PhD thesis in English

2. Diagonalization of Transition Amplitudesstep ∆ are more complex, and follow from the fact that the kinetic energy operatorcannot be exactly represented on finite real-space grids. For example, a typicalnaive discretization of the kinetic energy operator gives in our notation the followingHamiltonian matrix elements [48]⎧⎪⎨ 1/∆ 2 + V (n∆) if n = mH nm = −1/(2∆⎪⎩2 ) if |n − m| = 10 otherwise.(2.4)Note that in the absence of a potential term V in the Hamiltonian, the above definitioncorresponds to a tight-binding model [48]. This prescription leads to numericalresults for eigenvalues which in the ∆ → 0 limit converge to the exact continuumvalues as ∆ 2 . The errors associated with this approach have non-variational behavior,i.e. the obtained results are not always upper bounds of the exact energy levels.Several papers discuss this issue and analyze the behavior of errors in the direct diagonalizationapproach (for more details, see Refs. [49, 50] and references therein).The state-of-the-art in this approach is a set of systematically improved prescriptionsfor discretization of the kinetic energy operator, which speeds up convergenceto the continuum limit to higher powers of ∆ 2 . However, within this approach convergenceis always polynomial in ∆. Some recent results [51] also exist on extensionsof this approach that provide effective variational behavior of the discretized kineticenergy operator.As outlined in the Introduction, we focus on an alternative approach, based onsolving the eigenproblem of the corresponding transition amplitudes as proposed in[9]. The central equation isor in the discretized formN∑cut−1e −tĤ|ψ〉 = e −tE |ψ〉 , (2.5)m=−N cutA nm (t) 〈m∆|ψ〉 = e −t E(∆,L,t) 〈n∆|ψ〉 , (2.6)where A nm (t) = ∆·A(n∆, m∆; t) = ∆·〈n∆|e −tĤ|m∆〉. In this approach the time ofevolution t plays the role of an auxiliary parameter. This parameter is not relatedto the discretization, and in a continuous theory it does not affect the obtainedeigenvalues and eigenstates. However, in a discretized theory the numerically calcu-23