PhD thesis in English

2. Diagonalization of Transition AmplitudesW(x, δ; t) converges only for t < τ c . It is often difficult to analytically determinethe radius of convergence of the short time expansion of the transition amplitude.However, numerically this is a very simple problem, since outside of the radius ofconvergence the calculated approximative amplitudes rapidly tend to infinity (forlevels p for which the effective potential is not bounded from below; see Ref. [59]) orto zero with the increase of p. From Fig. 2.7 we easily estimate radius of convergenceto be τ c ≈ 1 for a quartic anharmonic potential V (x) = k 22x 2 + k 424 x4 , with parametersk 2 = 1, k 4 = 10. Such numerical determination of the radius of convergence for agiven level p is always done before practical use of the effective potential. Note thatwe are not interested in the precise value of τ c , just in its rough estimate which willallow us to safely use times of propagation below τ c .To conclude the section, let us stress that the effective action approach canbe used only for sufficiently smooth potentials, i.e. those that have derivatives ofthe required order, corresponding to the level p of effective action, as discussed inRef. [44]. For potentials that do not fulfill this condition (e.g. stepwise potentials),the effective action approach cannot be directly used. However, one can replacethe original potential with some of its smooth deformations, perform numericalcalculations, and at the end take the limit of the deformation parameter in whichthe original potential is recovered. The numerical results obtained in such a waymust be carefully cross-checked using other methods.2.4 Numerical results for one-dimensional modelsIn this section we apply the approach outlined above to several d = 1 modelsand demonstrate its substantial advantages for numerical studies of eigenstates ofvarious physical systems. We numerically analyze all sources of errors present inthis approach due to discretization parameters L and ∆, as well as the time ofpropagation parameter t. We present the obtained numerical results for energyeigenvalues and eigenstates. We also assess the quality of the obtained energy spectrathrough comparison with the semiclassical approximation for the density of states,which should be accurate at least for the higher regions of the spectrum.The first model we study is the quartic anharmonic oscillator with potentialV (x) = k 22 x2 + k 424 x4 . (2.29)36