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Schr\"odinger Models for Solutions of the Bethe-Salpeter Equation in Minkowski Space. II. Fermionic Bound-State Constituents
Mathematical Physics · Submitted 27 Aug 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This leads to the adaptation of explicit models that provide an overview accounting for the Bethe-Salpeter formalism's complexities.
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The abstract, sentence by sentence
In view of the obstacles encountered in any attempts to solve the Minkowski-space Bethe-Salpeter equation for bound states of two fermions, we study the possibility to model the bound-state features, at least at a qualitative level, by a Schr\"odinger description. Such a nonrelativistic potential model can be constructed by applying, to any given Bethe-Salpeter spectral data, "geometric spectral inversion" in its recently extended form, which tolerates also singular potentials. This leads to the adaptation of explicit models that provide an overview accounting for the Bethe-Salpeter formalism's complexities.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In view of the obstacles encountered in any attempts to solve the Minkowski-space Bethe-Salpeter equation for bound states of two fermions, we study the possibility to model the bound-state features, at least at a qualitative level, by a Schr\"odinger description. | contribution 0.49 | 0.41 | 0.11 | hedged possibility 0.86 |
| s1 | Such a nonrelativistic potential model can be constructed by applying, to any given Bethe-Salpeter spectral data, "geometric spectral inversion" in its recently extended form, which tolerates also singular potentials. | method 0.99 | 0.46 | 0.12 | asserted 0.85 |
| s2 | This leads to the adaptation of explicit models that provide an overview accounting for the Bethe-Salpeter formalism's complexities. | contribution 0.41 | 0.73 | 0.28 | asserted 0.97 |
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The paper
Original abstract
In view of the obstacles encountered in any attempts to solve the Minkowski-space Bethe-Salpeter equation for bound states of two fermions, we study the possibility to model the bound-state features, at least at a qualitative level, by a Schr\"odinger description. Such a nonrelativistic potential model can be constructed by applying, to any given Bethe-Salpeter spectral data, "geometric spectral inversion" in its recently extended form, which tolerates also singular potentials. This leads to the adaptation of explicit models that provide an overview accounting for the Bethe-Salpeter formalism's complexities.
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References
How the literature uses it
1 arXiv paper cite this in 1 sentence
Papers citing it
Cites it as background 1 paper
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Progress made using the Minkowski-space formulation of the BSE to study the $qq̅$ system is ongoing and can be traced via [ref] .
Spectra of heavy quarkonia in a Bethe-Salpeter-equation approach · Introduction · 0.94
What it cites · 6 sentences without a confident label
Uses the method or tool of 2 papers
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We now use our inversion theory [ref] to find the potential shape $f(r)$ in the Schrödinger Hamiltonian $H,$ Eq. ([ref]), that, for any spectral data set in Table [ref], generates those binding energies $E$ for the given values of the coupling parameter $v$.
Geometric spectral inversion for singular potentials · Construction of effective potentials by geometric spectral inversion · 0.99
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Meanwhile, we have at our disposal a geometric spectral inversion theory [ref] which, if $E=F(v)$ is the lowest eigenvalue of the Schrödinger Hamiltonian $H≡-Δ+v f(r),$ reconstructs from the given spectral curve $F(v)$ the underlying potential shape $f(r).$ By first analyzing the fermion data expressed in this manner, as we had done earlier [ref] for interacting bosons, we eventually made an elementary discovery: namely, when there exists a nonzero critical value $v_0$ of the coupling $v$ and the model Hamiltonian is written in the form $H=-Δ+(v-v_0) f(r),$ then the spectral data $F(v)$ found for the Bethe–Salpeter two-fermion problem are accurately represented as the eigenvalues of that Hamiltonian $H$ for which the potential shape has, for $a>0,$ the elementary form $f(r)=-a/r+b.$ As more eigenvalue data become available, we expect to be able to translate the essential features of such
Schroedinger models for solutions of the Bethe-Salpeter equation in Minkowski space · Conclusion · 0.88
Uses data or values from 4 papers
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We use as input BS findings from exchange of a single scalar, pseudoscalar, or massless vector boson reported in Refs. [ref] , and invert the data by our generalization to singular potentials [ref] of a previously formulated inversion technique [ref] .
Solution of the Bethe-Salpeter equation in Minkowski space for a two fermion system · Introduction: Motivation and Incentive · 0.97
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We use as input BS findings from exchange of a single scalar, pseudoscalar, or massless vector boson reported in Refs. [ref] , and invert the data by our generalization to singular potentials [ref] of a previously formulated inversion technique [ref] .
Solving Bethe-Salpeter equation for two fermions in Minkowski space · Introduction: Motivation and Incentive · 0.97
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We use as input BS findings from exchange of a single scalar, pseudoscalar, or massless vector boson reported in Refs. [ref] , and invert the data by our generalization to singular potentials [ref] of a previously formulated inversion technique [ref] .
Solutions of the Bethe-Salpeter equation in Minkowski space and applications to electromagnetic form factors · Introduction: Motivation and Incentive · 0.97
Cites as background 5 papers
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In their discussion [ref] of the homogeneous Bethe–Salpeter equation in Minkowski space, Carbonell and Karmanov consider bound states of two fermionic constituents of equal masses $m,$ bound by exchanging between these constituents a single boson of mass $μ$ and of either scalar, or pseudoscalar, or vector Lorentz nature.
Solution of the Bethe-Salpeter equation in Minkowski space for a two fermion system · Bethe–Salpeter bound-state energies as input data to spectral inversion · 0.80
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In their discussion [ref] of the homogeneous Bethe–Salpeter equation in Minkowski space, Carbonell and Karmanov consider bound states of two fermionic constituents of equal masses $m,$ bound by exchanging between these constituents a single boson of mass $μ$ and of either scalar, or pseudoscalar, or vector Lorentz nature.
Solving Bethe-Salpeter equation for two fermions in Minkowski space · Bethe–Salpeter bound-state energies as input data to spectral inversion · 0.80
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In their discussion [ref] of the homogeneous Bethe–Salpeter equation in Minkowski space, Carbonell and Karmanov consider bound states of two fermionic constituents of equal masses $m,$ bound by exchanging between these constituents a single boson of mass $μ$ and of either scalar, or pseudoscalar, or vector Lorentz nature.
Solutions of the Bethe-Salpeter equation in Minkowski space and applications to electromagnetic form factors · Bethe–Salpeter bound-state energies as input data to spectral inversion · 0.80
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