By Vakhrameev S.A.
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Additional info for A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems
The conduction band is discretized with N states and for the potential scattering model we then end up with N 2 coupled diﬀerential equations. It is fairly easy to solve systems up to 500 × 500 coupled equations on a standard workstation in a few hours. Two hints for a successful implementation: – Consider the diﬀerential equation for a coupling constant gk k (B) for given k , k. e. less than 1%). It is then quite appropriate to set gk k (B) ≡ 0 for larger values of B since the accuracy of the numerical solution will be limited anyway.
One frequently needs the symmetrized correlation function for the special case O1 = O2 . 27). 29) 2 ta1 (O) ta2 (O) cos(Ωa2 (t1 − t2 )) Ψ˜gs | Ta1 Ta2 |Ψ˜gs , = def (sym) Cgs (t1 , t2 ) = a1 ,a2 and Fourier transformation with respect to the time diﬀerence τ yields: (sym) Cgs (ω) = (sym) dτ eiωτ Cgs (τ ) ta1 (O) ta2 (O) Ψ˜gs | Ta1 Ta2 |Ψ˜gs × (δ(ω − Ωa2 ) + δ(ω + Ωa2 )) . 31) the following imaginary part of its Fourier transform: ta1 (O) ta2 (O) Ψ˜gs | Ta1 Ta2 |Ψ˜gs Im Rgs (ω) = π a1 ,a2 × (δ(ω − Ωa2 ) − δ(ω + Ωa2 )) .
Mod. Phys. 68, 13 (1996) 6. C. Hewson: The Kondo Problem to Heavy Fermions, 1st edn (Cambridge University Press, Cambridge 1993) 7. D. G. Wilson, Phys. Rev. D 48, 5863 (1993) 8. D. G. Wilson, Phys. Rev. D 49, 4214 (1994) 9. F. Wegner, Ann. Phys. (Leipzig) 3, 77 (1994) 10. F. Wegner, Phil. Mag. B 77, 1249 (1998) 11. F. Wegner, Phys. Rep. 348, 77 (2001) 12. F. Wegner, Nucl. Phys. B (Proc. ) 90, 141 (2000) 13. F. Wegner, cond-mat/0511660 14. D. Glazek, Renormalization of Hamiltonians. In: Lecture Notes of the First International School on Light–Cone Quantization (Iowa State University Press, Ames 1997) 15.
A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems by Vakhrameev S.A.
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