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arxiv: 2605.30536 · v2 · pith:FMBSXRKBnew · submitted 2026-05-28 · ✦ hep-th · hep-lat· quant-ph

Ambiguity problem of the Bootstrap Method in Quantum Mechanics

Takeshi Morita , Worapat Piensuk , Pushkar Soni This is my paper

Pith reviewed 2026-06-29 05:55 UTC · model grok-4.3

classification ✦ hep-th hep-latquant-ph
keywords bootstrap methodquantum mechanicsenergy spectrumambiguitypositivity conditionsmatrix modelsstatistical models
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0 comments X

The pith

The bootstrap method in quantum mechanics produces inconsistent spectra when potentials mix polynomial and exponential terms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the bootstrap method, which solves for energy eigenvalues through recursive relations and positivity constraints on moments, does not always return the physically correct spectrum. When the same potential is written using different functional forms, such as adding an exponential piece to a polynomial, the computed energies shift or become ambiguous. The same dependence on representation appears when extracting expectation values of operators that belong to different classes. Because the method is applied to quantum mechanics, statistical models, and matrix models, this ambiguity limits its reliability without extra rules. The authors outline three possible fixes.

Core claim

The bootstrap method suffers from an ambiguity problem: it fails to yield the correct spectrum when the potential contains different types of functions, such as polynomial and exponential terms. Similarly, the bootstrap method may break down when evaluating the expectation values of operators of different types. This issue can arise in a wide range of systems, including statistical models and matrix models.

What carries the argument

The bootstrap equations obtained from the Schrödinger equation together with positivity conditions on operator expectation values; these relations are meant to constrain the spectrum but turn out to depend on the functional representation chosen for the potential or operators.

If this is right

  • Energy spectra obtained by bootstrap become representation-dependent for any potential that cannot be written with a single functional class.
  • Expectation values extracted from the same bootstrap solution differ when operators of unlike types are used to close the equations.
  • The ambiguity appears in matrix models and statistical systems that employ similar positivity or moment methods.
  • Three concrete resolutions are proposed to restore uniqueness.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A canonical choice of basis or truncation scheme may be needed before applying bootstrap to general potentials.
  • Similar representation dependence could appear in other positivity-based or recursive numerical schemes used in quantum mechanics.
  • Direct numerical checks on simple mixed potentials would measure the magnitude of the discrepancy and test proposed fixes.

Load-bearing premise

The bootstrap equations and positivity conditions are assumed to produce a unique, correct spectrum independent of how the potential is expressed in terms of different functional forms.

What would settle it

Apply the bootstrap truncation to the potential x squared plus exponential of minus x squared, once using only polynomial moments and once including exponential moments, then compare both outputs against an independent numerical solution of the Schrödinger equation.

Figures

Figures reproduced from arXiv: 2605.30536 by Pushkar Soni, Takeshi Morita, Worapat Piensuk.

Figure 1
Figure 1. Figure 1: The spectrum of the Hamiltonian with the mixed potential (2.2) for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical bootstrap result for the anharmonic oscillator (2.1). The colored regions show [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Allowed regions for the mixed potential (2.2) with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Allowed regions of the mixed potential (2.2) for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The allowed regions of the mixed potential (2.2) for [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Allowed regions of the mixed potential (2.2) for [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Allowed regions of the mixed potential (2.2) for [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

The bootstrap method for quantum mechanics is a powerful tool for computing the energy eigenvalues of a Hamiltonian. However, we point out that this method suffers from an ambiguity problem: it fails to yield the correct spectrum when the potential contains different types of functions, such as polynomial and exponential terms. Similarly, the bootstrap method may break down when evaluating the expectation values of operators of different types. This issue can arise in a wide range of systems, including statistical models and matrix models. We propose three possible resolutions to this problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The manuscript claims that the bootstrap method for computing energy eigenvalues in quantum mechanics suffers from an ambiguity problem: it fails to yield the correct spectrum for potentials containing mixed functional forms (e.g., polynomial and exponential terms) and may break down when evaluating expectation values of operators of different types. The issue is asserted to arise in a wide range of systems including statistical and matrix models, and three possible resolutions are proposed.

Significance. If substantiated with explicit operator bases and spectrum comparisons, the identification of an intrinsic ambiguity would be significant for the bootstrap approach in QM, as it could restrict applicability to mixed potentials and require modifications to the standard positivity-constrained moment-matrix construction. The proposal of resolutions is a constructive element, but the absence of any derivations, numerical examples, or operator-algebra details in the manuscript prevents a full assessment of impact or novelty.

major comments (3)
  1. [Abstract] Abstract: the central claim that the bootstrap equations and positivity conditions produce inconsistent spectra for the same Hamiltonian when rewritten in polynomial versus exponential bases is load-bearing, yet no explicit operator set, commutation relations, moment matrix, or numerical spectrum comparison is provided to isolate this effect from truncation artifacts.
  2. [Abstract] Abstract: the assertion that the method 'fails to yield the correct spectrum' for mixed potentials requires demonstration that enlarging the operator algebra to include both functional types simultaneously still yields inconsistency; without this, the discrepancy may trace to incomplete closure under [H, O] = 0 rather than an inherent ambiguity.
  3. [Abstract] Abstract: no details are supplied on the three proposed resolutions (e.g., whether they modify the positivity conditions, enlarge the basis, or introduce auxiliary constraints), preventing evaluation of whether they restore uniqueness without new free parameters or circularity.
minor comments (1)
  1. [Abstract] The abstract states applicability to statistical and matrix models but provides no indication of how the ambiguity manifests or is resolved in those contexts.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The manuscript is a concise note identifying an ambiguity in the bootstrap method. We agree that explicit examples, operator details, and descriptions of the resolutions are required for a full assessment and will incorporate them in the revision.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the bootstrap equations and positivity conditions produce inconsistent spectra for the same Hamiltonian when rewritten in polynomial versus exponential bases is load-bearing, yet no explicit operator set, commutation relations, moment matrix, or numerical spectrum comparison is provided to isolate this effect from truncation artifacts.

    Authors: We agree that the claim requires explicit support to distinguish it from truncation effects. In the revised manuscript we will supply concrete operator sets for both bases, the relevant commutation relations, the explicit moment-matrix construction, and controlled numerical spectrum comparisons that demonstrate the inconsistency persists under systematic enlargement of the truncation. revision: yes

  2. Referee: [Abstract] Abstract: the assertion that the method 'fails to yield the correct spectrum' for mixed potentials requires demonstration that enlarging the operator algebra to include both functional types simultaneously still yields inconsistency; without this, the discrepancy may trace to incomplete closure under [H, O] = 0 rather than an inherent ambiguity.

    Authors: This distinction is important. The revision will include an explicit calculation in which the operator algebra is enlarged to contain both polynomial and exponential operators at once; we will show that the positivity constraints still produce inconsistent spectra, thereby isolating the ambiguity from simple closure issues. revision: yes

  3. Referee: [Abstract] Abstract: no details are supplied on the three proposed resolutions (e.g., whether they modify the positivity conditions, enlarge the basis, or introduce auxiliary constraints), preventing evaluation of whether they restore uniqueness without new free parameters or circularity.

    Authors: We will add a dedicated section describing each of the three resolutions in detail, specifying the modifications to the positivity conditions or basis, any auxiliary constraints introduced, and numerical tests confirming that uniqueness is restored without introducing free parameters or circular reasoning. revision: yes

Circularity Check

0 steps flagged

No circularity in observation of bootstrap ambiguity

full rationale

The paper's central claim is an observational identification that the standard bootstrap equations (from [H, O] = 0 and moment-matrix positivity) produce inconsistent results for mixed polynomial/exponential potentials or operator types. No derivation chain is advanced that reduces a 'prediction' or spectrum result to a fitted parameter or self-citation by construction; the text instead flags a practical limitation and lists proposed resolutions. The analysis therefore remains self-contained against external benchmarks of the bootstrap method and does not rely on load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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discussion (0)

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