Bishop, R.F.

Flynn, MF

Znojil, M

English

The University of Manchester - Institutional Repository

The University of Manchester ResearchThe triple problem of convergence in the perturbationexpansions with non-diagonal propagatorsDocument VersionAccepted author manuscriptLink to publication record in Manchester Research ExplorerCitation for published version (APA):Znojil, M., Flynn, MF., & Bishop, R. F. (1988). The triple problem of convergence in the perturbation expansionswith non-diagonal propagators. In D. Krupa (Ed.), Proc. of the Conf. "Hadron Structure '87" (Smolenice,Czechoslovakia; 16-20 November 1987) (pp. 252-256). (Physics and Applications; Vol. 14). VEDA Publ. Co.http://personalpages.manchester.ac.uk/staff/raymond.bishop/RFB_papers/[061] HadronStruc'87(1988)252Published in:Proc. of the Conf. "Hadron Structure '87" (Smolenice, Czechoslovakia; 16-20 November 1987)Citing this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:09. Jun. 2022HadronStructure:,"87Proceedingsof the ConferenceSmoleniceNovember 16-20, 1987Physics andApplicationsVol. 14Edited byD. KrupaINSTITUTE OF PHYSICS, EPRCSlovak Academy of SciencesBratislava 1988THE TRIPLE PROBLEM OF CONVERGENCE IN THE PERTURBATIONEXPANSIONS WITH NON-DIAGONAL PROPAGATORS;M. Znojila{ M. F. Flynnbl and R. F. Bishopbla/Nucl• Phys. Institute, ~ez, Czechoslovakiab/UMIST, Manchester, United KingdomLet us consider the standard perturbation theory of the Rayleigh-Schrodinger type, with the Hamiltonian splitH = Ho + g HIand pair of ansatzsE = Eo + g El + g2 E2 +If} :: 'ro') + ~ , 'f'" '> +-III121Their insertion in the Schr-cddngar-equation HI r> = E ''1'>leads to a RS hier~rchy of relations/31and141with k = 1, 2, ••• •In a textbook spirit, we may interpr=t El, E2, ••• asabbreviations,and, insertirg them in /4/, eliminate formally also the wavefun-etion corrections,/6/In this way, perturbation theory m~y be interpreted as ~ reduc-tion of the full problem to its simplified version /3/.The "Simplicity" of Ho is usually specified ac a possibili-ty of its compbete diagonalisation. In the modified RS /MRS/approachl, the "siJ'!plicity"ofHo is weakened: in a given"unper tur-bed " basis ,0), II) , ..., we admit all operatorsHo = T + 10> g < 0 I with a free parameter g and "invertible"matrix T, i.e., with such a matrix that we may obtain81so anexplicit form of the operator R /with, say, R = l/(E. - T) whereEo is a function of g/.The main MRS idea is simple - we have noticed that an expli-eit knowledge of R and V specifies alrea~ all the correctiobS/5/ and /6/, while e presence of a free parameter g enables usalso to get rid of the eigenvalue problem /3/1• Indeed, we maywrite, in an explicit manner,1'1'. > - RIO>, (01'(0 > , <01'/'.") •. ot ':,(E.) - ,,/ <0/ R.(£~)I() '>/7/253inversion of the truncated matrices N x N. 1In Ref. , theIn practice, it is useful to write g = g(Eo' and treat Eoas a free parameter itself.There is one important reason for using' non-diagonal T in thesplit /1/ - we may make H - H as small as necec sar-y foroa good convergence of the expansions /2/. There is a,priceto be paid of course - we must guarantee a quick pr-ac.t Lcel,con-vergence also in a transition T ~ R and in the correspondingMRS forms of prescriptions /5/ and /6/.1. The T .• R convergence.The simplest way how to define R is a brute-force numericalrelated N -> 00 convergence has been reduced to a continued-fractional convergence, by means of a restriction of T's totridiagonal matrices. In Ref.2, this proce dure has been-extended to 2s+1 - diagonal T's. An alternative, purely non-numeEical type of the T -~ R transi~ion3 represents oneof the possible final solutions of this problem - we may re-construct any trial T' into an "inve!"tible" one siTply by itS'fixed-point re-arrangement T' = T + corrections. Numerically,this has been illustrated elsewhere3 - we'may only summarizehere that there are no problems with the first, N ~ 00type of convergence in practice, since its "reslilduum"maysimply be ,incorporated in the perturbation itself.2542. The intermediete-summation convergence.Each MRS contribution, SEy, Ek ' is defined as a RS-type sumover intermediate states. Each incertion of R reprecents asingle summation in the RS formalism - here, the summationgoes over the two /left and right/ indices. The relateq"additional" convergence problem may again be eliminated inthe same manner as above - we may modify the input unperturbedpropagator R' /general matrix/ and use its 2t+l - diagonal partonly, R'~ H/t/, t < 00. Again, the related modification of~'..• T/t/ (= a general matrix now) is, in effect, again a merere-definition of the pe~turbation.The numerical tests of the above idea may again be foundelsewhere4 ~d illustrate, for the cut-offs t dedreasing frominfinity, an emergence of the RS-t~pe acymptotic-series diver-gence, especially for small t(= 0 or 1) In an opposit setting,the analysis of the t ~ ao limit supports a hypothesis of theMRS convergence - see Table 1here, which lists the "optimal0!rders" /giving the optimal asymptotic-series MRS results/foranharmonic oscillators as analysed in Ref.4.Table 1. An "optimal order" No as a function of t.t 0 1 3 5 1No 2 2 4 6 10255·13. The numerical indications of the MRS convergence. of energie~;~A .,For any coupling of anharmonici ty x4, we Jr.aychoose H wi th .~~;il~,:oJ<Janother coupling Ao as a matrix T. For a broad range of ~s,A similar pattern is obtained alsowe obtain results exemplified here in Figure 1.pRoe (! It I 0 JJFl~. ifor the very broad range of para-meters E • For the variable 1o ~we obtain the dependence illu-strated here in Figure 2 for A = 1.PRfC(S(~JJ.10 \\" 4.1,•.•_LO 1Jt-1.e.••l..J,.drJ.~fr----1d~,.t!tr,t.J.1 tJ vrJ.e."..z. •• ,( ()ptl..~.,.t-l" O,.d.~r,4z.~_---,_---,_.......--+~O-I of 40~ ~oJ. A.'-." e, ')4) 7\ , .We may see that the A < 1•part of the latter Figureis a cur.ve with an inf1ec--tion point which is alm08torder-independent. - Vevergence is very good forA (inflection) ~ 1 is a•believe that the MRS con-I ••.Jinflectionj d . t thatAo'~ en conJec ure"naturel" boundar~ of the convergenceasymptotic series.domain, or at least of a domain of a rel!able use of the MRS-----------------------------------------------------------References.1\1 M. Znojll, Phys. Rev. ~ /1987/ 2448./21 -" - Dubna , JINR communication E5 - 87 - 634./3/ ibid., E4 - 87 - 655,141 ibid., E4- 87 - 667.256

The triple problem of convergence in the perturbation expansions with non-diagonal propagators

https://pure.manchester.ac.uk/ws/files/31440397/FULL_TEXT.PDF

The triple problem of convergence in the perturbation expansions with non-diagonal propagators

Abstract

Similar works

Full text

Available Versions

The University of Manchester - Institutional Repository