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arXiv:hep-th/9810079

Examples of potentials with convergent Schwinger - DeWitt expansion

V. A. Slobodenyuk

Published Oct 12, 1998• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.hep-th/9810079

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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 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.
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.
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 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.
In a recent paper [Europhys. Lett. 95, 13001 (2011)] the question of whether the density of a time-dependent quantum system determines its external potential was reformulated as a fixed-point problem. This idea was used to generalize the existence and uniqueness theorems underlying time-dependent density-functional theory. In this work we extend this proof to allow for more general norms and provide a numerical implementation of the fixed-point iteration scheme. We focus on the one-dimensional case because it allows for a more in-depth analysis using singular Sturm-Liouville theory and at the same time provides an easy visualization of the numerical applications in space and time. We give an explicit relation between the boundary conditions on the density and the convergence properties of the fixed-point procedure via the spectral properties of the associated Sturm-Liouville operator. We show precisely under which conditions discrete and continuous spectra arise and give explicit examples. These conditions are then used to show that, in the most physically relevant cases, the fixed-point procedure converges. This is further demonstrated with an example.
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.