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The Petrov–Galerkin approach as a unifying framework for numerical solutions of the Lippmann–Schwinger equation.

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Title: The Petrov–Galerkin approach as a unifying framework for numerical solutions of the Lippmann–Schwinger equation.
Authors: Kuruoğlu, Zeki C.1 (AUTHOR) kuruoglu@bilkent.edu.tr
Source: European Physical Journal A -- Hadrons & Nuclei. May2026, Vol. 62 Issue 5, p1-13. 13p.
Subjects: Galerkin methods, Quantum scattering, Integral equations, Numerical solutions to equations, Approximation theory, Variational approach (Mathematics), Numerical analysis
Abstract: This paper aims to establish the Petrov–Galerkin approach as a unifying framework for various variational and degenerate kernel methods for the solution of the Lippmann–Schwinger (LS) equation of quantum scattering theory. Both transition-operator and resolvent-operator versions of the LS equation in momentum representation are considered. Within the PG formulation, commonly used schemes such as Galerkin, collocation, Schwinger variational and Bateman methods are shown to arise as specific realizations corresponding to particular choices of the approximation and test spaces. Interconnections of different realizations of the PG approach with finite-rank approximations of potential and free resolvent operators via oblique projectors are explored. In particular, new variants of Schwinger and Newton variational methods are obtained as special realizations of the PG approach. A total of sixteen distinct realizations of the Petrov–Galerkin and projection schemes are tested on a model s-wave scattering problem, and representative numerical results are presented to benchmark and compare the various formulations. [ABSTRACT FROM AUTHOR]
Copyright of European Physical Journal A -- Hadrons & Nuclei is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: The Petrov–Galerkin approach as a unifying framework for numerical solutions of the Lippmann–Schwinger equation.
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  Data: <searchLink fieldCode="DE" term="%22Galerkin+methods%22">Galerkin methods</searchLink><br /><searchLink fieldCode="DE" term="%22Quantum+scattering%22">Quantum scattering</searchLink><br /><searchLink fieldCode="DE" term="%22Integral+equations%22">Integral equations</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+solutions+to+equations%22">Numerical solutions to equations</searchLink><br /><searchLink fieldCode="DE" term="%22Approximation+theory%22">Approximation theory</searchLink><br /><searchLink fieldCode="DE" term="%22Variational+approach+%28Mathematics%29%22">Variational approach (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink>
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  Data: This paper aims to establish the Petrov–Galerkin approach as a unifying framework for various variational and degenerate kernel methods for the solution of the Lippmann–Schwinger (LS) equation of quantum scattering theory. Both transition-operator and resolvent-operator versions of the LS equation in momentum representation are considered. Within the PG formulation, commonly used schemes such as Galerkin, collocation, Schwinger variational and Bateman methods are shown to arise as specific realizations corresponding to particular choices of the approximation and test spaces. Interconnections of different realizations of the PG approach with finite-rank approximations of potential and free resolvent operators via oblique projectors are explored. In particular, new variants of Schwinger and Newton variational methods are obtained as special realizations of the PG approach. A total of sixteen distinct realizations of the Petrov–Galerkin and projection schemes are tested on a model s-wave scattering problem, and representative numerical results are presented to benchmark and compare the various formulations. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of European Physical Journal A -- Hadrons & Nuclei is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – SubjectFull: Galerkin methods
        Type: general
      – SubjectFull: Quantum scattering
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      – SubjectFull: Integral equations
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      – SubjectFull: Numerical solutions to equations
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      – SubjectFull: Approximation theory
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              Text: May2026
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