Iterative eigenvalue method using the Bloch wave operator formalism with Padé approximants and absorbing boundaries

Abstract

We present an iterative method for calculating eigenvalues and eigenvectors of large non-Hermitian matrices. The method uses an iterative procedure to solve the basic Bloch equation HOmega=OmegaHOmega of wave operator theory. It involves nonlinear transformations such as the translation of diagonal matrix elements in the complex plane and the use of Padé approximants to treat the strongly coupled states which constitute an intermediate space around the model space. In the particular case of Floquet eigenstates the further step of adding time-dependent absorbing boundaries significantly improves the convergence properties of the iterative calculations.

Global integration of the Schrödinger equation: a short iterative scheme within the wave operator formalism using discrete Fourier transforms

“…The total basis set describing the extended Hilbert space is then composed of N = N v × N t = 245760 states. In this framework, the dynamics is calculated by using the iterative scheme explained in section 2 (equations (5), (13), (17) and (18) with Ω (n+1) = P o + X (n+1) and X (n+1) = X (n) + δX (n) .) The integrals appearing in (18) are calculated using the numerical integrator presented in section 3 and Appendix A.…”

“…We think that this efficiency is essentially due to the use of a time-dependent effective Hamiltonian in the basic equations of the integration scheme ((13), (17) and (18)), especially when it is compared with previous iterative schemes. is present and gives the correct survival amplitude.…”

“…Actually the use of (43) in the integration scheme produces a very fast divergence of the iterative series caused by the crossings ofH (n) diag and H (n) ef f . Another approximate (but more sophisticated) integration scheme was proposed in refs [17] and [15]. By using a discrete Fourier basis set: |p with t|p = exp(i2πpt/T ) derived from the discrete time grid basis set by a FFT transformation, a generalised RDWA approximation leads to:…”

“…(n) ef f . Another approximate (but more sophisticated) integration scheme was proposed in refs [17] and [15]. By using a discrete Fourier basis set: |p with t|p = exp(i2πpt/T ) derived from the discrete time grid basis set by a FFT transformation, a generalised RDWA approximation leads to:…”

Constrained Adiabatic Trajectory Method

“…Nevertheless we note that the calculation of the final eigenstate |λ i , n =0 〉 need not be done by full diagonalization in the case of larger systems; it can be done more simply by using filter-diagonalization 38 methods, a Chebyshev expansion of a projection operator, or a wave operator diagonalization method. The wave operator option has been selected in our second example, using a recent new algorithm which performs the iterative RDWA integration of the Bloch equation ( H F Ω = Ω H F Ω) and which incorporates nonlinear Padé approximant techniques . The calculation is completed by testing for obedience to the initial condition 〈 k |λ( t = 0)〉 = δ k,i imposed by V opt .…”