REVIEW 3 major objections 5 minor 11 references
Matrix representation of the resolvent operator in square-integrable basis and physical application
T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [Section 3, Eqs. (22)-(25)] 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 3, Eqs. (17) and (21)] 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.
- [Abstract and Conclusion (computational cost claim)] 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.
minor comments (5)
- [Section numbering] 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.
- [References] Reference [9] is a duplicate of reference [4] (same authors, title, journal, and page numbers). Please remove the duplicate and renumber the references accordingly.
- [Equation (16)] 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.
- [Paragraph after Eq. (18)] 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.
- [Equation (35)] 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.
Circularity Check
No significant circularity: the resolvent and eigenvector-component formulas are derived identities, and the self-citations are historical or comparative rather than load-bearing.
full rationale
The derivation chain is self-contained. Equations (11) and (12) follow from the spectral decomposition of H and the inverse of a finite matrix in the chosen basis; Eqs. (17), (18), and (21) are obtained by applying the determinant identity (16) to that finite matrix inverse, with no fitted parameter and no imported ansatz. The byproduct formulas (22)-(25) are obtained by equating the two representations of G^N_{nm}(z) at z = eps_k, which is a legitimate finite-dimensional algebraic step rather than a renaming of an input. The statement that formula (23) was recently rediscovered by Mitnik and Mitnik and had been used earlier in the author's J-matrix work is a historical remark, not a premise of the derivation. The physical application compares resonance and bound-state energies against literature values, including external references [4-9] as well as the author's own Ref. [10]; this comparison does not make the central formulas depend on those references. The existence and completeness assumptions in Section 1 are explicitly stated assumptions, not hidden inputs, and the possible degeneracy caveat for formulas (22)-(25) is a validity condition rather than a circularity. Consequently, no circular step can be exhibited by equation or by definition, and the only relevant self-citation is minor and non-load-bearing.
Assumptions & free parameters
free parameters (3)
- lambda (basis scale parameter) =
1.0, 20 (chosen per example)
- N (basis dimension) =
5, 20, 60, 100 (per figure)
- analytic continuation fit function F(z) =
unspecified
assumptions (4)
- domain assumption H admits a resolvent (H-z)^{-1} in some region of the complex plane, i.e., the spectral set is not 'too large'.
- domain assumption The basis {psi_n} is complete and square-integrable, supporting a Hermitian matrix representation.
- standard math The determinant identity (16) relating inverse matrix elements to eigenvalues of submatrices holds for the generalized matrices H - z Omega.
- domain assumption The J-matrix asymptotic recursion (31) and boundary conditions (32) correctly describe the infinite tail of the wavefunction.
Cite this review
Pith. "Pith review of Matrix representation of the resolvent operator in square-integrable basis and physical application." pith.science (2026). https://pith.science/paper/OR2NEG27
@misc{pith2026241117736,
author = {Pith},
title = {Pith review of: Matrix representation of the resolvent operator in square-integrable basis and physical application},
year = {2026},
howpublished = {\url{https://pith.science/paper/OR2NEG27}},
note = {Machine review of arXiv:2411.17736}
}
read the original abstract
We obtain simple formulas for the matrix elements of the resolvent operator (the Green's function) in any finite set of square integrable basis. These formulas are suitable for numerical computations whether the basis elements are orthogonal or not. A byproduct of our findings is an expression for the normalized eigenvectors of a matrix in terms of its eigenvalues. We give a physical application as an illustration of how useful these results can be.
Reference graph
Works this paper leans on
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[1]
D. M. Mitnik and S. A. H. Mitnik, Wavefunctions from energies: Applications in simple potentials, J. Math. Phys. 61, 062101 (2020) and references therein
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[2]
A. D. Alhaidari, E. J. Heller, H. A. Yamani, and M. S. Abdelmonem (Eds.), The J-Matrix Method: Developments and Applications (Springer, Dordrecht, Netherlands, 2008)
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H. A. Yamani and M. S. Abdelmonem, Characterization of resonances using an exact model S-matrix, J. Phys. A: Math. Gen. 28 (1995) 2709
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S. A. Sofianos and S. A. Rakityansky, Exact method for locating potential resonances and Regge trajectories, J. Phys. A: Math. Gen. 30 (1997) 3725
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C. H. Maier, L. S. Cederbaum and W. Domcke, A spherical-box approach to resonances, J. Phys. B: At. Mol. Phys. 13 (1980) L119
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Show all 11 references
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[10]
A. D. Alhaidari, Unified algebraic treatment of resonance, Int. J. Mod. Phys. A 20 (2005) 2657
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[11]
H. A. Yamani, M. S. Abdelmonem, and A. D. Al-Haidari, Extracting density information from finite Hamiltonian matrices, Phys. Rev. A 62 (2000) 052103
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[12]
A. D. Alhaidari, Gauss Quadrature for Integrals and Sums, Int. J. Pure Appl. Math. Res. 3 (2023) 1 −12− Figures Captions Fig. 1: Plot of 1 ( )SE− for the system associated with 2( ) 7.5 rV r r e −= for 0= and 0Z = . We took the computational parameters 1 = and 60N = . The fig...
2023
Reviewed August 12, 2026 · model on record in the stance chip above.
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