ellmax = kmax = 100 and pmax much larger than 2(k+ell) using 50-digit precision; if the eigenvalue moves by more than the line thickness in Fig. 3, the claimed convergence is not real. Alternatively, compare the predicted BBBB tetramer energy at g+ = -10 hbar omega aho with an independent exact-diagonalization or Monte Carlo calculation; agreement to quoted precision would settle it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Lippmann-Schwinger equation and Green's function formalism on which all the eigenvalue equations are built."},{"cited_title":"Kanjilal and D","cited_arxiv_id":null,"evidence_quote":"Supplies the odd-parity derivative pseudopotential and the relation between the two-fermion and two-boson Green's functions used in the FFF matrix elements."},{"cited_title":"Girardeau, H","cited_arxiv_id":null,"evidence_quote":"Provides the Bose-Fermi mapping relation used to benchmark the FFF spectrum against the BBB spectrum."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier Lippmann-Schwinger treatment of trapped three-body systems whose alternative integral evaluation is used as a benchmark."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the two-component FFF' and FFF'F' eigenvalue equations that the authors adapt to benchmark their numerical approach."},{"cited_title":"D’Amico and M","cited_arxiv_id":null,"evidence_quote":"Previously derived three-boson eigenvalue equation that the BBB calculation extends and benchmarks against."},{"cited_title":"Derivation of recursive formulas for integrals of hermite polynomial products and their applications,","cited_arxiv_id":"2411.15541","evidence_quote":"Recursion formulas for integrals of Hermite polynomial products, the key computational tool that makes the scheme efficient."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Free-space three- and four-boson binding energies used to benchmark the trapped trimer and tetramer branches."},{"cited_title":"Ko´ scik and T","cited_arxiv_id":null,"evidence_quote":"Existing variational treatment of one-dimensional few-fermion systems that this direct calculation complements."}],"review_version":1}