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Complex-Dynamic Fractality and Its Applications in Life Sciences

where ^φn (Q )≡ φ_λ n1 (q_λ ⅛)₂ n 2 (q 2 y..q>_NN_N (qN ) and n ≡( nι^, n 2^,∙∙∙^{, n}N ) runs ^thr^ough all
possible eigenstate combinations. Inserting eq. (4) into eq. (2) and performing the
standard eigenfunction separation (e∙g∙ by taking a scalar product), we obtain the system
of equations for ψ_n (ξ) , which is equivalent to the starting existence equation:

[ho (^ξ) ⁺ V00 (^ξ)] ψ0 (^ξ ) ⁺ Σ V)n (^{ξ ψ} (^ξ ) = ηψθ (^ξ ) ^,            (⁵a)

n

[h0 (ξ) ⁺ Vnn (^ξ)] Ψn (^ξ) ⁺ Σ Vr_lr'' (^ξψn' (^ξ) = ηnψn (^ξ) ^- Vn0 (^ξ) ψ0 (^ξ) ^{,    (5b)}

n '≠ n

where n, n′ ≠ 0 (also everywhere below), η≡η₀=E -ε₀,

εn ≡∑εn,_t . Vnn∙(ξ) = ∑ V0'(ξ)+∑Vn"'
k                             k _                l > k

Vkn"ⁿ '(ξ )= ∫ dQΦn,( Q V^r_tr. (qk, ξ ) Φn ′(Q ). Vu' (ξ) = ∫ dQΦn,( Q V„ (q_k, q,) Φn∙( Q ), (7)
^ωq                                        ^ωq

and we have separated the equation for ψ₀ (ξ) describing the generalised “ground state”
of system elements, i∙ e∙ the state with minimum energy and complexity∙

Now we try to “solve” eqs∙ (5) by expressing ψ_n (ξ) through ψ₀ (ξ) from eqs∙
(5b) with the help of the standard Green function and substituting the result into eq∙
(5a), which gives the effective existence equation for ψ₀ (ξ) [9-13]:

h 0 (^ξ)ψ0 (^ξ) ⁺ Veff (^ξ; η^ψ0 (^ξ) = ηψ 0 (^ξ)^,                    (8)

where the effective (interaction) potential (EP), V_eff (ξ; η), is obtained as

V_iff (ξ; η ) = V00 (ξ) + V (ξη), V(ξ; η )ψ> (ξ)= ∫ dξV (ξ,ξ^,; η (ξ'),

Oξ

_v (_ξξη)=v ^v0 n (ξ)ψ0i(ξ) Vn 0 (ξ^,)ψ0* (ξ^,)
^, ’                       η^- ηn,^{- ε}n 0

and {ψ_n⁰_i (ξ)}, {η_n⁰_i } are complete sets of eigenfunctions and eigenvalues, respectively,
for a truncated system of equations obtained as “homogeneous” parts of eqs∙ (5b):

_ h0 (ξ)+Vnn (ξ)]ψn (ξ)+∑ Vnn'(ξ)ψ√(ξ) = ηnψn (ξ) ∙          (10)

n '≠ n

The eigenfunctions {ψ_0i (ξ)} and eigenvalues {η_i } found from eq∙ (8) are used

to obtain other state-function components:

ψni(ξ)= [dξ'gni(ξ,ξ')ψ_υi (ξ'), gn, (ξ,ξ^,) = Vn0(ξ^,)V^ψ,"^,(^ξ)₀^ψ-^,(^ξ) , (11)

J                                              ^η--i ηi^- ηni'- ^εn 0

Ωξ                                                                     i i

after which the total system state-function Ψ (ξ,Q) , eq∙ (4), is obtained as

(^ξ, ^Q) = ∑ ^ci ^φ0 (^Q) ^ψ0i (^ξ) ⁺ ∑ ^φn (^Q)ψn_t (^ξ)

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