PhD thesis in English

2. Diagonalization of Transition Amplitudes300250200k 4 = 12, k 2 = -10, L / ∆ = 8192k 4 = 12, k 2 = -1, L / ∆ = 8192k 4 = 48, k 2 = 1, L / ∆ = 8192k 4 = 48, k 2 = 1, L / ∆ = 100n(E)1501005000 200 400 600 800 1000 1200 1400 1600 1800EFigure 2.10: Cumulative distribution of the density of numerically obtained energyeigenstates for the quartic anharmonic (k 2 = 1) and double-well potential (k 2 = −1),for t = 0.02, p = 21 and the following values of diagonalization parameters: L = 10for k 4 = 12 and L = 8 for k 4 = 48. The discretization step is given on the graph bythe value of L/∆, top to bottom. Long-dashed lines give corresponding semiclassicalapproximations for the cumulative density of states.numerical results for the modified Pöschl-Teller modelV (x) = − χ22λ(λ − 1)cosh 2 χx , (2.35)which has only a finite set of discrete energy eigenlevels E k = −χ 2 (λ −1−k) 2 /2 forinteger k from the interval 0 ≤ k ≤ λ − 1. Energy eigenvalues and eigenfunctions ofthis model are analytically known, and we will use them to further test our method.Effective actions to very high order are available also for this potential [60], andwe use them for numerical diagonalization of the evolution operator. Naturally, thediagonalization will give as many eigenvalues and eigenvectors as the size of thematrix S, but only the first few can be interpreted as bound states of the potential,according to the above condition 0 ≤ k ≤ λ − 1.Fig. 2.11(top) gives the analysis of errors in the ground energy due to the spacecutoff, while Fig. 2.11(bottom) gives the corresponding analysis of L-errors for numericalcalculation of the energy level E 5 . As we can see, the behavior of errors is thesame as for the case of anharmonic oscillator, and we are again able to obtain high44