Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:hep-th/9710143 · Modern Physics Letters A

On Convergence of the Schwinger–De Witt Expansion

V. A. Slobodenyuk

Published Dec 7, 1997• 1 citation
arXivHTMLPDF

Abstract

The Schwinger–De Witt expansion for the evolution operator kernel of the Schrödinger equation is studied for convergence. It is established that divergence of this expansion which is usually implied for all continuous potentials, excluding those of the form V(q)=aq2+bq+c, takes place only if the coupling constant g is treated as an independent variable. But the expansion may be convergent for some kinds of potentials and for some discrete values of charge, if the latter is considered as fixed parameter. Class of such potentials is interesting because inside it the property of discreteness of the charge in nature is reproduced in theory in the natural way.

DOI
10.1142/s0217732397003010

Related papers

On Convergence of the Schwinger-DeWitt Expansion

V. A. Slobodenyuk

The Schwinger - DeWitt expansion for the evolution operator kernel of the Schrodinger equation is studied for convergence. It is established that divergence of this expansion which is usually implied for all continuous potentials, excluding ones of the form V(q)=aq^2+bq+c, really takes place only if the coupling constant g is treated as independent variable. But the expansion may be convergent for some kinds of the potentials and for some discrete values of the charge, if the latter is considered as fixed parameter. Class of such potentials is interesting because inside of it the property of discreteness of the charge in the nature is reproduced in the theory in natural way.

•Jan 1, 1997•1 citation

Examples of potentials with convergent Schwinger - DeWitt expansion

V. A. Slobodenyuk

Convergence of the Schwinger --- DeWitt expansion for the evolution operator kernel for special class of potentials is studied. It is shown, that this expansion, which is in general case asymptotic, converges for the potentials considered (widely used, in particular, in one-dimensional many-body problems), and besides, convergence takes place only for definite discrete values of the coupling constant. For other values of the charge divergent expansion determines the kernels having essential singularity at origin (beyond usual $δ$-function). If one consider only this class of potentials then one can avoid many problems, connected with asymptotic expansions, and one get the theory with discrete values of the coupling constant that is in correspondence with discreteness of the charge in the nature. This approach can be transmitted into the quantum field theory.

arXiv:hep-th/9810079•Oct 12, 1998•0 citations

Analytical Properties of Solutions of the Schrodinger Equation and Quantization of Charge

V. A. Slobodenyuk

The Schwinger--DeWitt expansion for the evolution operator kernel is used to investigate analytical properties of the Schrödinger equation solution in time variable. It is shown, that this expansion, which is in general asymptotic, converges for a number of potentials (widely used, in particular, in one-dimensional many-body problems), and besides, the convergence takes place only for definite discrete values of the coupling constant. For other values of charge the divergent expansion determines the functions having essential singularity at origin (beyond usual $δ$-function). This does not permit one to fulfil the initial condition. So, the function obtained from the Schrödinger equation cannot be the evolution operator kernel. The latter, rigorously speaking, does not exist in this case. Thus, the kernel exists only for definite potentials, and moreover, at the considered examples the charge may have only quantized values.

arXiv:hep-th/9506134•Jun 19, 1995•0 citations

Eigenvalue problem for bundle of operators and the sturm expansion method in quantum mechanics

S. I. Nikitin, A. I. Sherstyuk

Different variants of the generalized formulation of the eigenvalue problem related to a single-particle Schrödinger equation are considered. The nature of the spectrum of problems for a bundle of operators containing the Hamiltonian of the system is analyzed in relation to the form of the weighting operator. For rather general conditions, corresponding systems of eigenfunctions provide diagonal representations of the Green’s function and provide of a rather fast convergence of expansions over intermediate states. Cases of potentials with Coulomb asymptotics and short-range potentials are considered.

Optics and Spectroscopy•Apr 1, 2009•0 citations

The Asymptotic Behavior of Perturbation Expansions in Many-Body Problem and Superconducting State

Kenji Fukushima, Nobuyuki Fukuda

The convergence problem of perturbation expansions in the many-body problem is discussed by employing some simple models. The expectation value of rhe resolvent operator is analytic around the origin of the coupling constant so long as the system is enclosed in a finite volume, but as the volume tends to infinity with the density of particles kept constant, the question arises whether this limit is allowed to be taken prior to the sum of the perturbation series. Even if this is the case, the resulting series turns out to be usually divergent, but in some models we consider we can find a function whose asymptotic expansion coincides with this series. It is shown that for the repulsive interactions the function thus obtained gives a correct answer, while for the attractive interactions which lead to a superconducting state it is far from the true one. The asymptotic behavior of the perturbation series in the latter case is examined in some detail.

Progress of Theoretical Physics•Nov 1, 1962•4 citations

The Baker-Campbell-Hausdorff formula and the convergence of the Magnus expansion

Serge Klarsfeld, José Ángel Oteo

The authors show that for a wide class of dynamical systems (described by Hamiltonians of the form usually considered in time-dependent perturbation theory) the divergence of the Magnus expansion in the Schrodinger picture for large time intervals is due to pole singularities inherent to the Baker-Campbell-Hausdorff formula.

Journal of Physics A Mathematical and General•Nov 7, 1989•37 citations
The Schwinger–De Witt expansion for the evolution operator kernel of the Schrödinger equation is studied for convergence. It is established that divergence of this expansion which is usually implied for all continuous potentials, excluding those of the form V(q)=aq2+bq+c, takes place only if the coupling constant g is treated as an independent variable. But the expansion may be convergent for some kinds of potentials and for some discrete values of charge, if the latter is considered as fixed parameter. Class of such potentials is interesting because inside it the property of discreteness of the charge in nature is reproduced in theory in the natural way.
The Schwinger - DeWitt expansion for the evolution operator kernel of the Schrodinger equation is studied for convergence. It is established that divergence of this expansion which is usually implied for all continuous potentials, excluding ones of the form V(q)=aq^2+bq+c, really takes place only if the coupling constant g is treated as independent variable. But the expansion may be convergent for some kinds of the potentials and for some discrete values of the charge, if the latter is considered as fixed parameter. Class of such potentials is interesting because inside of it the property of discreteness of the charge in the nature is reproduced in the theory in natural way.
Convergence of the Schwinger --- DeWitt expansion for the evolution operator kernel for special class of potentials is studied. It is shown, that this expansion, which is in general case asymptotic, converges for the potentials considered (widely used, in particular, in one-dimensional many-body problems), and besides, convergence takes place only for definite discrete values of the coupling constant. For other values of the charge divergent expansion determines the kernels having essential singularity at origin (beyond usual δδδ-function). If one consider only this class of potentials then one can avoid many problems, connected with asymptotic expansions, and one get the theory with discrete values of the coupling constant that is in correspondence with discreteness of the charge in the nature. This approach can be transmitted into the quantum field theory.
The Schwinger--DeWitt expansion for the evolution operator kernel is used to investigate analytical properties of the Schrödinger equation solution in time variable. It is shown, that this expansion, which is in general asymptotic, converges for a number of potentials (widely used, in particular, in one-dimensional many-body problems), and besides, the convergence takes place only for definite discrete values of the coupling constant. For other values of charge the divergent expansion determines the functions having essential singularity at origin (beyond usual δδδ-function). This does not permit one to fulfil the initial condition. So, the function obtained from the Schrödinger equation cannot be the evolution operator kernel. The latter, rigorously speaking, does not exist in this case. Thus, the kernel exists only for definite potentials, and moreover, at the considered examples the charge may have only quantized values.
Different variants of the generalized formulation of the eigenvalue problem related to a single-particle Schrödinger equation are considered. The nature of the spectrum of problems for a bundle of operators containing the Hamiltonian of the system is analyzed in relation to the form of the weighting operator. For rather general conditions, corresponding systems of eigenfunctions provide diagonal representations of the Green’s function and provide of a rather fast convergence of expansions over intermediate states. Cases of potentials with Coulomb asymptotics and short-range potentials are considered.
The convergence problem of perturbation expansions in the many-body problem is discussed by employing some simple models. The expectation value of rhe resolvent operator is analytic around the origin of the coupling constant so long as the system is enclosed in a finite volume, but as the volume tends to infinity with the density of particles kept constant, the question arises whether this limit is allowed to be taken prior to the sum of the perturbation series. Even if this is the case, the resulting series turns out to be usually divergent, but in some models we consider we can find a function whose asymptotic expansion coincides with this series. It is shown that for the repulsive interactions the function thus obtained gives a correct answer, while for the attractive interactions which lead to a superconducting state it is far from the true one. The asymptotic behavior of the perturbation series in the latter case is examined in some detail.