PhD thesis in English

2. Diagonalization of Transition AmplitudesTable 2.4: Low-lying energy levels of a d = 2 anharmonic potential (2.36), obtainedusing the level p = 21 effective action. The discretization parameters are L = 14,∆ = 0.14, and t = 0.2.k E k , k 2 = −0.1025, k 4 = k exp4 E k , k 2 = −0.1025, k 4 = 10 3 k exp40 -1.1279858856602 1.12872978314351 -1.1169327267787 2.61613484978342 -1.1169327267787 2.61613484978343 -1.0842518375067 4.34765152798104 -1.0842518374840 4.34765152798125 -1.0311383813261 4.65284518520136 -1.0311383813261 6.27045529036717 -0.95910186300510 6.27045529036718 -0.95910186300478 6.75898824914119 -0.86968170695135 6.7589882491412found to fully agree with the scaling law t p for sufficiently fine discretization. Again,the discretization errors conform to the universal dependence given in Eq. (2.19).Here there is an additional factor of 2 in the cosh term due to the dimensionality ofthe system.Table 2.4 gives the numerically obtained energy eigenvalues for different sets ofparameters of the potential (2.36). Motivated by the values of the experimentalparameters [12, 64], we introduce and use the constant k exp4 = 1.95×10 −3 . From theanalysis of discretization errors and errors related to the use of a chosen effectiveaction level p, we can estimate the errors in found energy eigenvalues to be of theorder 10 −15 . The results in the Table 2.4 are obtained by numerical diagonalizationbased on the C SPEEDUP code [60] and the use of the LAPACK [62] library. Theestimated error in energy eigenvalues is smaller than the (relative) error which canbe achieved in typical C simulations, which is of the order 10 −14 . This is easilyverified, since for several different values of discretization parameters we get thesame stable results shown in the table. Therefore, this table gives certain digitsin all energy eigenvalues, and the error can be cited as implicit (half of the lastdigit). This is good example for practical applications, where we have managedto eliminate all types of errors below the limit that can be seen due to inherentnumerical errors of computer simulation. However, if such complete elimination oferrors is not possible due to the limitations in computer memory or computation49