Complex tridiagonal quantum Hamiltonians and matrix continued fractions

Miloslav Znojil

REVIEW 3 major objections 3 minor 1 cited by

Complex tridiagonal quantum Hamiltonians and matrix continued fractions

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Pith's one-line read The paper claims that singular values of a complex symmetric tridiagonal Hamiltonian can be computed as eigenvalues of a Hermitian block-tridiagonal partner via a matrix continued fraction, whose convergence is governed by real fixed…

desk verdict Useful finite-dimensional singular-value trick; the N=∞ convergence claim is asserted, not demonstrated. read the letter →

arxiv 2504.16424 v2 pith:DH3GUULQ submitted 2025-04-23 math-ph math.MPquant-ph

Miloslav Znojil This is my paper · ORCID

classification math-phmath.MPquant-ph MSC 15A1847B3640A1581Q12

keywords non-HermitianHamiltonianscomplexsymmetricmatricestridiagonalsingularvaluesmatrixcontinuedfractionsblock-tridiagonalHermitianoperatorfixed-pointconvergencequantumresonances

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 is about unstable quantum states described by non-Hermitian tridiagonal Hamiltonians whose energy eigenvalues are complex. It proposes replacing the difficult task of finding those complex energies by the easier task of finding the real, non-negative singular values of the Hamiltonian. Its central claim is Lemma 1: for a complex symmetric tridiagonal Hamiltonian matrix, the singular values are exactly the eigenvalues of an auxiliary Hermitian block-tridiagonal matrix, so they can be obtained from a matrix continued fraction expansion of the auxiliary resolvent. A fixed-point analysis of the recurrence then tells when the continued fraction converges, and an illustrative discrete Schrödinger equation with a complex local potential shows fast convergence in a diagonal-dominated case and failure when the fixed points become complex. A sympathetic reader would care because the method gives real spectral data for open quantum systems without solving a complex eigenvalue problem.

What carries the argument

The central object is the complex symmetric tridiagonal matrix Hamiltonian $H$ of Eq. (7), together with its Hermitian partner $\mathbb{H}$ of Eq. (13), a block-tridiagonal matrix with $2\times2$ blocks. The argument runs on the matrix continued fraction recurrence (16), obtained by replacing scalar entries in the standard analytic continued fraction factorization of $H-E$ by Hermitian two-by-two blocks, and then on its reduction to the three coupled scalar iteration maps (20) with real parameters $u,x,y$. The convergence of the whole scheme is controlled by fixed points of these maps, and Lemma 2 packages the fixed-point condition as the quartic equation $P(u)=0$; this real-root condition is what makes the method either converge quickly or fail, as the two illustrative examples demonstrate.

What would settle it

Run the iteration (20) with a parameter triple for which the quartic polynomial $P(u)$ has no real root—for example $\sigma=1$, $\gamma=1/2$, $\beta=2$—and check whether the sequences $u,x,y$ settle to a fixed point. If they nevertheless converge to the correct singular values, the fixed-point criterion would not be necessary; if they diverge, the stated limitation is confirmed. A broader settlement would compare the singular values from the MCF computation with a direct full SVD of finite truncations of $H$.

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Extended reading notes

Core claim

For every complex symmetric tridiagonal Hamiltonian $H$ of the form (7), with real $\alpha_k, \beta_k, \gamma_k$ and $N \leq \infty$, the singular values $\sigma_n$ of $H$ coincide with the eigenvalues of the Hermitian block-tridiagonal matrix $\mathbb{H}$ in (13), whose $2\times2$ blocks are $A_k = \begin{pmatrix} 0 & \beta_k+i\gamma_k \\ \beta_k-i\gamma_k & 0 \end{pmatrix}$ and $B_k = \begin{pmatrix} 0 & \alpha_k \\ \alpha_k & 0 \end{pmatrix}$. The proof goes through the doubled matrix $\widetilde{H} = \begin{pmatrix} 0 & H \\ H^\dagger & 0 \end{pmatrix}$ and a renumbering of basis elements. Since $\mathbb{H}$ is Hermitian, its resolvent factorizes into a matrix continued fraction through the recurrence (16) for two-by-two matrices $F_k$, and the paper reduces this recurrence to the coupled scalar maps (20) for $u,x,y$. Lemma 2 gives a quartic polynomial $P(u)$ whose real roots are the candidate fixed points of these maps, so convergence of the continued fraction is decided by the existence of a real attracting fixed point. The paper's examples show quick numerical convergence for one parameter set and non-convergence for a nearby set.

Load-bearing premise

The proof of convergence assumes that for large indices the recurrences can be replaced by constant-parameter maps with real parameters that possess a real attracting fixed point, and this fails for some complex potentials (the $\beta=2$, $\gamma=1/2$ example), so the method is not guaranteed to converge in general.

Editorial extensions

If this is right

Singular values of complex-symmetric tridiagonal resonance Hamiltonians can be obtained as real eigenvalues of a Hermitian block-tridiagonal operator, bypassing direct complex spectral computations.
The matrix continued fraction recurrence gives a practical numerical scheme whose convergence rate is governed by the fixed-point map (20), with the quartic polynomial (23) as a closed-form diagnostic.
The method inherits a sharp boundary: when the fixed-point equation has no real root, as for $\sigma=1$, $\gamma=1/2$, $\beta=2$, the continued fraction fails, so not every complex local potential can be treated in this way.
The Hermitian partner $\mathbb{H}$ has all resolvent poles on the real half-axis, so the singular values are directly readable from the secular equation $\det F_1^{-1}(\sigma)=0$ in the MCF formalism.

Reading between the lines

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

If the fixed-point condition is sharpened, the transition between real and complex roots of $P(u)$ could provide a boundary in parameter space separating computable from non-computable regimes; mapping this boundary numerically would be a direct test of the method's domain.
The block-tridiagonal partner has the structure of a tight-binding model on a doubled chain, so the convergence of the continued fraction may correspond to a localization transition of the associated transfer-matrix map, connecting the result to transport phenomenology.
A natural extension is to $M\times M$ blocks: the quartic fixed-point condition would become a higher-degree algebraic condition, and one could test whether real-root obstructions of the same kind control convergence there.
Since the iteration maps (20) are low-dimensional, the derivative at the stable fixed point could yield a quantitative bound on the contraction rate, giving an a priori stopping criterion for practical computations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: the finite-dimensional identity in this paper is real and worth borrowing — singular values of a complex symmetric tridiagonal matrix come out of a Hermitian block-tridiagonal matrix via a terminating matrix continued fraction. The advertised N = ∞ convergence, however, rests on a constant-coefficient placeholder, not on the actual variable-coefficient recurrence. Don't rely on the convergence claim without checking it.

What's genuinely new: the explicit MCF recurrence (16), the reduction of the 2x2 block map to the scalar maps (20), and the quartic fixed-point equation (23). For fixed finite N, the construction is sound: the recurrence terminates at F_{N+1}=0, and the spectral equivalence with the block matrix [[0,H],[H†,0]] after renumbering is exactly the Pushnitski–Stampach idea. That's a clear, legitimate extension.

The soft spots are in the infinite-dimensional part. Lemma 1 states N ≤ ∞, but the proof only handles finite matrices. For N = ∞ the author appeals to a pragmatic assumption of discrete non-degenerate spectrum, which is a framing, not a proof. The convergence section replaces the k-dependent β_k and γ_k by constants and studies fixed points of the asymptotic map. That tells you about the constant map, not why the original recurrence converges. There is no error bound on truncating at F_{M+1}=0 and no stability calculation — the stable fixed point in the example is simply asserted, and Table 1 iterates the constant map rather than an MCF for a discretized Hamiltonian (7) with growing diagonal. The β=2, γ=1/2 example shows a failure but gives no general criterion. These are real gaps.

Who should read it: people computing resonances in tridiagonal models who want a Hermitian reformulation; the finite-N machinery could be useful. It deserves a serious referee, but the N=∞ claims need heavy revision — either restrict Lemma 1 to finite N or provide a genuine contraction argument. I'd bring it to a reading group with a caveat.

Referee Report

3 major / 3 minor

Summary. The paper studies complex symmetric tridiagonal Hamiltonians H of the form (7), interpreted as discretizations of one-dimensional Schrödinger operators with complex local potentials. It proposes to compute the singular values σ_n of H as eigenvalues of an auxiliary Hermitian block-tridiagonal operator H defined in (13)–(14), and to evaluate the resolvent of H by a matrix continued fraction (MCF) recurrence (16). The main claims are Lemma 1 (spectral equivalence for N≤∞), a fixed-point-based convergence analysis in §4.2 culminating in Lemma 2, and a numerical illustration in Table 1 purporting to show quick convergence. The finite-dimensional block construction is a reordering of the Hermitian dilation [[0,H],[H†,0]] and is correct; the burden of the paper lies in the infinite-dimensional and convergence claims.

Significance. If established, the method would give a practical route from complex resonance spectra to real singular values through a Hermitian block-tridiagonal eigenproblem, and the explicit 2×2 parametrization (17)–(19) is a genuinely useful computational reduction. Credit should be given for the clean finite-dimensional construction and for the clear scalar continued-fraction background in §3.3. However, the advertised contributions—the N=∞ spectral equivalence and the fixed-point proof of MCF convergence—are not established as written, and the numerical illustration exercises a simplified constant-coefficient map rather than the MCF for the variable-coefficient Hamiltonian (7).

major comments (3)

[4, Lemma 1] The statement that Lemma 1 holds at any finite or infinite Hilbert-space dimension N≤∞ is not supported by the proof. The proof defines singular values as eigenvalues of the dilation (15) and then cites the Pushnitski–Stampach renumbering, but for N=∞ the equivalence requires operator-theoretic hypotheses on H (for example, a precise definition of singular values for non-compact operators and a self-adjoint realization of the dilation). The manuscript explicitly steps back from such hypotheses ('we will proceed in a more pragmatic manner') and merely assumes a discrete non-degenerate spectrum. Thus the infinite-dimensional part of Lemma 1 remains an assertion, and the abstract's claim about the discretized Schrödinger operator (7) is not justified. If the intended scope is finite N or a rigorously controlled truncation, that restriction should be stated and the claims adjusted accordingly.
[4.2, MCF convergence] The convergence analysis replaces the k-dependent MCF recurrence (16) by the constant-coefficient scalar maps (20) and studies their real fixed points, but it does not prove convergence of the original backward recurrence for the variable-coefficient Hamiltonian (7). No relation between the asymptotic constant map and the actual k-dependent iteration is established, and no truncation-error bound for the boundary condition F_{M+1}=0 is supplied. Moreover, the stability of the selected fixed point is asserted rather than proved: unlike the scalar case in (11)–(12), no Jacobian or spectral-radius computation is given for the three-dimensional map (20). The β=2, γ=1/2 example is informative, but it does not provide a general criterion separating convergent from divergent regimes, and the paper itself acknowledges this limitation in §4.3.
[4.3, Table 1] Table 1 iterates the constant-coefficient maps (20) with α=σ=1, β=4, γ=1/2; it does not compute the MCF (16) for a discretized Hamiltonian (7) with k-dependent diagonal elements β_k, γ_k. The caption should state this explicitly, because as presented the table invites the reading that the MCF for the physical application converges quickly. Consequently the abstract's statement that 'numerical MCF convergence is found quick' is not demonstrated by the numerical evidence in the paper; the example shows only convergence of the simplified constant-map iteration.

minor comments (3)

[4.2, Lemma 2] Lemma 2's proof is described only as 'standard elimination ... Gröbner basis'; since the quartic (23) and the relation (24) are load-bearing algebraic inputs, the derivation should be supplied in an appendix or made reproducible.
[4, preliminaries] The definition of singular values is cited to a Wikipedia article; for the operator-theoretic claims in Lemma 1, a mathematical reference with precise hypotheses on singular values of non-compact operators should be used.
[4.1, Eq. (17)] The notation in (17) would be clearer if it explicitly stated that u_k, v_k, x_k, y_k are the parameters of F_k^{-1}, not of F_k; the subsequent map (18)–(19) is easier to follow when this convention is highlighted.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the singular-value construction and MCF recurrence are derived from external linear-algebra identities, with self-citations only in background roles.

full rationale

The central derivation is self-contained relative to its stated inputs. Lemma 1 defines singular values via the Hermitian dilation (15), H-tilde = [[0,H],[H†,0]], and then obtains the block-tridiagonal form (13)–(14) by a basis renumbering credited to Pushnitski and Stampach [11]; this is an external linear-algebra identity, not an assumption of the conclusion. The MCF recurrence (16) is obtained by applying the already-proved scalar factorization H−E=UFL of Eq. (2) to the block-tridiagonal operator, and the fixed-point equations (20) and quartic (23) are algebraic consequences of that recurrence after setting constant coefficients; none of these objects is fitted to the quantities later called predictions. Table 1 iterates the derived map (20), so it is a numerical check of the derived recurrence, not a prediction from a fitted parameter. The self-citations ([19], [22], [28]) appear as examples of continued-fraction/MCF practice and as historical comparison; they are not the sole or load-bearing justification of Lemma 1 or Lemma 2. The genuine weakness is the N=∞ convergence step: Lemma 2 is conditional ('if it exists'), and Section 4.3 itself shows the β=2, γ=1/2 case has only complex fixed points, making the constant-coefficient analysis insufficient to prove convergence for the variable-coefficient Hamiltonians (7). But that is a gap in the mathematical support for the infinite-dimensional claim, not a circular reduction: the claimed conclusion is not built into the premises by definition or by a fitted input. Accordingly, no circularity step is identified.

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

No constants are fitted to data; the example parameters are inputs, not fitted values. No new physical entities are introduced. The key uncharged assumptions are the discrete-spectrum domain assumption for infinite N, the constant-coefficient tail approximation, and the transfer of the scalar fixed-point stability criterion to the matrix iteration.

assumptions (4)

standard math Singular values of a matrix H are eigenvalues of the block matrix [[0,H],[H†,0]] after suitable basis renumbering.
Lemma 1, standard linear algebra, also cited to Pushnitski and Stampach [11].
domain assumption The infinite-dimensional complex symmetric Hamiltonian of the form (7) has discrete, non-degenerate spectrum and admits well-defined singular values as eigenvalues of the block matrix (15).
Section 4 states that the paper proceeds pragmatically and assumes a discrete spectrum because boundedness would be counterproductive. This is load-bearing for the N=infinity version of Lemma 1.
ad hoc to paper For k >> 1 the coefficients alpha_k, beta_k, gamma_k and the spectral parameter can be treated as constants, reducing the recurrence to the constant-coefficient map (20).
Sections 3.3 and 4.2 use this asymptotic rescaling to derive the fixed-point criteria. The paper does not prove it holds for generic potentials, and the beta=2 example shows the constant map can lack real fixed points.
domain assumption The fixed-point stability condition |df'/df| < 1 is sufficient to guarantee practical convergence of the iterated two-by-two MCF maps.
Section 4.2 applies the scalar stability idea to the matrix case by inspection of fixed points; no general proof is given that the contraction condition transfers to the matrix iteration.

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Cite this review

Pith. "Pith review of Complex tridiagonal quantum Hamiltonians and matrix continued fractions." pith.science (2026). https://pith.science/paper/DH3GUULQ

@misc{pith2026250416424,
  author       = {Pith},
  title        = {Pith review of: Complex tridiagonal quantum Hamiltonians and matrix continued fractions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DH3GUULQ}},
  note         = {Machine review of arXiv:2504.16424}
}

abstract

Quantum resonances described by non-Hermitian tridiagonal-matrix Hamiltonians $H$ with complex energy eigenvalues are considered. The method of evaluation of quantities $\sigma_n$ known as the singular values of $H$ is proposed. Its basic idea is that the quantities $\sigma_n$ can be treated as eigenvalues of an auxiliary self-adjoint operator $\mathbb{H}$. As long as such an operator can be given a block-tridiagonal matrix form, we finally expand its resolvent in terms of a matrix continued fraction (MCF). In an illustrative application, a discrete version of conventional Hamiltonian $H=-d^2/dx^2+V(x)$ with complex local $V(x) \neq V^*(x)$ is considered. The numerical MCF convergence is found quick, supported also by a fixed-point-based formal proof.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Resonances and continued-fraction Green's functions in non-Hermitian Bose-Hubbard-like quantum models
quant-ph 2025-05 conditional novelty 5.0 of 10

For doubly-infinite tridiagonal non-Hermitian Hamiltonians, the singular values can be expressed as poles of a Green's function built from two matrix continued fractions.

Reference graph

Works this paper leans on

28 extracted references · 20 canonical work pages · cited by 1 Pith paper

[1]

F. M. Fern´ andez, On a class of non-Hermitian Hamiltonians with tr idiagonal matrix representation. Ann. Phys. (NY) 443 (2022) 169008

work page 2022
[2]

F. G. Scholtz, H. B. Geyer and F. J. W. Hahne, Quasi-Hermitian op erators in quan- tum mechanics and the variational principle. Ann. Phys. (NY) 213 (1 992) 74

work page
[3]

Messiah, Quantum Mechanics

A. Messiah, Quantum Mechanics. North Holland, Amsterdam, the Netherlands, 1961

work page 1961
[4]

C. M. Bender, Making sense of nonhermitian Hamiltonians. Rep. Pr og. Phys. 70 (2007) 947 - 1018

work page 2007
[5]

Mostafazadeh, Pseudo Hermitian representation of quantu m mechanics

A. Mostafazadeh, Pseudo Hermitian representation of quantu m mechanics. Int. J. Geom. Meth. Mod. Phys. 7 (2010) 1191 - 1306

work page 2010
[6]

Bagarello, J.-P

F. Bagarello, J.-P. Gazeau, F. Szafraniec and M. Znojil, Eds., Non -Selfadjoint Op- erators in Quantum Physics: Mathematical Aspects. Wiley, Hoboke n, 2015

work page 2015
[7]

Moiseyev, Non-Hermitian Quantum Mechanics

N. Moiseyev, Non-Hermitian Quantum Mechanics. Cambridge, CUP , 2011

2011
[8]

A. D. Dente, R. A. Bustos-Mar´ un, H. M. Pastawski, Dynamical regimes of a quantum SW AP gate beyond the Fermi golden rule. Phys. Rev. A 78 (2008) 06 2116

work page 2008

Show all 28 references

[9]

Garmon, G

S. Garmon, G. Ordonez, K. Noba, Weak-coupling bound states in semi-inﬁnite topo- logical waveguide QED. Preprint arXiv:2503.07211
[10]

https://en.wikipedia.org/wiki/Singular value
[11]

Pushnitski, F

A. Pushnitski, F. ˇStampach, An inverse spectral problem for non-self-adjoint Jac obi matrices. Int. Math. Res. Notices 2024 (2024) 6106 - 6139

2024
[12]

S. R. Garcia, E. Prodan, M. Putinar, Mathematical and physica l aspects of complex symmetric operators. J. Phys. A: Math. Theor. 47 (2014) 35300 1

2014
[13]

S. R. Garcia, M. Putinar, Complex symmetric operators and app lications. Trans. Amer. Math. Soc. 358 (2006) 1285 - 1315

2006
[14]

S. R. Garcia, M. Putinar, Complex symmetric operators and app lications II. Trans. Amer. Math. Soc. 359 (2007), 3913 - 3931

2007
[15]

N. I. Akhiezer, The classical moment problem and some related q uestions in analysis. SIAM, Philadelphia, PA, 2021. 14

2021
[16]

Haydock, The inverse of a linear operator

R. Haydock, The inverse of a linear operator. J. Phys. A: Math ., Nucl. and Gen. 7 (1974) 2120

1974
[17]

Singh, S

V. Singh, S. N. Biswas and K. Datta, Anharmonic oscillator and th e analytic theory of continued fractions. Phys. Rev. D 18 (1978) 1901

1978
[18]

A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mech anics. IOPP, Bristol, 1994

1994
[19]

Znojil, Fixed point perturbation theory and the potential r2 + λr 2/ (1 + gr 2)

M. Znojil, Fixed point perturbation theory and the potential r2 + λr 2/ (1 + gr 2). I. Analysis of convergence. J. Phys. A: Math. Gen. 17 (1984) 3441 - 3448

1984
[20]

Stoer, R

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis. Spr inger-Verlag, New York, 1980

1980
[21]

S. Graﬃ, V. Grecchi, A matrix continued-fraction solution for t he anharmonic- oscillator eigenvalues. Lett. Nuovo Cimento 12 (1975) 425 - 431

1975
[22]

Znojil, Generalized method of a resolvent operator expansio n

M. Znojil, Generalized method of a resolvent operator expansio n. J. Math. Phys. 18 (1977) 717 - 719

1977
[23]

Giannozzi, G

P. Giannozzi, G. Grosso, S. Moroni, G. P. Parravicini, The ordina ry and matrix continued fractions in the theoretical analysis of Hermitian and rela xation operators. Appl. Numer. Math. 4 (1988) 273 - 295

1988
[24]

C. M. M. Nex, The block Lanczos algorithm and the calculation of m atrix resolvents. Comp. Phys. Comm. 53 (1989) 141 - 146

1989
[25]

Bagarello, F

F. Bagarello, F. Gargano, F. Roccati, Tridiagonality, supersym metry and non self- adjoint Hamiltonians, J. Phys. A: Math. Theor. 52 (2019) 355203

2019
[26]

NIST Digital Library of Mathematical Functions: http://dlmf.nis t.gov/18
[27]

Pushnitski, F

A. Pushnitski, F. ˇStampach, A functional model and tridiagonalisation for symmetric anti-linear operators. J. Oper. Theory, to appear (arXiv preprin t arXiv:2402.01237)

arXiv
[28]

Znojil, Symmetrically anharmonic oscillators

M. Znojil, Symmetrically anharmonic oscillators. Phys. Rev. D 24 ( 1981) 903. 15

1981

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

F. M. Fern´ andez, On a class of non-Hermitian Hamiltonians with tr idiagonal matrix representation. Ann. Phys. (NY) 443 (2022) 169008

work page 2022

[2] [2]

F. G. Scholtz, H. B. Geyer and F. J. W. Hahne, Quasi-Hermitian op erators in quan- tum mechanics and the variational principle. Ann. Phys. (NY) 213 (1 992) 74

work page

[3] [3]

Messiah, Quantum Mechanics

A. Messiah, Quantum Mechanics. North Holland, Amsterdam, the Netherlands, 1961

work page 1961

[4] [4]

C. M. Bender, Making sense of nonhermitian Hamiltonians. Rep. Pr og. Phys. 70 (2007) 947 - 1018

work page 2007

[5] [5]

Mostafazadeh, Pseudo Hermitian representation of quantu m mechanics

A. Mostafazadeh, Pseudo Hermitian representation of quantu m mechanics. Int. J. Geom. Meth. Mod. Phys. 7 (2010) 1191 - 1306

work page 2010

[6] [6]

Bagarello, J.-P

F. Bagarello, J.-P. Gazeau, F. Szafraniec and M. Znojil, Eds., Non -Selfadjoint Op- erators in Quantum Physics: Mathematical Aspects. Wiley, Hoboke n, 2015

work page 2015

[7] [7]

Moiseyev, Non-Hermitian Quantum Mechanics

N. Moiseyev, Non-Hermitian Quantum Mechanics. Cambridge, CUP , 2011

2011

[8] [8]

A. D. Dente, R. A. Bustos-Mar´ un, H. M. Pastawski, Dynamical regimes of a quantum SW AP gate beyond the Fermi golden rule. Phys. Rev. A 78 (2008) 06 2116

work page 2008