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Exact treatment of generalized modifications of finite-dimensional systems by the LRM approach

LRM (Low Rank Modification) is a mathematical method that produces eigenvalues and eigenstates of generalized eigenvalue equations. It is similar to the perturbation expansion in that it assumes the knowledge of the eigenvalues and eigenstates of some related (unperturbed) system. However, unlike perturbation expansion, LRM produces correct results however large the modification of the original system. LRM of finite-dimensional systems is here generalized to the combined (external and internal) modifications. Parent n-dimensional system A n containing n eigenvalues λ i and n eigenstates |Φi⟩ is described by the generalized n × n eigenvalue equation. In an external modification system A n interacts with another ρ-dimensional system B ρ which is situated outside the system A n . In an internal modification relatively small σ-dimensional subsystem of the parent system A n is modified. Modified system C n+ρ that contains external as well as internal modifications is described by the generalized (n + ρ) × (n + ρ) eigenvalue equation. This system has (n + ρ) eigenvalues εs and (n + ρ) corresponding eigenstates |Ψs⟩ . In LRM this generalized (ρ + n) × (ρ + n) eigenvalue equation is replaced with a (nonlinear) (ρ + σ) × (ρ + σ) equation which produces all eigenvalues εs∉{;λi}; and all the corresponding eigenstates |Ψs⟩ of C n + ρ. Another equation produces remaining solutions (if any) that satisfy εs∈{;λi}; . Those two equations produce exact solution of the modified system C n + ρ. If (ρ + σ) is small with respect to n, this approach is numerically much more efficient than a standard diagonalization of the original generalized eigenvalue equation. Unlike perturbation expansion, LRM produces exact results, however large modification of the parent system A n .

Interaction of quantum systems ; Low Rank Modification ; Diagonalization ; Generalized eigenvalue equation ; Generalized modification of quantum systems

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