PhD thesis in English

2. Diagonalization of Transition Amplitudes∆-dependence of errors within our calculation scheme highly outperforms the polynomialdependence in ∆ 2 obtained by the direct diagonalization of the Hamiltonian.This is true even for short times of propagation t. Although interaction terms inthe potential affect the numerical values of errors, diagonalization of the transitionamplitudes still substantially outperforms diagonalization of the Hamiltonian, andis the preferred method. This success is a consequence of the non-perturbative behaviorof the spatial discretization error within this calculation scheme. This leadsus to the key conclusion that discretization parameters can be always optimized sothat presented approach of solving eigenvalue problem of space-discretized transitionamplitudes highly outperforms direct diagonalization of the space-discretizedHamiltonian. The continuum limit ∆ → 0 is far more easily approached in thefirst case and the corresponding discretization errors are substantially smaller forthe same discretization coarseness. From the numerical point of view, as the valueof parameter ∆ directly determines the size of the matrix to be diagonalized, thecomputational cost for the same precision is significantly reduced.Further analysis of various errors in the ground energy calculation for parametersk 2 = 1, k 4 = 48 of the anharmonic potential (2.29) is given in Fig. 2.9. Thedependence of the error related to the introduction of the space cutoff L is illustratedin Fig. 2.9(top), while Fig. 2.9(bottom) gives the dependence of ground energy errorson the time of propagation parameter t for various values of the discretization step ∆.On both graphs we see the results obtained with effective actions of different levelsp. Fig. 2.9(bottom) clearly shows that the errors due to the time of propagationare proportional to t p , as expected when we use the effective action of the level p.The errors in eigenvalues are of the same order as errors in calculation of individualmatrix elements, and for this reason we see the typical t p behavior of ground andhigher energy eigenvalues. It is already now evident that the use of higher-ordereffective actions increases the accuracy of numerically calculated energy eigenstatesfor many orders of magnitude.The L-dependence of the error is analytically known [55, 56]. The saturation oferrors on the top graph for a given level p corresponds to a maximal precision thatcan be achieved with that p, i.e. denotes the value of L for which errors introduced byother sources become larger than the error due to the finite value of the space cutoff.This can be easily seen if we combine the data from both graphs. For example, thelevel p = 9 effective action has the saturated value of the error of the order of10 −14 . For t = 0.02 we find that the error due to the time of propagation is of the38