PhD thesis in English

2. Diagonalization of Transition Amplitudes12010080t = 0.015t = 0.020t = 0.040t = 0.125E k60402000 10 20 30 40 50 60 70 80 90kFigure 2.4: Harmonic oscillator dispersion relation. The solid line gives the exactlinear dispersion E k = k +1/2. The points correspond to numerically calculated energyeigenvalues E k as function of level k. We show the results of the diagonalizationof transition amplitudes for several values of t. In this plot L = 12, ∆ = 0.25.where E k = π2 (k+1) 2and k = 0, 1, 2, . . . As expected, the universal term gives the8L 2dominant ∆ dependence. One obtains similar analytical results for the case of theharmonic oscillator.The non-perturbatively small effect of spatial discretization is the reason whythe new method highly outperforms direct diagonalization of the Hamiltonian andleads to much smaller errors for the same size of discretization step ∆. In additionto this the free parameter associated with the method, the time of evolution t, canbe used to further minimize errors. As illustrated in Fig. 2.2, while keeping ∆ fixed,we can adjust time t to obtain much smaller errors and practically reproduce theexact spectrum of the theory. This is also evident in Fig. 2.3, where we see thatby adjusting t, errors can be reduced by orders of magnitude for fixed value ofdiscretization step ∆.We next consider a harmonic oscillator. Fig 2.4 shows how the presented methodmay be used to obtain energy eigenvalues to high levels. The numerical calculationsagree well with the well known linear dispersion of the harmonic oscillator.Figs. 2.5(top) and 2.5(bottom) display respectively the ∆ and t dependence of thedeviations |E k (∆, L, t) − E k |, showing agreement with the analytically derived estimateof the discretization error given in Eq. (2.18). In order to achieve such a29