PhD thesis in English

2. Diagonalization of Transition Amplitudes250200L = 14.4, ∆ = 0.14L = 14.4, ∆ = 0.24L = 10, ∆ = 0.25L = 5, ∆ = 0.251501005003500030000250000 50 100 150 200 250L = 14.4, ∆ = 0.14L = 14.4, ∆ = 0.24L = 10, ∆ = 0.25L = 5, ∆ = 0.25200001500010000500000 50 100 150 200 250 300Figure 2.16: Distribution of the density of numerically obtained energy eigenstates(top) and cumulative distribution of the density of numerically obtained energyeigenstates (bottom) for a d = 2 anharmonic potential (2.36), calculated with thelevel p = 21 effective action. The parameters are k 2 = 1, k 4 = k exp4 , t = 0.2, whilediscretization parameters are given on the graph, corresponding to the curves topto bottom. Long-dashed lines on both graphs give the corresponding semiclassicalapproximations.integrating the expression (2.38), which can be calculated analytically. The comparisonof numerical and semiclassical cumulative density of states in Fig. 2.16(bottom)verifies our conclusions from Fig. 2.16(top), and again sets the same limit of reliableenergy levels for chosen discretization parameters.52