REVIEW 2 major objections 4 minor 7 references
On the application of the Rayleigh-Ritz method to a projected Hamiltonian
T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read For a projected Hamiltonian, the Rayleigh-Ritz method gives lower bounds in most cases and exact zero eigenvalues—the reverse of the usual upper-bound rule.
desk verdict A sound, narrowly scoped correction to Ding et al.'s projected-Hamiltonian variational claim, overstating the generality of the lower-bound pattern. 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 central object is the projected Hamiltonian $H_D=\sum_{k=1}^{D} E_k |\psi_k\rangle\langle\psi_k|$, a finite-rank operator on an infinite-dimensional Hilbert space, treated by the RRM through the secular determinant $|H_D - W S|=0$ with overlap matrix $S_{ij}=\langle u_i|u_j\rangle$. Two mechanisms carry the argument. First, because $H_D$ has rank $D$, its $N\times N$ matrix representation has at least $N-D$ zero eigenvalues for any $N>D$, and these zeros are obtained exactly. Second, for a trial state orthogonal to every eigenstate except one, the Cauchy-Schwarz inequality shows the expectation value lies below or above the eigenvalue depending on the sign of that eigenvalue; this is what allows lower bounds to appear. The non-orthogonal polynomial basis $u_i(x)=x^i(1-x)$ supplies the explicit numerical demonstration.
What would settle it
For the same particle-in-a-box projected Hamiltonian with $D=3$, repeat the Rayleigh-Ritz calculation with a different complete basis that respects the boundary conditions, for example $u_i(x)=x^i(1-x)^2$; if the lowest eigenvalue $W_1$ for $N=2$ or $N=4$ lies above $E_1=\pi^2/2$, the paper's claim that lower bounds occur in most cases would be shown to be basis-dependent rather than generic.
Extended reading notes
Core claim
The central discovery is that the Rayleigh-Ritz method, applied to a projected Hamiltonian $H_D=\sum_{k=1}^{D} E_k |\psi_k\rangle\langle\psi_k|$ formed from the $D$ lowest eigenstates of a Hermitian operator, does not obey the standard variational upper-bound theorem. For the toy model $H=-\frac12 \frac{d^2}{dx^2}$ on $[0,1]$ with basis $u_i(x)=x^i(1-x)$, the RRM eigenvalues $W_k$ approach the nonzero eigenvalues $E_k$ from below for most $N$, while the remaining $N-D$ eigenvalues are exactly zero. Exceptions do occur: for $D=3$ and $N=1$ the lowest RRM value is an upper bound, and for $D=4$ and $N=3$ the second eigenvalue is an upper bound; for $N>3$ the convergence is from below. The paper proves that for a trial state orthogonal to all but one eigenstate, the expectation value lies below or above the eigenvalue according to the sign of that eigenvalue (via the Cauchy-Schwarz inequality), and it shows numerically that the pattern survives an energy shift and arbitrary real coefficients $\alpha_k$ in the projection. The author's conclusion is that for this kind of projected Hamiltonian the RRM yields both upper and lower bounds, with the lower-bound direction predominant.
Load-bearing premise
The paper's empirical conclusion that the Rayleigh-Ritz eigenvalues converge from below 'in most cases' for projected Hamiltonians rests on the assumption that the particle-in-a-box model with polynomial trial functions is representative of general behavior; no theorem guarantees this, and the observed upper bounds for small basis sizes show the effect is not universal.
Editorial extensions
If this is right
- For any $N>D$, the projected Hamiltonian's $N-D$ zero eigenvalues are obtained exactly by the RRM, even though the corresponding wavefunctions only become exact in the $N\to\infty$ limit.
- The direction of the variational bound is tied to the sign of the target eigenvalue: via the Cauchy-Schwarz argument, a trial state overlapping only one eigenstate gives a lower bound for positive eigenvalues and an upper bound for negative ones.
- Adding a constant energy shift can flip the bound direction; with the shift $-5$, the lowest RRM eigenvalue becomes an upper bound while the higher ones remain lower bounds.
- If the projected identity $I_D$ replaces the full identity in the overlap matrix, the RRM gives upper bounds for $1\le N<D$ and exact eigenvalues when $N=D$.
- The usual RRM upper-bound proofs fail for projected Hamiltonians because the trial functions cannot be expressed in the finite eigenbasis of $H_D$; the numerical evidence indicates a new, weaker bound direction applies.
Reading between the lines
- The exact reproduction of the $N-D$ zero eigenvalues suggests a general rank-imposed constraint: for any finite-rank operator, a variational subspace of dimension $N>D$ will always contain a null space of dimension at least $N-D$, independent of basis quality; this could serve as a diagnostic in truncated configuration-interaction or density-matrix embeddings.
- Because the bound direction depends on shifting the zero of energy, the apparent 'lower-bound rule' is not an intrinsic property of the projected operator but an artifact of its spectrum's position; adding a large positive constant would restore the familiar upper-bound regime, so any test of this behavior must specify the energy reference.
- If the ensemble-state variational principle built on projected Hamiltonians inherits this lower-bound direction, variational optimization over ensemble weights could converge from below, which would change how convergence is monitored in such calculations.
- A natural next step is to test the pattern on degenerate or quasi-degenerate eigenstates; the rank argument and Cauchy-Schwarz bound still apply, but the ordering of the eigenvalues may become ambiguous.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the Rayleigh-Ritz method (RRM) applied to a projected Hamiltonian H_D = sum_{k=1}^D E_k |psi_k><psi_k|, where |psi_k> are eigenstates of an infinite-dimensional Hamiltonian H. Using a particle-in-a-box toy model and the polynomial basis u_i(x)=x^i(1-x), the author numerically shows that the RRM eigenvalues W_k of H_D often converge from below to the exact eigenvalues E_k, contrary to the usual upper-bound property of the RRM for the full Hamiltonian H. Exceptions are reported for small trial-space dimensions N. The paper also discusses the effect of an energy shift and the behavior of the projected identity operator.
Significance. The observation is useful: it shows that the Rayleigh-Ritz upper-bound theorem does not automatically apply to finite-rank projections of a Hamiltonian, which is directly relevant to the ensemble variational principle proposed by Ding et al. The numerical tables are consistent with the rank argument that H_D has at most D nonzero RRM eigenvalues and the remaining ones are exactly zero; the zero-eigenvalue multiplicity is correctly explained by orthogonality of the corresponding Ritz vectors to the eigenvectors spanning H_D. The paper makes no parametric fits and uses textbook RRM theory, so the core computation is sound. The main weakness is that the 'most cases' claim is an extrapolation from a single basis set and a single Hamiltonian, with no theorem or second numerical experiment to delimit its scope.
major comments (2)
- [Abstract; Section 3.2, Tables 2-4; Section 3.3, Table 7] The assertion that 'the RRM eigenvalues approach those of the projected Hamiltonian from below in most cases' is supported only by numerical experiments for one Hamiltonian (particle in a box) and one basis set u_i(x)=x^i(1-x). No theorem or independent numerical test is provided, and the paper itself records exceptions (e.g., W1>E1 for N=1 in Table 4). Since this claim appears in the abstract and in the conclusions, the authors should either add a proof or a second basis/Hamiltonian experiment, or explicitly restrict the claim to 'in the cases studied here.'
- [Section 3.4 and Eq. (18)] The treatment of the projected identity operator is unclear and appears to conflict with the spectral properties of compressions of a projection. For I_D = sum_{k=1}^D |psi_k><psi_k|, its compression to an N-dimensional subspace has eigenvalues in [0,1], so they are lower bounds to the nonzero eigenvalue 1, not upper bounds; yet the text states 'we obtain upper bounds for 1 <= N < D'. Equation (18) uses a secular determinant |H5 - W S5| with S5 defined as the projected overlap, while the standard RRM uses the full overlap matrix S. The relation of this determinant to the RRM for I_D (or for H) should be clarified and the statement about upper bounds should be corrected or qualified.
minor comments (4)
- [Equation (7)] Equation (7) appears to be misprinted: the Rayleigh quotient denominator is missing; it should read <psi|H_D|psi> = E_n |<psi_n|psi>|^2, or the same expression in quotient form.
- [Throughout] The name 'Cauchy-Schwartz' should be 'Cauchy-Schwarz'.
- [Equation (13)] In Eq. (13), the summation on the right-hand side runs over D, but the left-hand side is a sum over N Ritz vectors; since W_i=0 for i>D, the equality is correct only if the summation indices are understood consistently, and this should be stated explicitly.
- [Section 3.2] The statement that 'the remaining roots vanish W_k = 0, D < k <= N as expected' would be more precise if it noted that the matrix (H_D) has rank at most D, so the generalized eigenproblem has at least N-D zero eigenvalues.
Circularity Check
No significant circularity: the RRM eigenvalues of the projected Hamiltonian are computed numerically, not fitted, and the self-citations are not load-bearing.
full rationale
I find no circular step. The paper applies the standard Rayleigh-Ritz secular equation (Eq. 4) to a finite-rank projected Hamiltonian H_D (Eq. 5) and reports computed eigenvalues in Tables 2-4, 6, and 7. These eigenvalues are outputs of numerical diagonalization, not inputs used to define the claim. The standard upper-bound property of the RRM for the unprojected Hamiltonian is supported by the external, textbook result of MacDonald (Ref. [3]); the self-citations (Refs. [4] and [6]) are used only to motivate the toy model and the polynomial basis, not to establish the projected-Hamiltonian lower-bound behavior. The paper explicitly acknowledges counterexamples (e.g., W1 > E1 for D = 3, N = 1; W2 > E2 for D = 4, N = 3), so the 'approach from below in most cases' statement is an empirical generalization rather than a conclusion forced by construction. The main weakness is that the generalization rests on one basis set and one Hamiltonian with no proof of universality; that is a correctness or evidence concern, not circularity. The analysis is self-contained with respect to its actual computational claims.
Assumptions & free parameters
assumptions (3)
- standard math The Hamiltonian H with H = -1/2 d^2/dx^2 and Dirichlet boundary conditions has eigenvalues E_k = k^2 pi^2 / 2 and eigenfunctions psi_k(x) = sqrt(2) sin(k*pi*x).
- domain assumption The basis functions u_i(x) = x^i(1-x), i=1,2,..., form a complete basis for the relevant function space.
- standard math The Rayleigh-Ritz secular determinant |H - W S| = 0 yields the variational eigenvalues.
Cite this review
Pith. "Pith review of On the application of the Rayleigh-Ritz method to a projected Hamiltonian." pith.science (2026). https://pith.science/paper/N45UYOWS
@misc{pith2026241114490,
author = {Pith},
title = {Pith review of: On the application of the Rayleigh-Ritz method to a projected Hamiltonian},
year = {2026},
howpublished = {\url{https://pith.science/paper/N45UYOWS}},
note = {Machine review of arXiv:2411.14490}
}
read the original abstract
We apply the well known Rayleigh-Ritz method (RRM) to the projection of a Hamiltonian operator chosen recently for the extension of the Rayleigh-Ritz variational principle to ensemble states. By means of a toy model we show that the RRM eigenvalues approach to those of the projected Hamiltonian from below in most cases but a few ones. We also discuss the effect of an energy shift and the projection of the identity operator.
Reference graph
Works this paper leans on
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[1]
F. L. Pilar, Elementary Quantum Chemistry (McGraw-Hill, New York , 1968)
1968
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[2]
A. Szabo and N. S. Ostlund, Modern Quantum Chemistry (Dover P ublica- tions, Inc., Mineola, New York, 1996)
work page 1996
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[3]
J. K. L. MacDonald, Phys Rev. 43, 830 (1933). 7 Table 1: Lowest RRM eigenvalues for the Hamiltonian (9) N W 1 W2 W3 W4 1 5 3 4 . 934874810 21 51 . 06512518 5 4 . 934802217 19 . 75077640 44 . 58681182 100 . 2492235 7 4 . 934802200 19 . 73923669 44 . 41473408 79 . 99595777 9 4 . 934802200 19 . 73920882 44 . 41322468 78 . 97848206 11 4 . 934802200 19 . 73920...
work page 1933
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[4]
F. M. Fern´ andez, On the Rayleigh-Ritz variational method, arXiv:2206.05122 [quant-ph]
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[5]
Ground and Excited States from Ensemble Variational Principles
L. Ding, C-L. Hong, and C. Schilling, Quantum 8, 1525 (2024). arXiv:2401.12104v2 [quant-ph]
work page Pith review arXiv 2024
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[6]
F. M. Fern´ andez, J. Math. Chem. 62, 2083 (2024). arXiv:2405.10340 [quant- ph]
arXiv 2024
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[7]
T. Apostol, Calculus, Second ed. (John Wiley & Sons, New York, 19 67). 8 Table 2: RRM for the projected Hamiltonian with D = 1 N W 1 Wk, 1<k ≤N 1 4 . 927671482 3 4 . 934799721 0 5 4 . 934802200 0 7 4 . 934802200 0 Table 3: RRM for the projected Hamiltonian with D = 2 N W 1 W2 Wk, 2<k ≤N 1 4 . 927671482 3 4 . 934799721 19 . 40270646 0 5 4 . 934802200 19 . ...
Reviewed August 12, 2026 · model on record in the stance chip above.
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