Polynomials Faber

The most successful strategy for approximating the Liouville-von Neumann propagator is to interpolate the operator with polynomial operators. To this end, Newton and Faber polynomials have been suggested to globally approximate the propagator,126,127,225,232-234 as in Eq. [95]. For short-time propagation, short-iterative Arnoldi,235 dual Lanczos,236 and Chebyshev... 

Energy-Separable Faber Polynomial Representation of Operator Functions Theory and Application in Quantum Scattering. 

Huang Y., Kouri, D.J. and Hoffman, D.K. (1994) General, energy-separable Faber polynomial representation of operator-functions - theory and application in quantum scattering J. Chem. Phys. 101, 10493-10506. 

Usually, the propagator (7(r, to) is approximated by various schemes [55,60,137], and there are plenty of wonderful articles that have explained each in detail, such as the split operator method and higher order split operator methods [11, 36, 130], Chebyshev polynomial expansion [131], Faber polynomial expansion [51, 146], short iterative Lanczos propagation method [95], Crank-Nicholson second-order differencing [10,56,57], symplectic method [14,45], recently proposed real Chebyshev method [24,44,125], and Multi-configuration Time-Dependent Hartree (MCTDH) Method [ 12,73,81-83]. For details, one may refer to the corresponding references. 

The time evolution operator exp(—/HAf/ft) acting on ( ) propagates the wave function forward in time. A number of propagation methods have been developed and we will briefly describe the following the split operator method [91,94,95], the Lanzcos method [96] and the polynomial methods such as Chebychev [93,97], Newtonian [98], Faber [99] and Hermite [100,101]. A classical comparison between the three first mentioned methods was done by Leforestier et al. [102]. 

W. Huisinga, et al, Faber and Newton polynomial integrators for open-system density matrix propagation, /. Chem. Phys. 110 (12) (1999) 5538-5547. 

Interestingly, in Refs. 14 and 28 respectively the Faber and Newton polynomials of the operator H — iu were used to expand the Green s function defined by... 

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