The relativistic rotated harmonic oscillator

A. Balmaseda; D. Krejcirik; J.M. P\'erez-Pardo

REVIEW 2 major objections 5 minor 35 references

The relativistic rotated harmonic oscillator

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The relativistic rotated harmonic oscillator has a real discrete spectrum, but for any nonzero rotation its eigenfunctions fail to form a basis, with eigenprojector norms growing at the exact rate log sqrt((1+|sin θ|)/(1-|sin θ|)).

desk verdict Solid new result on eigenprojector blow-up for a relativistic rotated oscillator, but the upper pseudospectral inclusion in Theorem 1.2 rests on a false estimate in Proposition 5.2 and needs repair. read the letter →

arxiv 2411.16494 v2 pith:SR4FZXMU submitted 2024-11-25 math.SP math-phmath.MPquant-ph

A. Balmaseda , D. Krejcirik , J.M. Pérez-Pardo This is my paper · ORCID

classification math.SPmath-phmath.MPquant-ph MSC 34L4034L1547A1081Q12

keywords pseudospectrumresolventestimaterelativisticharmonicoscillatornon-self-adjointDiracoperatorrotatedDavieswildbasisproperties

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

This paper constructs a relativistic counterpart of the complex-rotated harmonic oscillator, a Dirac operator on spinors, and shows that despite a real, discrete spectrum, its eigenfunctions behave wildly for any nonzero rotation angle. The central result pins the growth of the spectral projectors to the exact exponential rate log sqrt((1+|sin θ|)/(1-|sin θ|)), which forces the eigenfunctions to form neither a Riesz nor a Schauder basis. It further proves that the pseudospectra extend far from the spectrum in a wedge determined by the rotation angle, making the operator numerically pathological and not similar to a self-adjoint one. The interest is that this is an exactly solvable non-self-adjoint Dirac model with the same illusory simplicity as the non-relativistic rotated oscillator.

What carries the argument

The load-bearing object is the supersymmetric square identity $H_θ^{2}$ = (S_θ + $m^{2}$) I + i α1 α2, where S_θ = -$e^{{-iθ}}$ ∂$x^{2}$ + $e^{{iθ}}$ $x^{2}$ is the rotated Schrödinger oscillator; the extra matrix term is diagonal, so $H_θ^{2}$ decomposes into two copies of S_θ plus constants. This reduces the spectral analysis of the Dirac operator to the already understood non-relativistic operator S_θ. The proof of Theorem 1.1 then uses explicit biorthonormal eigenfunctions built from the rotated Hermite functions φ_n, with the norm of the eigenprojectors bounded in terms of ||φ_n||^2, whose exponential growth is quoted from the existing literature on S_θ. For the pseudospectral statements, the same identity transfers known pseudospectral inclusions for S_θ to H_θ.

What would settle it

Compute the spectral projectors P_n^± numerically for a fixed rotation angle, say θ = π/4 and m = 1, for n up to a few hundred, and check whether log ||P_n^±||/n approaches log $\sqrt$((1+sin θ)/(1-sin θ)) ≈ 0.4407; a systematic deviation from this value would refute Theorem 1.1. Alternatively, test the resolvent claim (1.10) by evaluating ||(H_θ ∓ m - r $e^{{iϑ}}$)^{-1}|| along the ray ϑ = θ/4 in the wedge (−θ/2, θ/2) as r grows: if the norms stay bounded instead of diverging, the pseudospectral picture fails.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for the relativistic rotated harmonic oscillator H_θ = -iα1 $e^{{-iθ/2}}$ ∂x - α2 $e^{{iθ/2}}$ x + m α3 on $L^{2}$(R)^4, the spectral projectors P_n^± onto the eigenspaces of ±√(2n+$m^{2}$) satisfy lim_{n→∞} (log ||P_n^±||)/n = log $\sqrt$((1+|sin θ|)/(1-|sin θ|)). This is the exact Dirac analogue of the known growth for the rotated Schrödinger oscillator S_θ. As a consequence, whenever θ≠0 the eigenfunctions of H_θ form neither a Riesz nor a Schauder basis. The same supersymmetric identity $H_θ^{2}$ = (S_θ + $m^{2}$) I + i α1 α2 yields the spectrum σ(H_θ) = {±√(2n+$m^{2}$)} and, via known pseudospectral bounds for S_θ, a two-sided description of the ε-pseudospectra showing they are highly non-trivial.

Load-bearing premise

The whole argument leans on the quoted estimate (2.8) that the norm of the rotated Hermite functions grows at the exponential rate log sqrt((1+|sin θ|)/(1-|sin θ|)), and on the pseudospectral inclusion (2.9) for S_θ; both are taken from the existing literature rather than reproved here, and if either is wrong or misquoted the central theorems fail.

Editorial extensions

If this is right

For any θ ≠ 0, H_θ is not similar via a bounded and boundedly invertible transformation to a self-adjoint or normal operator, so the usual quantum-mechanical interpretation of observables fails for this model.
The eigenfunctions of H_θ do not form a Riesz or Schauder basis in L^2(R)^4, so expansions in eigenfunctions are unstable in a strong sense.
The ε-pseudospectrum of H_θ contains points arbitrarily far from the spectrum within the wedge |arg(z^2 - m^2)| ≤ θ - δ for any δ > 0, meaning numerical computation of eigenvalues and resonances is unstable.
Along rays inside the wedge (−θ/2, θ/2), the resolvent norm grows to infinity as the spectral parameter goes to infinity, as stated in equation (1.10).
In the non-relativistic limit c → ∞, the renormalized operator converges in norm-resolvent sense to a constant shift of the rotated Schrödinger oscillator S_θ, connecting the model to the well-known non-relativistic case.

Reading between the lines

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

A testable prediction left open by the paper is that for m = 0 the transition angle for pseudospectral growth is exactly θ/2, by analogy with the transition angle θ for S_θ; the numerical pseudospectra shown in the paper could be checked directly for this threshold.
The same supersymmetric transfer should work for other non-self-adjoint Dirac operators built from S_θ-type quadratic operators, for instance in higher dimensions or with external potentials, giving explicit exponential growth rates for their eigenprojectors.
If the exponential growth of eigenspace norms is generic for rotated relativistic oscillators, then any attempt to use such operators as quasi-Hermitian quantum models would need to confront this instability rather than assume a hidden self-adjointness.

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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

The main result—the exponential growth of the spectral projectors in Theorem 1.1—is new, correctly proved, and a worthwhile contribution to the non-self-adjoint Dirac literature. The construction of the model via H_theta^2 = S_theta + m^2 + i alpha1 alpha2 is transparent, and the non-relativistic limit is a nice addition. The lower pseudospectral inclusion in Theorem 1.2 follows cleanly from known results on S_theta and is not in question.

The soft spot is real and is exactly where the stress-test points. In Proposition 5.2 the authors claim the estimate ||Bpsi|| ≤ |tan(theta/2)| ||(A - i Im z)psi|| for all psi. That is false for small |Im z|. Take psi = (phi_0,0,0,0)^T with phi_0 the harmonic-oscillator ground state. Then A psi = 0 and ||B psi|| = sqrt(2) sin(theta/2) ||psi||, so the inequality would force |Im z| ≥ sqrt(2) cos(theta/2). The hypothesis of Proposition 5.2 with m = Re z = 0 only requires |Im z| > 0; for theta = pi/4 and z = i/5 the inequality crumbles. Since the second inclusion of Theorem 1.2 rests on this estimate, that containment is not established as written. The authors may still be right—numerically the pseudospectrum looks confined—but the proof needs a genuinely different argument, not a cosmetic repair. Theorem 1.1 and the first inclusion of Theorem 1.2 do not depend on this step and stand on their own.

The imported Schrödinger estimates (2.8)–(2.9) are another caveat, though not a flaw: the paper leans on established results about S_theta rather than reproving them, which is acceptable but leaves the Dirac results dependent on that external input. A self-contained treatment would have been stronger, though longer.

Who gets value: readers working on non-self-adjoint Dirac operators, pseudospectra, and wild basis behaviour will want this model and Theorem 1.1. The paper deserves a serious referee, but not a quick accept: the author should be asked to fix Proposition 5.2 or downgrade the upper inclusion to a conjecture. I would take it to review with that expectation.

Referee Report

2 major / 5 minor

Summary. The manuscript introduces the rotated relativistic harmonic oscillator H_θ on L^2(R)^4, defines it as a closed operator with compact resolvent, and analyzes its spectrum, eigenfunctions, and pseudospectra. The main positive results are the explicit spectrum σ(H_θ) = {±√(2n+m^2)}, the construction of biorthonormal eigenfunctions via the supersymmetric square H_θ^2 = S_θ + m^2 + iα_1α_2, the exponential growth of spectral projectors in Theorem 1.1, and two pseudospectral inclusions in Theorem 1.2. The proofs import the known estimates (2.8) and (2.9) for the rotated Schrödinger operator S_θ.

Significance. If the main results stand, the paper gives a valuable explicit non-self-adjoint Dirac model: it provides an exact relativistic analogue of Davies' rotated oscillator, exhibits wild basis properties, and gives detailed pseudospectral information without any parameter fitting. The explicit ladder-operator construction and the projector asymptotics in Theorem 1.1 are particularly clean and appear correct. However, the upper pseudospectral inclusion in Theorem 1.2 rests on a false intermediate estimate, so the two-sided description of the pseudospectra and the claimed optimality discussion are not established as written.

major comments (2)

[5, Proposition 5.2] The key estimate in the proof of Proposition 5.2 is not a consequence of (3.2)–(3.3). For k = Im z, the exact identities give ||(A − i Im z)ψ||² = cos²(θ/2)(base + τ) + (Im z)²||ψ||² and ||Bψ||² = sin²(θ/2)(base − τ), where base = ||ψ'||² + ||xψ||² and τ = −||ψ_1||² + ||ψ_2||² − ||ψ_3||² + ||ψ_4||². Comparing these shows that the claimed inequality ||Bψ|| ≤ |tan(θ/2)| ||(A − i Im z)ψ|| for all ψ requires (Im z)² ≥ 2 cos²(θ/2), not merely the angle condition stated in the proposition. The stated hypothesis permits arbitrarily small |Im z|: for instance, with m = 0, θ = π/4, z = i/5, and ψ = (φ_0, 0, 0, 0)^T where φ_0 is the normalized harmonic-oscillator ground state, one has ||Bψ||² = 2 sin²(θ/2) while |tan(θ/2)|² ||(A − i Im z)ψ||² = tan²(θ/2)(Im z)², so the inequality fails by a large margin. Therefore the application of Theorem 5.1 in Proposition 5.2 is invalid, and the second inclusion of Theorem 1.2 is not proved as written.
[5, after Eq. (5.2)] The optimality discussion is directly dependent on Proposition 5.2. Since that proposition's proof is invalid, the claims that the second inclusion of Theorem 1.2 is optimal for small θ, and the associated discussion of the transition angle f(θ), are currently unsupported. The first inclusion of Theorem 1.2 and Theorem 1.1 are independent of this issue and are not affected.

minor comments (5)

[3.1, Eq. (3.2)–(3.3)] The notation involving the symbol '~' is confusing: it appears both as an operator and as a scalar coefficient. Please define a named quantity, for example τ(ψ), so that the displayed identities are unambiguous.
[Figure 1 caption] The caption contains 'complex plain', which should be 'complex plane'.
[Keywords and MSC] The keyword 'oscilator' is a typo for 'oscillator', and 'MCS 2020' should be 'MSC 2020'.
[References] Reference [30] contains 'and et al.' in the author list; this should be cleaned up before publication.
[4.3, Lemma 4.5] The use of concavity of the logarithm is fine, but the line would be easier to read if the quantities a = |u_1^1|² + |u_1^3|² and b = |u_1^2|² + |u_1^4|² were introduced explicitly before the inequality.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central claims are derived from established results on the rotated Schrödinger operator, not from this paper's own conclusions.

full rationale

The derivation chain is input-independent. Theorem 1.1 estimates the Hθ eigenprojectors by reducing them to the explicit eigenfunctions (4.7)–(4.8) and inherits the known asymptotic (2.8) for the rotated Schrödinger eigenfunctions; this asymptotic is quoted from the survey [26] but the surrounding text cites the independent origins [4], [31], [18, Prop. 14.13], and [2]. Theorem 1.2's lower inclusion follows from the supersymmetric identity (1.8) plus the known pseudospectral inclusion (2.9), while the upper inclusion is a relative-bound calculation from (3.2)–(3.3). No fitted parameter is relabeled as a prediction, and no uniqueness theorem is imported from prior work by the present authors. The paper explicitly leaves the sharp transition angle as an open problem at the end of Section 5, which is a stated limitation rather than a circular move. The only self-citation is the survey [26], whose relevant estimates are documented as originating in independent earlier works, so it is not load-bearing. If Proposition 5.2 were challenged, that would be a correctness issue, not a circularity of the kind defined here.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central argument reduces the Dirac operator to the known rotated Schrödinger operator via the square identity; nothing is fitted. The only axioms are standard operator-theory theorems and the established S_theta results quoted in Section 2.2. No new physical entities are introduced. In particular, the paper does not fit free parameters to data.

assumptions (3)

standard math Known spectral and pseudospectral facts for the rotated Schrödinger operator S_theta, especially Eqs. (2.8) and (2.9), are valid as stated.
Invoked in Section 2.2 and used in Lemmas 4.4-4.5 and Theorem 1.2; originally proved in [4,31,18,2] and collected in the survey [26].
standard math Kato's perturbation theorem IV.1.16 (Theorem 5.1 in the paper) is applicable to the operators considered.
Used to define H_theta as a closed operator in Theorem 3.2 and to prove the outer pseudospectral bound in Proposition 5.2.
standard math The non-relativistic limit theorem from Thaller [33, Thm. 6.1 and Corol. 6.2] extends to the non-self-adjoint c-scaled operator in Theorem 4.6.
The paper states the proof is a direct application and notes that the self-adjointness in Thaller's argument is not essential; the spectra of the relevant intermediate operators are real.

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Pith. "Pith review of The relativistic rotated harmonic oscillator." pith.science (2026). https://pith.science/paper/SR4FZXMU

@misc{pith2026241116494,
  author       = {Pith},
  title        = {Pith review of: The relativistic rotated harmonic oscillator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SR4FZXMU}},
  note         = {Machine review of arXiv:2411.16494}
}

read the original abstract

We introduce a relativistic version of the non-self-adjoint operator obtained by a dilation analytic transformation of the quantum harmonic oscillator. While the spectrum is real and discrete, we show that the eigenfunctions do not form a basis and that the pseudospectra are highly non-trivial.

Figures

Figures reproduced from arXiv: 2411.16494 by the authors.

**Figure 2.** Numerical calculation of the pseudospectrum of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Visualisation of the (lack of) optimality of Theorem 1.2 for [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Aldaya, J

V. Aldaya, J. Bisquert, and J. Navarro-Salas, The quantum relativistic harmonic oscillator: Gener- alized Hermite polynomials , Phys. Lett. A 156 (1991), 381–385

work page 1991

[2] [2]

Resolvent estimates for one-dimensional Schr\"odinger operators with complex potentials

A. Arnal and P. Siegl, Resolvent estimates for one-dimensional Schr¨ odinger operators with complex potentials, J. Funct. Anal., to appear; preprint on arXiv:2203.15938 (2022)

work page Pith review arXiv 2022

[3] [3]

Bagarello, Examples of pseudo-bosons in quantum mechanics , Phys

F. Bagarello, Examples of pseudo-bosons in quantum mechanics , Phys. Lett. A 374 (2010), 3823– 3827

work page 2010

[4] [4]

Boulton, The non-self-adjoint harmonic oscillator, compact semigr oups and pseudospectra , J

L. Boulton, The non-self-adjoint harmonic oscillator, compact semigr oups and pseudospectra , J. Operator Theory 47 (2002), 413–429

work page 2002

[5] [5]

Spectral analysis of Dirac operators for dislocated potentials with a purely imaginary jump

L. Boulton, D. Krejˇ ciˇ r ´ ık, and T. Nguyen Duc,Spectral analysis of Dirac operators for dislocated potentials with a purely imaginary jump , arXiv:2409.06480 [math.SP] (2024)

work page Pith review arXiv 2024

[6] [6]

M. C. Cˆ amara and D. Krejˇ ciˇ r ´ ık,Complex-self-adjointness, Anal. Math. Phys. 13 (2023), 6

work page 2023

[7] [7]

Cassano, O

B. Cassano, O. O. Ibrogimov, D. Krejˇ ciˇ r ´ ık, and F.ˇStampach, Location of eigenvalues of non-self- adjoint discrete dirac operators , Ann. Henri Poincar´ e21 (2020), 2193–2217

work page 2020

[8] [8]

Cuenin, Estimates on complex eigenvalues for Dirac operators on the half-line, Integral Equ

J.-C. Cuenin, Estimates on complex eigenvalues for Dirac operators on the half-line, Integral Equ. Oper. Theory 79 (2014), 377–388. [9] , Eigenvalue bounds for Dirac and fractional Schr¨ odinger op erators with complex potentials , J. Funct. Anal. 272 (2017), 2987–3018

work page 2014