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By Tomislav P. Zivkovic

ISBN-10: 1616688904

ISBN-13: 9781616688905

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Hence if all modified eigenvalues are needed, LRM will be numerically more efficient if ρ << n . If a single or only a few selected eigenvalues are required, LRM is comparatively even more efficient since it requires only O ( ρ 2 n) operations per eigenvalue, while all known numerical methods require substantially more than O (( n + ρ ) 2 ) ≈ O ( n 2 ) eigenvalue. operations per Tomislav P. 12a). Since ε s is known, this calculation is of the order O( ρ 3 ) . Since ρ << n < n this operation count is negligible with respect to the operation count required to obtain the eigenvalue ε s in the first place.

These eigenvalues thus arranged satisfy λ s + r −σ h s −σ h ≤ ε s ≤ λ s + rh +σ s . 2. Cardinal Solutions of the Modified System Let me now consider solutions of the modified system C n in more details. In analogy to the external modifications, in the case of internal modifications one can also have cardinal ( ε s ∉ {λi } ) and singular ( ε s ∈ {λi } ) solutions of this system. 1b). 34). 36a) is a σ × σ matrix equation. 34b). 2a) is hence a root of the σ × σ determinant g (ε ) : g (ε ) ≡ β Ω a (ε ) − Va + ε Pa = 0 .

In general, only in the case when the modification operators V and P increase the symmetry of the initially non-interacting systems A n and B ρ , combined system C n + ρ may have some weakly singular eigenstates. Otherwise combined system has no such eigenstates. Above two theorems produce all solutions of the combined system C n + ρ . All cardinal eigenvalues ε s ∉ {λi } of the combined eigenvalue equation are roots of the function h (ε ) . Once a particular root ε s of h (ε ) is found, the corresponding eigenstate (or eigenstates) is given by eq.

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Exact Treatment of Finite-Dimensional and Infinite-Dimensional Quantum Systems (Mathematics Research Developments) by Tomislav P. Zivkovic

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