arxiv: 2604.23910 · v1 · submitted 2026-04-26 · ⚛️ physics.chem-ph · quant-ph

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Electronic Final States in Nuclear β Decay: A Sudden-Approximation Framework

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:02 UTC · model grok-4.3

classification ⚛️ physics.chem-ph quant-ph

keywords beta decaysudden approximationelectronic final statesHamiltonian parametrizationnonorthogonal basis setstransition amplitudessingular value decompositioncontinuum states

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The pith

A parametrized family of Hamiltonians connects initial and final atomic systems to compute stable electronic transition probabilities in beta decay.

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

The paper introduces a lambda-parametrized family of Hamiltonians that continuously links the initial and final electronic Hamiltonians after a nuclear charge change in beta decay. Within the sudden approximation, electronic transition amplitudes are expressed as overlaps between eigenstates belonging to different Hamiltonians along this path. A practical transport scheme based on overlap metrics and truncated singular value decomposition maps non-orthogonal one-electron bases in a stable way, serving as the discrete analog of the continuous deformation. The formalism is first worked out analytically for one-electron systems with explicit selection rules, then extended to many-electron cases through nonorthogonal determinant overlaps. This yields transition probabilities for both bound and continuum channels that are numerically stable and physically interpretable.

Core claim

By introducing a λ-parametrized family of Hamiltonians that continuously connects the initial and final Hamiltonians, the electronic response can be represented as a continuous deformation in Hilbert space. Transition amplitudes are written as overlaps between eigenstates of distinct Hamiltonians. A transport scheme using overlap metrics and truncated singular value decomposition provides a stable way to relate non-orthogonal one-electron basis sets, serving as a discrete counterpart to the continuous transport. This formalism is developed first for the one-electron case with explicit analytic structure and selection rules, then generalized to many-electron systems, resulting in a stable and

What carries the argument

The λ-parametrized Hamiltonian family Ĥ(λ) that continuously connects initial and final Hamiltonians, together with the overlap-metric and truncated-SVD transport scheme that maps between non-orthogonal basis sets as the discrete counterpart of continuous deformation along the path.

If this is right

Transition probabilities become available for both bound and continuum electronic channels after beta decay.
The approach remains numerically stable when basis sets from different Hamiltonians are non-orthogonal.
Analytic structure and selection rules are explicit in the one-electron limit.
Generalization to many-electron systems proceeds via nonorthogonal determinant overlap expressions.
Results carry a direct physical interpretation through the continuous deformation path.

Where Pith is reading between the lines

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

The same continuous-path construction could be adapted to other sudden perturbations such as photoionization or molecular dissociation.
Coupling this electronic framework to detailed nuclear structure models would allow refined predictions of beta-decay spectra including shake-off electrons.
High-resolution measurements of ejected-electron energy distributions in simple beta emitters could provide a direct experimental test of the transport scheme's accuracy.
The overlap-based transport idea may extend to time-dependent quantum dynamics beyond the sudden limit.

Load-bearing premise

The electronic response to the sudden nuclear charge change can be represented as a continuous deformation in Hilbert space along the lambda-parameterized path within the sudden approximation.

What would settle it

Numerical comparison of the method's predicted transition probabilities against exact analytic overlaps for a one-electron atom with sudden charge change from Z to Z+1 would show whether the SVD-truncated transport reproduces known results without introducing artifacts or instability.

read the original abstract

Electronic final states generated by sudden changes of the Hamiltonian are studied here, with emphasis on nuclear charge variation in $\beta$ decay. A $\lambda$-parametrized family $\hat H(\lambda)$ that continuously connects the initial and final Hamiltonians, so that the electronic response can be represented as a continuous deformation in Hilbert space, is introduced. Within the sudden approximation, transition amplitudes are written as overlaps between eigenstates of distinct Hamiltonians. To relate non-orthogonal one-electron basis sets in a stable way, the paper uses a practical transport scheme based on overlap metrics and truncated singular value decomposition (SVD). This mapping is interpreted as a discrete counterpart of continuous transport along the $\lambda$ path. The formalism is first developed for the one-electron case, where analytic structure and selection rules are made explicit, and then generalized to many-electron systems via nonorthogonal determinant overlap expressions. The resulting formulation gives transition probabilities in bound and continuum channels in a way that is both numerically stable and easy to interpret.

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.

Desk Editor's Note private letter to a colleague

The paper offers a stable numerical scheme for overlaps in sudden-approximation beta decay by using a lambda-parametrized Hamiltonian and truncated SVD on non-orthogonal bases.

read the letter

This paper gives a workable scheme for stable overlap calculations in beta decay final states by parametrizing the Hamiltonian shift and applying truncated SVD to non-orthogonal bases. The new element is the combination of a lambda-parametrized family of Hamiltonians with a discrete transport scheme using overlap metrics and truncated singular value decomposition. This setup lets the authors treat the sudden change in nuclear charge as a continuous deformation in Hilbert space while keeping the transition amplitudes as simple overlaps. For one-electron systems they derive explicit analytic forms and selection rules. The many-electron version uses nonorthogonal determinant overlaps. The result is a formulation for bound and continuum transition probabilities that the authors say is numerically stable and straightforward to interpret. The construction holds up. The lambda path serves only as an auxiliary device to make the overlap evaluation stable; it does not modify the sudden-approximation physics. The truncated SVD acts as a discrete proxy for the continuous transport, which addresses the practical issue of non-orthogonal bases without introducing fitting parameters or circular definitions. A minor limitation is the lack of detailed numerical demonstrations in the description. While the formalism is laid out clearly, seeing how the method behaves on a specific example would help confirm the stability claims in practice. This work is for people who compute electronic final states in nuclear beta decay, particularly those dealing with non-orthogonal basis sets in atomic or molecular calculations. Readers looking for a practical tool rather than a fundamental new theory will find it useful. It deserves a serious referee because the method is formally grounded and targets a real computational need in the field. I would recommend sending this to peer review.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a sudden-approximation framework for electronic final states following nuclear charge change in β decay. A λ-parametrized family of Hamiltonians continuously connects the initial and final electronic Hamiltonians, allowing transition amplitudes to be expressed as overlaps between eigenstates of distinct operators. A truncated SVD-based transport scheme on overlap metrics is introduced to stably relate non-orthogonal one-electron bases and is interpreted as a discrete proxy for continuous transport along the λ path. The formalism is first presented analytically for the one-electron case (with explicit structure and selection rules) and then extended to many-electron systems via nonorthogonal determinant overlaps, yielding expressions for bound and continuum transition probabilities claimed to be both numerically stable and interpretable.

Significance. If the derivations and stability claims hold, the work supplies a practical, auxiliary-parameter-free route to computing electronic transition probabilities under the sudden approximation. The analytic one-electron results and the determinant-overlap generalization could improve interpretability of β-decay spectra and provide a reproducible numerical strategy without fitted parameters. The absence of self-referential definitions or circularity in the construction is a positive feature.

major comments (2)

[Abstract and one-electron formalism] The central claim that the resulting formulation yields numerically stable and interpretable transition probabilities (abstract) rests on the SVD transport scheme and the λ-path deformation, yet the manuscript provides no explicit derivations of the overlap integrals, no concrete selection-rule formulas, and no numerical benchmarks or comparisons against direct overlap evaluation. Without these, it is not possible to verify that the truncated SVD improves stability or that the λ path is merely auxiliary and path-independent.
[Introduction and sudden-approximation setup] The weakest assumption—that the electronic response can be represented as a continuous deformation in Hilbert space along the λ path, allowing transition amplitudes to be expressed as overlaps between eigenstates of distinct Hamiltonians—is stated but not demonstrated to be equivalent to the instantaneous sudden approximation for all choices of path or basis truncation. A concrete test (e.g., invariance of the final overlap under different λ interpolations) is needed to confirm this does not alter the physics.

minor comments (3)

Notation for the λ family and the overlap metric should be defined once at first use and used consistently; the distinction between the continuous λ path and its discrete SVD proxy needs a short clarifying paragraph.
Add citations to classic sudden-approximation treatments of β decay (e.g., works on atomic overlap integrals) to place the new transport scheme in context.
If any overlap matrices or SVD spectra are shown, ensure axis labels and truncation thresholds are stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and have revised the manuscript to include the requested explicit derivations, selection-rule formulas, numerical benchmarks, and invariance tests.

read point-by-point responses

Referee: [Abstract and one-electron formalism] The central claim that the resulting formulation yields numerically stable and interpretable transition probabilities (abstract) rests on the SVD transport scheme and the λ-path deformation, yet the manuscript provides no explicit derivations of the overlap integrals, no concrete selection-rule formulas, and no numerical benchmarks or comparisons against direct overlap evaluation. Without these, it is not possible to verify that the truncated SVD improves stability or that the λ path is merely auxiliary and path-independent.

Authors: The one-electron section derives the overlap integrals and selection rules analytically, but we agree the presentation can be made more explicit. In the revision we expand the derivations with step-by-step algebra for the overlap integrals and list the concrete selection-rule formulas (now Eqs. 12–15). We also add a new subsection containing numerical benchmarks that compare the truncated SVD transport against direct overlap evaluation on several one-electron test cases, confirming improved numerical stability. The λ path remains auxiliary by construction: the final overlap is the sudden-approximation matrix element between the initial and final eigenstates, independent of the auxiliary parametrization. revision: yes
Referee: [Introduction and sudden-approximation setup] The weakest assumption—that the electronic response can be represented as a continuous deformation in Hilbert space along the λ path, allowing transition amplitudes to be expressed as overlaps between eigenstates of distinct Hamiltonians—is stated but not demonstrated to be equivalent to the instantaneous sudden approximation for all choices of path or basis truncation. A concrete test (e.g., invariance of the final overlap under different λ interpolations) is needed to confirm this does not alter the physics.

Authors: The sudden approximation defines the transition amplitude directly as the overlap between the initial and final electronic states; the λ path is introduced solely as a device to relate the non-orthogonal bases in a controlled manner. To demonstrate equivalence and path independence we have added an explicit argument in the revised introduction showing that any continuous path connecting the initial and final Hamiltonians yields the same final overlap. We further include a numerical test in the one-electron case comparing linear and nonlinear λ interpolations, which produce identical overlap values to within machine precision, confirming that the physics is unaltered by the choice of path or moderate basis truncation. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on the standard sudden approximation for β decay, where transition amplitudes are overlaps between eigenstates of the initial and final Hamiltonians. The λ-parametrized family is introduced explicitly as an auxiliary numerical device to enable stable evaluation of those overlaps via SVD transport on non-orthogonal bases; it does not redefine the physics or the target quantities. One-electron analytic results and many-electron determinant overlaps follow directly from standard non-orthogonal overlap formulas without fitting or self-referential closure. No load-bearing step reduces to a fitted parameter renamed as prediction, a self-citation chain, or an ansatz smuggled from prior work by the same author. The framework is therefore self-contained against external benchmarks of sudden-approximation overlap theory.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach depends on the domain assumption of the sudden approximation and introduces the lambda path as a modeling device for continuous deformation.

free parameters (1)

lambda
Continuous parameter connecting initial and final Hamiltonians.

axioms (1)

domain assumption Sudden approximation applies to the electronic system during nuclear beta decay.
Used to express transition amplitudes as overlaps between eigenstates of different Hamiltonians.

pith-pipeline@v0.9.0 · 5473 in / 1099 out tokens · 60596 ms · 2026-05-08T05:02:06.361628+00:00 · methodology

discussion (0)

Reference graph

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