PhD thesis in English

2. Diagonalization of Transition Amplitudesnential growth in the size of analytic expressions for the effective potential with theincrease of the level p, as discussed in Ref. [44]. Therefore, the required CPU timefor construction of the matrix to be diagonalized in the presented approach growsexponentially with the level p, while in other methods the construction of such amatrix does not require a significant amount of time. However, the time for exactdiagonalization far outweighs the time needed for construction of even large matriceswith moderate levels p of the order 10-20. The significant benefit of practicallyeliminating errors associated with the time of propagation therefore fully justifiesthe use of the effective action approach. Of course, in practical applications one hasto study the complexity of the algorithm and to choose the optimal level p whichwill sufficiently reduce the errors, while keeping the complexity of the calculation onthe acceptable level.2.6 Conclusions and outlookIn this Chapter, we have dealt with the thorough understanding and optimizationof the method of the calculation of the properties of quantum systems based on thediagonalization of transition amplitudes, previously introduced in Ref. [9]. First,we have focused on analyzing the errors associated with real-space discretizationand finite size effects. In particular, we have shown that within this calculationscheme spatial discretization leads to a universal and non-perturbatively small discretizationerror. This highly outperforms the usual polynomial behavior of errorsin approaches based on the diagonalization of space-discretized Hamiltonians. Akey problem in practical applications of this approach - accurate calculation of transitionamplitudes, matrix elements of the space-discretized evolution operator, hasbeen resolved using recently introduced effective action approach [44], which givessystematic short-time expansion of the evolution operator.The derived analytical estimates for all types of errors, including errors due to theapproximative calculation of transition amplitudes, provide us with a way to chooseoptimal discretization parameters and to reduce overall errors in energy eigenvaluesand eigenstates for many orders of magnitude, as was demonstrated for several oneandtwo-dimensional models. We have shown that numerical diagonalization of thespace-discretized evolution operator can be successfully applied for studies of manyinteresting lower dimensional models. The approach allows exact numeric calculationof a large number of energy eigenvalues and eigenstates of the system. Due to55