PhD thesis in English

||2. Diagonalization of Transition Amplitudes| E 0(p) (∆, L, t) - E 0exact10 -5 110 -1010 -1510 -2010 -25p = 1p = 5p = 9p = 13p = 17p = 211 2 3 4 5 6L| E 5(p) (∆, L, t) - E 5exact10 -5 110 -1010 -1510 -2010 -25p = 1p = 5p = 9p = 13p = 17p = 211 2 3 4 5 6 7 8LFigure 2.11: Deviations |E (p)k(∆, L, t) − Eexact k | as a function of L for k = 0 (top)and k = 5 (bottom), for the modified Pöschl-Teller potential. Energy eigenvaluesare obtained using effective action levels p = 1, 5, 9, 13, 17, 21 and t = 0.1, with theparameters χ = 0.5, λ = 15.5, ∆ = 0.02.accuracy results. Fig. 2.12 gives the time dependence of errors in ground energyobtained by numerical diagonalization using different levels p of effective actions.The scaling of errors proportional to t p is evident from the graph, as well as the discretizationerrors due to the finite discretization step ∆. To ensure that the effectivepotential is bounded from below, in this case we have to remove higher-order powersof discretized velocity δ from the effective potential near x = 0, since such termshave non-vanishing negative coefficients in the vicinity of x = 0, due to a peculiarnature of the potential. In practical applications, one can use e.g. p = 1 effective45