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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Method of continued fractions ： ウィキペディア英語版 Method of continued fractions The method of continued fractions is a method developed specifically for solution of integral equations of quantum scattering theory like Lippmann-Schwinger equation or Faddeev equations. It was invented by Horáček and Sasakawa 〔Horáček J., Sasakawa T. “Method of continued factions with application to atomic physics”, Phys. Rev. A 28, 2151-2156 (1983).〕 in 1983. The goal of the method is to solve the integral equation :$|\backslash psi\backslash rangle\; =\; |\backslash phi\backslash rangle\; +\; G\_0\; V|\backslash psi\backslash rangle$ iteratively and to construct convergent continued fraction for the T-matrix :$T=\; \backslash langle\; \backslash phi\; |V\; |\backslash psi\backslash rangle\; .$ The method has two variants. In the first one (denoted as MCFV) we construct approximations of the potential energy operator $V$ in the form of separable function of rank 1, 2, 3 ... The second variant (MCFG method〔Horáček J., Sasakawa T. “Method of continued factions with application to atomic physics. II”, Phys. Rev. A 30, 2274-2277 (1984).〕) constructs the finite rank approximations to Green's operator. The approximations are constructed within Krylov subspace constructed from vector $|\backslash phi\backslash rangle$ with action of the operator $A=G\_0\; V$. The method can thus be understood as resummation of (in general divergent) Born series by Padé approximants. It is also closely related to Schwinger variational principle. In general the method requires similar amount of numerical work as calculation of terms of Born series, but it provides much faster convergence of the results. ==Algorithm of MCFV== The derivation of the method proceeds as follows. First we introduce rank-one (separable) approximation to the potential :$V\; =\; \backslash frac\; +\; V\_1\; .$ The integral equation for the rank-one part of potential is easily soluble. The full solution of the original problem can therefore be expressed as :$|\backslash psi\backslash rangle\; =\; |\backslash phi\backslash rangle\; +\; \backslash frac|\backslash psi\_1\backslash rangle,\; \backslash qquad\; T\; =\; \backslash frac,$ in terms of new function $|\backslash psi\_1\backslash rangle$. This function is solution of modified Lippmann-Schwinger equation :$|\backslash psi\_1\backslash rangle\; =\; |\backslash phi\_1\backslash rangle\; +\; G\_0\; V\_1|\backslash psi\_1\backslash rangle\; ,$ with $|\backslash phi\_1\backslash rangle\; =\; G\_0\; V|\backslash phi\backslash rangle\; .$ The remainder potential term $V\_1$ is transparent for incoming wave :$V\_1|\backslash phi\backslash rangle\; =\; \backslash langle\backslash phi|V\_1\; =0\; ,$ i. e. it is weaker operator than the original one. The new problem thus obtained for $|\backslash psi\_1\backslash rangle$ is of the same form as the original one and we can repeat the procedure. This lreads to recurrent relations :$V\_i\; =\; V\_-\backslash frac\backslash rangle\backslash langle\backslash phi\_|V\_\}|\backslash phi\_\backslash rangle\}$ :$|\backslash phi\_i\backslash rangle\; =\; G\_0\; V\_|\backslash phi\_\backslash rangle\; .$ It is possible to show that the T-matrix of the original problem can be expressed in the form of chain fraction :$T\; =\; \backslash cfrac\}\},$ where we defined :$\backslash beta\_i\; =\; \backslash langle\backslash phi\_|V\_|\backslash phi\_\backslash rangle\; ,\; \backslash qquad\; \backslash gamma\_i\; =\; \backslash langle\backslash phi\_|V\_|\backslash phi\_\backslash rangle.$ In practical calculation the infinite chain fraction is replaced by finite one assuming that :$\backslash beta\_N\; =\; \backslash beta\_=\backslash dots=0\; ,\; \backslash qquad\; \backslash gamma\_N\; =\; \backslash gamma\_=\backslash dots=0.$ This is equivalent to assuming that the remainder solution :$|\backslash psi\_N\backslash rangle\; =\; |\backslash phi\_N\backslash rangle\; +\; G\_0\; V\_N|\backslash psi\_N\backslash rangle\; ,$ is negligible. This is plausible assumption, since the remainder potential $V\_N$ has all vectors $|\backslash phi\_i\backslash rangle,\; i=0,1,...,N-1$ in its null space and it can be shown that this potential converges to zero and the chain fraction converges to the exact T-matrix. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Method of continued fractions」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.