PhD thesis in English

2. Diagonalization of Transition Amplitudes10 150.05 0.1 0.15 0.2 0.25 0.3| E 0 (∆, L, t) - E 0 |10 1010 5110 -510 -1010 -1510 -2010 -25Ht = 0.005t = 0.01t = 0.02t = 0.03t = 0.0410 -110 -210 -310 -40.1 0.2 0.3Figure 2.8: Plot of |E 0 (∆, L, t) − E 0 | for an anharmonic oscillator (2.29) given as afunction of ∆ for different values of time of evolution t. The parameters used are L =6, k 2 = 1, and anharmonicity k 4 = 48. Dashed lines correspond to the discretizationerror in Eq. (2.19). For comparison, we also plot the corresponding deviations ofnumerical results (designated by H) obtained using direct diagonalization of thecorresponding space-discretized Hamiltonian.∆For this potential the effective actions have been previously derived up to p = 144[60], and here we will use various levels p to illustrate the dependence of errors onthe level p used in calculations. We will study the interesting regime of the strongcoupling k 4 , since there are various other techniques that can be successfully usedfor small couplings.Fig. 2.8 displays |E 0 (∆, L, t) − E 0 | as a function of discretization step ∆ for thecase of an anharmonic oscillator with potential (2.29). The parameters used in theplot are L = 6, k 2 = 1, and anharmonicity k 4 = 48. The transition amplitude matrixelements were calculated using p = 18 effective actions [44]. The high precision valuefor the exact ground energy that we compare to was calculated in Ref. [61]. As wecan see, even though we are dealing with a relatively strong anharmonicity, thenumerically calculated values stay right on the dashed black lines corresponding tothe universal discretization error just as in the case of the previously consideredintegrable models. This is in complete agreement with our analytical derivation ofthe discretization error.As can be seen from Fig. 2.8 the numerical results clearly demonstrate that the37