PhD thesis in English

2. Diagonalization of Transition Amplitudes10 10 110 -10 5 6 7 8 9 10 11 12E k (∆, L, t) - E k10 -2010 -3010 -4010 -5010 -6010 -70∆ E 0 , t = 0.1∆ E 0 , t = 10∆ E 14 , t = 0.1∆ E 14 , t = 10Figure 2.6: Deviations E k (∆, L, t) − E k for a harmonic oscillator as a function ofspace cutoff L for different values of time of evolution t. The discretization step is∆ = 0.1. Solid thin lines give the dominant behavior of Eq. (2.23). The dashedthick lines correspond to the error estimate in Eq. (2.20).Lamplitudes. For large t we have A(x, x; t) ≈ |ψ 0 (x)| 2 e −E0t . Integrating this we findan approximate result for the ground energy of a system with cutoff LE 0 (L, t) ≈ − 1 ∫ Lt ln dxA(x, x; t) , (2.22)In the L → ∞ limit we recover the exact ground energy, so that a simple estimateof finite size effects on E 0 is given byE 0 (L, t) − E 0 ≈ 1 ∫dx |ψ 0 (x)| 2 . (2.23)t−L|x|>LAlthough the above equation is just a rough estimate of the errors introduced by aspace cutoff L, Fig. 2.6 shows that it is in good agreement with numerical resultsfor the harmonic oscillator. In order to clearly demonstrate L-dependence of errorsin this graph, we have used small value of the discretization step ∆, such thatdiscretization errors can be neglected. The dashed lines in the figure represent errorestimates given by Eq. (2.20).Using the data from Fig. 2.6 we can now fully explain the saturation of errorsobserved in Fig. 2.5(bottom). The value of the cutoff L used to obtain this data32