{"id":"67c308c0-9515-4ac4-8f34-baafd35b4b6c","arxiv_id":"2411.17736","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper gives eigenvalue-only formulas for resolvent matrix elements and eigenvector component squares in both orthogonal and non-orthogonal bases, with a J-matrix application to resonances.","lead":"This paper derives formulas for computing matrix elements of the Green's function (resolvent) in any finite square-integrable basis, orthogonal or not, and uses them to find resonance and bound-state energies for model potentials. The formulas recast known spectral representations using only eigenvalues, and the paper applies them to J-matrix scattering calculations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eigenvector formulas (22)-(25) silently assume a simple spectrum; for degenerate matrices they reduce to 0/0 and are undefined, contradicting the advertised generality.","rationale":"The reader's weakest_assumption focuses on the infinite-dimensional completeness and resolvent-existence assumption used when passing from the operator to finite matrices. That is a legitimate concern for the physical application, but it does not touch the finite-N algebraic identities that constitute the central claim. The load-bearing gap I identify is internal to the finite-dimensional derivation: formulas (22)-(25) are derived by taking residues at the eigenvalues of H, and residues only isolate individual eigenvector components when the eigenvalues are simple. For degenerate spectra the individual components are not even well-defined without specifying the basis of the eigenspace, and the formulas collapse to 0/0. This is a concrete, checkable limitation of the advertised byproduct, and it should be stated as a hypothesis. The main resolvent identities (17) and (21) are likely correct for non-degenerate matrices, and the issue is fixable by adding a simple-spectrum condition, so the existing CONDITIONAL verdict remains appropriate. I also note a secondary concern: formula (17) requires invertibility of the non-principal submatrix Omega^{(n,m)} in the non-orthogonal case, which is not guaranteed even when Omega is positive definite, but the degeneracy problem is the more fundamental and more likely to affect users.","tokens_in":11973,"tokens_out":13823,"duration_ms":115351,"concrete_test":"Apply formula (23) to the 2x2 identity matrix H = I_2, whose eigenvalues are eps_0 = eps_1 = 1. The numerator det(H^{(0,0)} - 1) = 0 and the denominator prod_{j != 0} (eps_j - 1) = 0, so the expression is 0/0 and gives no value. Yet the true squared component |Gamma_{0,0}|^2 is basis-dependent inside the degenerate eigenspace (it can be 1 or 0 for different valid orthonormal eigenbases). A complementary test: perturb H(delta) = diag(1, 1+delta) and let delta -> 0; the limit of the right-hand side of (23) is not unique unless the eigenvector basis is held fixed, confirming that eigenvalue data alone cannot determine individual eigenvector components when eigenvalues coalesce.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central byproduct claim is that formulas (22)-(25) give normalized eigenvector components from eigenvalues alone. These formulas are obtained by evaluating two expressions for G^N_{nm}(z) at z = eps_k, i.e., by extracting the residue at a pole of the resolvent. That extraction is legitimate only when eps_k is a simple eigenvalue. If H has an eigenvalue of multiplicity r > 1, the residue is the projection onto the degenerate eigenspace, not the individual product Gamma_{n,k} Gamma_{m,k}. On the right-hand sides of (21)-(25), the factor prod_{j != k} (eps_j - eps_k) appears in the denominator and vanishes under degeneracy; the numerator also vanishes because, by the interlacing theorem, a submatrix H^{(n,m)} inherits the degenerate eigenvalue. Thus formulas (21)-(25) are indeterminate for repeated eigenvalues. The paper never states the required non-degeneracy condition. Since repeated eigenvalues are common in physical systems with symmetry, the advertised generality of 'an expression for the normalized eigenvectors of a matrix in terms of its eigenvalues' fails for a substantial class of matrices.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives finite-dimensional matrix representations of the resolvent (Green's function) for a self-adjoint operator in a square-integrable basis, allowing for non-orthogonal bases. The central claims are formulas (17) and (21), which express the finite resolvent matrix element G^N_{nm}(z) in terms of the eigenvalues of the Hamiltonian matrix and its (N-1)-dimensional submatrices, and a byproduct set of formulas (22)-(25) giving squared eigenvector (or generalized eigenvector) components from eigenvalues alone. The paper applies these results to the J-matrix method for scattering resonances, bound-state energies, and the density of states of a Coulomb-plus-short-range potential, reporting agreement with published data. The derivation in Sections 2 and 3 is plausible and the final forms match known determinantal identities, but several key steps are asserted without proof, and the eigenvector byproduct is not valid as stated for degenerate spectra.","tokens_in":12173,"tokens_out":4576,"duration_ms":43899,"significance":"If the formulas are correct, they are useful: they provide an alternative to eigenvector computation for selected Green's function matrix elements, and the explicit treatment of non-orthogonal bases is a practical extension. The physical application reproduces known resonance parameters (Table 1, Figs. 3, 5, and 6), which gives confidence in the algebra. The paper credits earlier work on formula (23) and is transparent about prior use in the J-matrix context. However, the advertised byproduct—'an expression for the normalized eigenvectors of a matrix in terms of its eigenvalues'—is stated without the necessary non-degeneracy restriction, and the claimed computational-cost reduction is not quantified. These gaps affect the paper's central claims, though they appear to be fixable within the manuscript's scope.","major_comments":[{"comment":"The formulas for squared eigenvector components are obtained by evaluating two expressions for G^N_{nm}(z) at z = ε_k, i.e., by extracting the residue at a pole of the resolvent. This is legitimate only when ε_k is a simple eigenvalue. If the spectrum is degenerate, the residue of the resolvent is the projection onto the degenerate eigenspace, and the right-hand sides of (22)-(25) become indeterminate 0/0 because both the numerator and the denominator contain the factor ∏_{j≠k}(ε_j - ε_k), which vanishes. The paper never states a non-degeneracy assumption, so the advertised byproduct fails for matrices with repeated eigenvalues, which are common in symmetric physical systems. The authors should either state and justify the simple-spectrum assumption explicitly or provide a limiting procedure for degenerate cases.","section":"Section 3, Eqs. (22)-(25)"},{"comment":"The derivation of the central formulas is not supplied. Equation (17) is introduced with 'it is easy to show' and equation (21) with 'we can finally write,' but no proof or intermediate steps are given. Since these formulas are the main results of the paper, the derivation should be presented at least in outline, using identity (16) and the determinant identities explicitly. The reader should not have to reconstruct the argument from the typeset expressions, which are also partially garbled.","section":"Section 3, Eqs. (17) and (21)"},{"comment":"The paper repeatedly claims that formulas (17) and (21) 'reduce the computational cost' because they use eigenvalues rather than eigenvectors. This is not self-evident: computing the eigenvalues of the (N-1)×(N-1) submatrices H^{(n,m)} (or H^{(n,m)} - ε_j I) for all required pairs (n,m) is in general more expensive than a single full diagonalization of the N×N matrix H, which costs O(N^3). For a fixed (n,m) the cost may be comparable, and the application in Section 4 indeed uses only a single component, but the general claim needs quantification or moderation.","section":"Abstract and Conclusion (computational cost claim)"}],"minor_comments":[{"comment":"The Introduction promises a conclusion in Section 5, but the concluding section is labeled '4. Conclusion' immediately after '4. Physical application.' Please renumber the sections so that the conclusion is Section 5.","section":"Section numbering"},{"comment":"Reference [9] is a duplicate of reference [4] (same authors, title, journal, and page numbers). Please remove the duplicate and renumber the references accordingly.","section":"References"},{"comment":"The display of identity (16) is garbled (e.g., 'c C C C' and misplaced brackets). Please typeset the adjugate/determinant identity properly so that its role in the subsequent derivation is clear.","section":"Equation (16)"},{"comment":"The statement that formula (18) is avoided for n≠m because of the 'functional dependence of the eigenvalue ω_i^{(n,m)}(z)' could be expanded: the point is that recomputing these z-dependent eigenvalues for each energy is costly, whereas formula (21) uses fixed submatrix eigenvalues. A sentence explaining this trade-off would help the reader.","section":"Paragraph after Eq. (18)"},{"comment":"In Eq. (35), the symbols τ_i and ε_i are used without reintroduction after their definition in Eq. (17). Please define these explicitly as the eigenvalues of the truncated overlap and Hamiltonian matrices for the specific (N-1,N-1) case.","section":"Equation (35)"}],"recommendation":"major_revision","confidential_remarks":"The paper is essentially a technical derivation with an application; the novelty is moderate because Eq. (23) is explicitly acknowledged as known, but the non-orthogonal versions and the unified treatment may be useful. The main concern is the unqualified eigenvector-byproduct claim in the degenerate case, which is a genuine correctness gap. The missing derivations of (17) and (21) and the unquantified cost claim also need attention. I believe the issues are fixable and do not warrant rejection, provided the authors add the non-degeneracy assumption (or a degenerate-case treatment) and expand the derivations."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the non-orthogonal version of the resolvent formulas: equations (17), (24), and (25) express finite Green's function matrix elements and squared generalized eigenvector components through eigenvalues of the Hamiltonian and overlap submatrices. I checked the orthonormal counterparts: (12), (18), (21)-(23) are essentially known spectral identities, and the paper credits the recent Mitnik-Mitnik result and its own earlier J-matrix work honestly. The physical application is credible: the J-matrix scattering calculation reproduces known resonance positions, bound states, and density of states to the expected precision, with the basis parameters stated.\n\nThe main soft spot is the overclaim in the abstract and Section 3 about 'the normalized eigenvectors of a matrix in terms of its eigenvalues.' The formulas give squared components, not the eigenvectors themselves, and they silently assume a simple spectrum. Look at (22)-(25): the right-hand side has prod_{j != k}(eps_j - eps_k) in the denominator. For a degenerate eigenvalue, both numerator and denominator vanish, and the residue extraction at a pole of order >1 is not legitimate. The paper never states the non-degeneracy condition. That is a real flaw, not a nitpick, and it should be fixed by adding the assumption and discussing the degenerate case separately. It matters because symmetric physical systems routinely have repeated eigenvalues.\n\nTwo smaller issues. First, the key non-orthogonal formula (17) and the orthonormal (21) are introduced with 'it is easy to show' and no derivation; given that the paper's contribution is exactly these identities, a reader shouldn't have to reconstruct them. Second, there is no code or data, and the density-of-states section relies on an unspecified analytic-continuation fit function F(z). That weakens reproducibility but is not fatal for a derivation paper.\n\nOverall: the mathematics is defensible, the new non-orthogonal piece is real, and the application is a useful demonstration. With the degeneracy caveat stated and the derivations filled in, it would be a solid contribution to the J-matrix / spectral-methods literature. I'd send it to a serious referee, mainly to verify the derivations and push on the degenerate case. I wouldn't cite it in my own near-term work, but I'd put it on the reading-group list if anyone cares about spectral representations.","headline":"A competent derivation of resolvent formulas for non-orthogonal bases, with a real overclaim about eigenvectors and a missing degeneracy caveat that should be fixed before publication.","tokens_in":12710,"tokens_out":2497,"would_cite":false,"duration_ms":23083,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Finite-basis resolvent matrix elements can be written purely in terms of eigenvalues, which yields Green's functions, eigenvector magnitudes, and physical spectra without constructing eigenvectors.","keywords":["resolvent operator","Green's function","eigenvalue-only formulas","square-integrable basis","non-orthogonal basis","generalized eigenvalue problem","resonance energies","density of states"],"falsifier":"Take a small Hermitian matrix $H$ (say dimension 4 or 5) with well-separated eigenvalues, compute the matrix element $G^N_{nm}(z)$ by direct inversion of $H-zI$, and compare it with formulas (17), (18), and (21) at several complex $z$ and off-diagonal pairs $(n,m)$; if any of the eigenvalue-only expressions disagrees beyond round-off, the central identity is false.","tokens_in":11755,"feed_emoji":"🧮","tokens_out":13556,"duration_ms":106380,"temperature":0.7,"pith_summary":"This paper derives exact formulas for the matrix elements of the resolvent operator --- the inverse $(H-z)^{-1}$, whose matrix elements are the Green's function --- in a finite set of square-integrable (normalizable) basis functions. The central point is that these matrix elements can be evaluated purely from eigenvalues: eigenvalues of the finite Hamiltonian matrix, of its $(N-1) \\times (N-1)$ submatrices, and, in the non-orthogonal case, of the overlap matrix. That removes the need to construct eigenvectors when computing Green's functions, lowering the numerical cost. A byproduct is a recipe for the squared magnitudes of eigenvector components from eigenvalue sets alone, in both orthogonal and non-orthogonal bases. The paper demonstrates the formulas by computing resonance energies, bound-state energies, and the density of states for model short-range potentials.","feed_headline":"Green's function entries come from eigenvalues alone","feed_subtitle":"Eigenvalue-only formulas give Green's functions, eigenvector magnitudes, and resonances in any square-integrable basis.","key_machinery":"The load-bearing object is the finite resolvent matrix element $G^N_{nm}(z)$, together with determinant identity (16), which expresses the cofactor-to-determinant ratio of a matrix as a quotient of products of the eigenvalues of the matrix and of its $(n,m)$ submatrix. This identity is what converts the inverse-matrix form of the Green's function into ratios of eigenvalue products, eliminating eigenvectors from the calculation. In a non-orthogonal basis the same machinery runs on the generalized eigenvalue problem $H\\Gamma = \\Omega\\Gamma \\varepsilon$, with the overlap matrix $\\Omega$ supplying the non-orthogonality corrections; in an orthonormal basis the overlap matrix is the identity and the formulas simplify accordingly.","core_discovery":"The central claim is that the finite-basis resolvent matrix element $G^N_{nm}(z)$ can be written as a product and quotient of eigenvalue factors only. Formulas (17) and (21) give $G^N_{nm}(z)$ in terms of the eigenvalues of the Hamiltonian matrix $H$, the eigenvalues of the submatrix $H^{(n,m)}$ obtained by deleting row $n$ and column $m$, and, for non-orthogonal bases, the eigenvalues $\\tau$ of the overlap matrix $\\Omega$ and its submatrices. For orthonormal bases, the diagonal element is formula (19) and the off-diagonal element is formula (21). Evaluating these formulas at $z=\\varepsilon_k$ produces the squared eigenvector-component magnitudes (22)--(25), so the entries of the (generalized) eigenvector matrix are fixed by spectra alone. The physical application builds the finite Green's function from these eigenvalue-only expressions to extract scattering phase shifts, resonance positions, bound states, and density of states for short-range potentials, with computed resonances matching published values.","pith_inferences":["Not stated in the paper: the same ratio-of-eigenvalue structure should apply to other functions of $H$ that are defined through resolvents, such as spectral projections or the unitary time-evolution operator, when restricted to a finite basis.","Not stated in the paper: formulas (22)--(25) suggest an inverse-spectral use, namely constraining a matrix from its eigenvalue sets alone, which could matter when only spectral measurements are available.","Not stated in the paper: the numerical examples avoid degenerate or nearly degenerate eigenvalues, so treating the zero-determinant limit of the eigenvalue products is a natural stress test that the examples do not cover."],"forward_implications":["Any finite-basis calculation that already knows the spectrum can produce diagonal and off-diagonal Green's matrix elements without computing eigenvectors, so resolvent-based quantities become cheaper in large truncated-basis calculations.","For non-orthogonal bases, the generalized eigenvalue form of the formulas means non-orthogonality does not force an explicit eigenvector solve: the overlap matrix eigenvalues carry the correction.","Squared eigenvector component magnitudes follow from eigenvalue data alone, which gives a direct way to interpret truncated-basis wavefunctions, for example where a state is concentrated.","Resonance energies, bound-state energies, and density of states extracted from the finite Green's function are consistent with published values for the tested model potentials, so the formulas can serve as a practical scattering-toolbox ingredient."],"supporting_citations":[{"why":"Supplies the recently rediscovered eigenvalue-only formula for diagonal eigenvector components that the paper re-derives and extends to off-diagonal and non-orthogonal cases.","marker":"[1]"},{"why":"Supplies the recursion relations and boundary conditions that turn the finite Green's function into scattering phase shifts in the application.","marker":"[3]"},{"why":"Provides reference resonance energies for one of the model potentials, used as the benchmark the computed resonance is compared against.","marker":"[8]"},{"why":"Provides the table of resonance energies used as the comparison baseline for the charge-dependent and angular-momentum-dependent resonances.","marker":"[10]"},{"why":"Supplies the analytic-continuation techniques used to extract the density of states from the finite Green's function.","marker":"[11]"},{"why":"Supplies the Gauss quadrature method used to evaluate the potential matrix elements in the application.","marker":"[12]"}],"fun_headline_variants":["Green's functions from eigenvalues alone, no vectors","Eigenvalues alone give resolvent matrix entries","Spectra determine Green's function, eigenvector magnitudes, and resonances","Finite-basis Green's function without explicit eigenvectors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Hamiltonian has a resolvent in some region of the complex plane and that the chosen square-integrable basis is complete, so that the finite-$N$ formulas converge to the true operator Green's function as $N$ is increased.","fun_headline_variants_meta":{"raw":{"variants":["Green's functions from eigenvalues alone, no vectors","Eigenvalues alone give resolvent matrix entries","Spectra determine Green's function, eigenvector magnitudes, and resonances","Finite-basis Green's function without explicit eigenvectors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000889,"raw_usage":{"total_tokens":3774,"prompt_tokens":824,"completion_tokens":2950,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":440,"completion_tokens_details":{"reasoning_tokens":2885}},"tokens_in":440,"tokens_out":2950,"duration_ms":21576,"temperature":1.0,"reasoning_tokens":2885,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:09:10.101732+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small Hermitian matrix $H$ (say dimension 4 or 5) with well-separated eigenvalues, compute the matrix element $G^N_{nm}(z)$ by direct inversion of $H-zI$, and compare it with formulas (17), (18), and (21) at several complex $z$ and off-diagonal pairs $(n,m)$; if any of the eigenvalue-only expressions disagrees beyond round-off, the central identity is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the recently rediscovered eigenvalue-only formula for diagonal eigenvector components that the paper re-derives and extends to off-diagonal and non-orthogonal cases."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the recursion relations and boundary conditions that turn the finite Green's function into scattering phase shifts in the application."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides reference resonance energies for one of the model potentials, used as the benchmark the computed resonance is compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the table of resonance energies used as the comparison baseline for the charge-dependent and angular-momentum-dependent resonances."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the analytic-continuation techniques used to extract the density of states from the finite Green's function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Gauss quadrature method used to evaluate the potential matrix elements in the application."}],"review_version":1}