By Tomislav P. Zivkovic

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6a). In analogy to the perturbation expansion, one assumes that eigenvalues λi and eigenstates Φ i of the system A n are known. From those quantities one can construct Hermitean operator Ω (ε ) which depends on a real parameter ε : Φi Φi n Ω (ε ) = i( ∑ λ ε i≠ ) ε − λi . 9). Operator Ω (ε ) is associated with the parent system A n . Since { Φ } is i a complete set in X n , in the case ε ∉ {λi } one has Ω (ε ) ψ ≠ 0 for each a nontrivial state ψ ∈ X na . Operator Ω (ε ) is in this case nonsingular in X n and hence rank (Ω (ε ) ) = n .

1a) where in a base α Tomislav P. 5⎥⎥ . 2 ⎤ Pba = ⎢ ⎥. 26c) In the above expressions Vab is a transpose of Vba , Pab is a transpose of Pba , and parameter β is chosen to equal unity. 27a) Finite-Dimensional Systems Ψ1 ⎛ 1 . 11912 ⎜ ⎜ − 0 . 39826 ⎜ ⎜ − 0 . 51397 ⎜ 0 . 73945 ⎟ . 26a). 6c), as required. One can solve the same eigenvalue equation by the LRM approach. 1a) that describes parent system A 3 . This system has three eigenvalues and three corresponding eigenstates. 28a) Tomislav P. 39004⎟ .

No other assumption about those operators is made. 5a) are real. 1b), the corresponding eigenstates Θ r can be orthonormalized according to: Θ r S b Θ t = δ rt , r , t = 1, ,ρ . 5b) Since the states Θ r form a complete set in X ρb , this implies ρ ∑Θ r Θ r S b = I bρ . 5c) r where I bρ is a projection operator on the space X ρb . Inclusion of the interaction between initially noninteracting systems B ρ (ρ + n ) -dimensional system C n+ ρ . Each state in this combined system is contained in the (ρ + n ) -dimensional space X nc+ ρ , and An creates a combined orthogonal sum of spaces X ρb and X na .

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