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Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models

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Pith's one-line read This paper claims that the singular values of a doubly infinite non-Hermitian tridiagonal Hamiltonian are the poles of a Green's function built from a pair of matrix continued fractions.

desk verdict Lemma 2 overstates its result: poles of G(z) only capture singular values whose eigenvectors have a nonzero central-block component, and the presented 3x3 example shows a missing singular value as a zero instead. read the letter →

arxiv 2505.05850 v2 pith:UWXQHYRK submitted 2025-05-09 quant-ph physics.atom-ph

Miloslav Znojil This is my paper · ORCID
classification quant-phphysics.atom-ph
keywords non-HermitianHamiltonianssingularvaluesmatrixcontinuedfractionsGreen'sfunctionsBose-Hubbardmodeltridiagonalmatricesresonancesexceptionalpoints
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 tackles the hard problem of locating complex resonance energies of non-Hermitian quantum Hamiltonians. It argues that for tridiagonal Hamiltonians with diagonals growing at both infinities, a cheaper target works: the real singular values of H, which can be recovered as poles of an auxiliary Green's function. The Green's function is assembled from two independent matrix continued fractions, one running up the index ladder and one running down. If this construction is correct, resonance physics in Bose-Hubbard-like many-boson models becomes accessible to a semi-analytic computation that avoids direct complex diagonalization.

What carries the argument

The load-bearing object is the pair of two-by-two matrix continued fractions in recurrences (41) and (42), together with the central block $F_0(z)$ of Eq. (43). The recurrences are the block-tridiagonal analogues of the scalar continued fractions that define Green's functions for ordinary Jacobi matrices; they solve the factorization of $H - \sigma$ into $U F L$. Their convergence, expected under the two-sided growth condition (15), is what makes $G(z)$ well defined. The sparse structure of the 2x2 blocks ($A_k$ has zeros on the diagonal, $B_k$ and $C_k$ have a single nonzero entry) is invoked as the reason convergence should be quick.

What would settle it

Take a finite Bose-Hubbard-like matrix $H[M,N]$ with diagonals satisfying $|a_k| \sim |k|^p$ for large $|k|$ and constant off-diagonals, compute the poles of $G(z)$ from recurrences (41)-(43) at increasing $M,N$, and compare with the singular values of a direct SVD of the same finite matrix; if the pole sets do not converge to the singular values as $M,N$ grow, or if the recurrences blow up before convergence, Lemma 2 is falsified.

Watch

Extended reading notes

Core claim

The central claim is Lemma 2: after Hermitization by embedding H and H† into a self-adjoint block operator and permuting basis states, the Hamiltonian becomes block-tridiagonal with 2x2 blocks. Factoring $H - \sigma = U F L$ and solving the block recurrences yields $F_0(z) = [A_0 - z - C_0 F_{-1}(z) B_{-1} - B_0 F_1(z) C_1]^{-1}$, where $F_1$ and $F_{-1}$ are the limits of the recursions $F_k = 1/(A_k - z - B_k F_{k+1} C_{k+1})$ and $F_{-j} = 1/(A_{-j} - z - C_{-j} F_{-j-1} B_{-j-1})$. The Green's function $G(z) = \det F_0(z)$ then has poles exactly at the singular values $\sigma_n$ of $H$. The paper presents this as a generalization of the one-sided continued-fraction method, needed because Bose-Hubbard-like Hamiltonians are doubly infinite with growth at both ends.

Load-bearing premise

The whole construction assumes that the two matrix continued fractions in recurrences (41) and (42) converge as M and N tend to infinity under the two-sided growth condition (15); the paper offers only an analogy-based expectation, not a proof or numerical demonstration.

Editorial extensions

If this is right

  • Resonance energies for non-Hermitian tridiagonal models can be located without finding complex eigenvalues: one computes real singular values as poles of $G(z)$.
  • The method removes the one-sided limitation of earlier continued-fraction treatments and covers doubly infinite Bose-Hubbard-like Hamiltonians with two-sided diagonal growth.
  • Because the 2x2 blocks are sparse, the matrix continued fractions are expected to converge quickly, making the approach competitive with standard singular-value algorithms.
  • The same construction carries over to odd numbers of bosons and to asymmetric cut-offs $M \neq N$, since the recurrences treat the two sides independently.
  • Onset of divergence of the continued fractions is tied to a physical regime change, typically toward a continuous spectrum.

Reading between the lines

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

  • Inference: if the two-sided convergence is proved rigorously, the construction gives a deterministic, parameter-free way to compute singular values of non-Hermitian many-body Hamiltonians, complementing stochastic and Krylov-based approaches.
  • Inference: the fixed-point basis of the one-sided convergence proof in the companion work suggests the two-sided recurrences may also be stable under small random perturbations, which would make the method usable for ensemble studies of non-Hermitian random tridiagonal matrices.
  • Inference: a numerical scaling test, fixing the off-diagonal structure and varying the growth rate of $a_k$ while monitoring the truncation depth needed for a target pole accuracy, would turn the expected convergence into a quantitative criterion.
  • Inference: in PT-symmetric $M=N$ models, tracking where poles of $G(z)$ coalesce could locate higher-order exceptional points, connecting singular-value poles to the EP-unfolding phenomenology.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a semi-analytic method for locating the singular values of non-Hermitian tridiagonal (Jacobi-like) Hamiltonians of Bose-Hubbard type. After reviewing the scalar continued-fraction Green's function for one-sided tridiagonal problems, the author block-tridiagonalizes the Hermitized operator [0 H; H† 0], introduces two 2×2 matrix continued fractions in Eqs. (41)–(42), and states in Lemma 2 that the Green's function is G(z)=det F0(z) with F0 defined by Eq. (43). The abstract and introduction claim that the singular values of H can be sought as poles of this Green's function. The algebraic derivation of the Schur-complement/F0 formula is standard, but the paper supplies no proof of convergence in the doubly infinite limit and, more importantly, the pole claim is false as stated when a singular vector has zero projection on the central block.

Significance. If the pole statement were correct, the paper would offer a parameter-free, semi-analytic route to singular values of large or doubly infinite non-Hermitian tridiagonal models, extending the one-sided result of Ref. [6] and giving a practical tool for the cited Bose-Hubbard-like models. The manuscript is self-contained in its algebraic manipulations and contains no fitted parameters, which is a genuine strength. However, the advertised central claim is not merely missing a proof: it fails for a simple 3×3 example embedded in the paper's own class of models. The useful algebraic core can likely be repaired, but the present version does not establish the stated result.

major comments (3)
  1. [Section 4.4, Lemma 2 and Eq. (43)] The statement that the singular values of H are poles of G(z)=det F0(z) is false when a singular vector vanishes at the central site. For H=[[0,1,0],[2,0,3],[0,4,0]], which is of the form (14) with M=N=1 and all a_k=0, the recurrences (41)-(43) give F0(z)=diag(z/(13-z^2), z/(17-z^2)), hence G(z)=z^2/[(13-z^2)(17-z^2)]. The singular values of H are 0, sqrt(13), sqrt(17); G has poles at ±sqrt(13) and ±sqrt(17), but z=0 is a double zero, not a pole. The singular vector for σ=0 is (3,0,-2), whose central component vanishes, so the Hermitized eigenvector has zero projection on the central 2×2 block. The proof of Lemma 2, which says that 'the idea is the same as in Lemma 1', silently assumes this projection is nonzero. The same phenomenon already occurs in the scalar Lemma 1, e.g. for H=[[0,1,0],[1,0,1],[0,1,0]], whose zero-energy eigenvector (1,0,-1) has f0(z)=z/(2-z^2) with a zero, not a pole, at z=0. Lemma 2 therefore needs an explicit nonvanishing condition on the central projection, or a reformulation in terms of zeros/poles of the Schur complement, together with a treatment of the exceptional eigenvectors.
  2. [Section 4.4, Eqs. (41)-(43)] Convergence of the two matrix continued fractions in the limit M,N→∞ is load-bearing but is never established. The text only says that convergence 'may be expected' by analogy with the one-sided case of Ref. [6] and that the sparse 2×2 blocks make convergence 'quick'. No proof is given for the doubly infinite two-sided setting, and no numerical test is reported. The one-sided proof in Ref. [6] does not automatically cover the asymmetric two-sided case with non-Hermitian off-diagonal elements b_k, c_{k+1}. Since G(z) is defined through these infinite recurrences, the paper must either prove convergence under assumption (15) or explicitly restrict the claims to finite M,N.
  3. [Section 4.3 and Eq. (39)] The paper does not state the invertibility and regularity assumptions under which the matrix recurrences (38), (41), and (42) are well defined. The scalar assumption (22) is formulated for the f_j, but the analogous non-vanishing conditions for the 2×2 matrices F_j, and for the denominators appearing in Eqs. (41)-(43), are missing. As the counterexample above shows, the denominators can vanish at a singular value, and the limiting formula then produces a zero rather than a pole. A precise statement of the regularity domain of z, including what happens at points where intermediate F_j are singular, is needed for Lemma 2 to be meaningful.
minor comments (6)
  1. [Section 4.4] The sentence 'the convergence may be expected quick here' is grammatically awkward and unsupported; it should be rewritten as a precise quantitative claim or removed.
  2. [Section 3.2, Eq. (22)] The regularity assumption (22) is written for scalar continued fractions; the corresponding matrix invertibility conditions for the F_j in Eqs. (41)-(42) should be stated explicitly.
  3. [Section 4.4, Eq. (43)] G(z) is called a Green's function but is defined as the determinant of a 2×2 block of the resolvent of the Hermitized operator; the terminology should be defined and distinguished from the full resolvent.
  4. [Footnote 11] There is a typo in the footnote: 'atytempts' should be 'attempts'.
  5. [References] Reference [7] is a Wikipedia page; a standard textbook reference for singular values would be more appropriate.
  6. [General] The paper contains no numerical illustration of Eq. (43), even for a small finite matrix; a finite-dimensional example that confirms the pole locations, or that explicitly exhibits the problematic zero case, would greatly improve the presentation.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: G is a block-elimination identity; cited convergence support from the author's [6] is not load-bearing.

full rationale

The central formula G(z)=det F0(z) with F0 given by Eq. (43) is obtained by block LU/Schur-complement elimination of the block-tridiagonal Hermitized matrix (40), using the matrix-continued-fraction recurrences (41) and (42). The Hermitization step is explicitly attributed to Pushnitski and Stampach [5], and the permutation V in Eq. (32) is an explicit isospectral transformation. No fitted parameter, no data subset, and no target singular value enters the recurrences; the pole condition is a derived consequence of the factorization, not an input. The paper does cite the author's own preprint [6] for convergence of matrix continued fractions ('an explicit confirmation of the expectation convergence can also be found provided in the most recent update of preprint [6]'), but that citation is not load-bearing: Section 4.4 labels convergence only as an expectation ('one may expect that the convergence may be expected quick here'), and the algebraic identity in Lemma 2 is derived before any convergence guarantee is invoked. This is a minor self-citation used as supporting evidence for a practical assumption, not a circular step. The main rigour concerns are correctness-type gaps, not circularity: Lemma 2's proof says only 'the idea of the proof is the same as in Lemma 1' and silently assumes a nonzero central-site component of every eigenvector; the convergence of the two-sided recurrences (41)-(42) is also left as an unproved expectation. These are limitations of proof or false-lemma risks, but they do not make the derivation equivalent to its own inputs.

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

No free parameters are fitted. The construction relies on domain assumptions (tridiagonality, two-sided growth, regularity) and on an unproven convergence assumption for the matrix continued fractions. No new entities are introduced.

assumptions (4)
  • domain assumption The Hamiltonian H[M,N] has a tridiagonal Jacobi-matrix form (14) with diagonal elements growing to infinity on both sides, satisfying the asymptotic condition (15).
    Restricts to the Bose-Hubbard-like class of models; used in Lemmas 1 and 2 and in the recurrences (20)-(21) and (41)-(42).
  • domain assumption Regularity assumption (22): the continued-fraction denominators 1/f_j and 1/F_j remain nonzero at generic complex z outside the spectrum.
    Needed to define the Green's function via the recurrences; excludes exceptional-point degeneracies.
  • domain assumption Convergence of the scalar continued fractions (20)-(21) and of the matrix continued fractions (41)-(42) in the infinite limits M,N to infinity.
    The paper does not prove convergence for the doubly-infinite case; it relies on the asymptotic growth (15) and analogy with the one-sided case of Ref. [6]. This is the load-bearing assumption for the central claim.
  • standard math Standard linear algebra: LU factorization of (block-)tridiagonal matrices, Schur-complement determinant relations, and invertibility of the bidiagonal factors U and L.
    Used in the proofs of Lemmas 1 and 2 and in the derivation of the recurrences.

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Cite this review

Pith. "Pith review of Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models." pith.science (2026). https://pith.science/paper/UWXQHYRK

@misc{pith2026250505850,
  author       = {Pith},
  title        = {Pith review of: Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWXQHYRK}},
  note         = {Machine review of arXiv:2505.05850}
}
read the original abstract

With resonances treated as eigenstates of a non-Hermitian quantum Hamiltonian, the task of localization of the complex energy eigenvalues is considered. The paper is devoted to the reduced version of this task in which one only computes the real quantities called singular values. It is shown that in such an approach (and under suitable constraints including the tridiagonality of the Hamiltdonian) the singular values can be sought as poles of an auxiliary Green's function expressible in terms of a doublet of matrix continued fractions. A family of multi-bosonic Bose-Hubbard-like complex Hamiltonians is recalled for illustration purposes.

Figures

Figures reproduced from arXiv: 2505.05850 by the authors.

Figure 1
Figure 1. The shape of potential V(x) of Eq. (50) (we choose [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. The shape of the real part of potential W(x) of Eq. (51) [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. The shape of the imaginary part of potential W(x) of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗

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