PhD thesis in English

2. Diagonalization of Transition Amplitudestime, the analysis of errors presented in Fig. 2.13 allows us to reliably estimatenumerical errors in energy eigenvalues.Fig. 2.14 shows the numerically obtained ground state for this two-dimensionalpotential for the case of k 2 < 0. The ground state has the expected Mexican-hatshape. The figure gives a three-dimensional plot of the ground state on the left,and the corresponding density plot on the right, with values of the wave functionmapped to colors. Fig. 2.15 gives density plots of k = 1, 2, 3, 4 eigenfunctions forthe same values of parameters. The discretization is sufficiently fine (∆ = 0.25) sothat all features of calculated eigenfunctions are clearly visible.As in the one-dimensional case, we will calculate the density of states ρ sc (E)in semiclassical approximation, and use it as a criterion for the reliability of highenergyeigenstates. In d = 2, the density of states is given by a simple formulaρ sc (E) = 1 ∫ ∫2πdxdy Θ(E − V (x, y)) . (2.37)For the quartic anharmonic potential (2.36) the density of states can be analyticallycalculatedρ sc (E) = − 3 k 2k 4+√9 k 2 2k 2 4+ 6Ek 4. (2.38)Fig. 2.16(top) shows the comparison of semiclassical approximation for the densityof states, and the histogram for numerically obtained energy eigenvalues of the-15|ψ|0.020.010.020.010y-10-5050.020.010010-10-50x510-10 -5 05y1015-15 -10 -5 0 5 10 15xFigure 2.14: Ground state (as 3-D plot on the left, and as a density plot on theright) of a d = 2 anharmonic potential (2.36) obtained using p = 21 effective action.The parameters are k 2 = −0.1025, k 4 = k exp4 , L = 20, ∆ = 0.25, t = 0.2.50