PhD thesis in English

2. Diagonalization of Transition AmplitudesTable 2.2: Low-lying energy levels of the double-well potential, obtained by diagonalizationusing level p = 18 effective action. The parameters used: k 2 = −1,k 4 = 12, L = 16, ∆ = 0.1, t = 0.05. The errors are estimated by comparisonwith the diagonalization results obtained from higher-order effective actions, finerdiscretizations, larger space cutoffs, and lower values of the propagation time t.k E k |∆E k | δE k0 0.328826502590357561530(2) 7 × 10 −22 2 × 10 −211 1.41726810105965210733(23) 5 × 10 −21 4 × 10 −212 3.0819506284815341204(849) 3 × 10 −20 1 × 10 −203 5.019323060355788021(7990) 2 × 10 −19 4 × 10 −204 7.186203252338934478(3958) 5 × 10 −19 8 × 10 −205 9.54285734251209386(72421) 2 × 10 −18 2 × 10 −196 12.06403774639116375(04211) 4 × 10 −18 4 × 10 −197 14.7314279571006902(462590) 1 × 10 −17 7 × 10 −198 17.5310745155383834(413592) 3 × 10 −17 2 × 10 −189 20.4519281359123716(968554) 5 × 10 −17 3 × 10 −18numerical results with some known properties of the physical system. One suchproperty is density of states, defined formally asρ(E) =∞∑δ(E − E k ) , (2.30)k=0assuming that the system has a discrete spectrum. This highly relevant physicalquantity can be directly calculated using the numerically obtained spectra. On theother hand, it can be also analytically calculated using semiclassical approximation.This approximation is valid at least in the high-energy region, and we can use itto assess the quality of our numerical results. In semiclassical approximation, thedensity of states in d spatial dimensions is calculated asρ sc (E) =∫ d⃗xd⃗pδ(E − H(⃗x, ⃗p)) . (2.31)(2π)dreplacing the discrete spectrum with a continuous distribution of energy defined bythe classical Hamilton function H(⃗x, ⃗p). After integration over momenta, we obtain42