By Walter E. Meyerhof

For undergraduate physics scholars or for nuclear engineers.

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Note that complex ‘Hamiltonians’ could also follow from the Dyson device which is based upon the so-called adiabatic theorem from formal scattering theory [145]–[157]. This amounts to a modification consisting of including the damping factor i η ˆ ˆ −→ Ve ˆ −iη which is from G(ω) directly into the total interaction potential V ˆ a part of . Alternatively, one could use the well-known rotated coordinate method (RCM) [207]–[210], which is also called the complex scaling method, to construct a complex ‘Hamiltonian’ ˆ , which is non-Hermitean1.

9), associated with ˆ , we shall also consider the Copyright © 2005 IOP Publishing Ltd. 21) where the frequency ω is real as before. The exact quantum-mechanical spectrum ˆ of the operator R(u) is defined through the following Green function Ê(u) = ( ˆ 0 |R(u)| 0 ). 6) to write down at once the following approximations ˆ R(u) ≈ K k=1 πˆ k u − uk Ê(u) ≈ K k=1 dk . 8), an infinite set {cn }∞ n=0 is needed. However, N−1 N−1 generally any set of time signals {cn }n=0 or auto-correlation functions {Cn }n=0 is of a finite length N < ∞.

E. the signal c(t) would cease to have any direct physical meaning, since c(t) −→ ∞ as t −→ ∞. This situation will remain incurable as long as one keeps adding up directly the partial sums of the time signal, as a linear combination of exponentials {exp (−i ωk t)}. However, such a divergent set of partial sums could still be computed with a finite result |c(t)| < ∞ by means of certain sequence-to-sequence nonlinear transformations that are capable of converting divergent into convergent sequences through the concept of the anti-limit [72].