General, energy-separable Faber polynomial representation of operator functions: Theory and application in quantum scattering

Abstract

A general, uniformly convergent series representation of operator-valued functions in terms of Faber polynomials is presented. The method can be used to evaluate the action of any operator-valued function which is analytic in a simply connected region enclosed by a curve, Lγ. The three most important examples include the time-independent Green’s operator, G+(E)=1/[E−(H−iε)], where H may be Hermitian or may also contain a negative imaginary absorbing potential, the time-dependent Green’s or evolution operator, exp(−iHt/ℏ), and the generalized collision operator from nonequilibrium statistical mechanics, 1/[E−(ℒ−iε)], where ℒ is the Liouvillian operator for the Hamiltonian. The particular uniformly convergent Faber polynomial expansion employed is determined by the conformal mapping between the simply connected region external to the curve Lγ, which encloses the spectrum of H−iε (or ℒ−iε), and the region external to a disk of radius γ. A locally smoothed conformal mapping is introduced containing a finite number of Laurent series terms. This results in an equal number of terms in the recursion of the Faber polynomials and avoids a serious memory problem in a calculation for a large system. In addition, this conformal mapping uniquely determines a scaled Hamiltonian, which when combined with the radius γ, ensures a completely stable recursion relation for calculating the Faber polynomials of the operator of interest (i.e., the Hamiltonian or Liouvillian). We earlier showed that for Lγ chosen to be an ellipse, the Faber polynomial expansion provides the generalization to non-Hermitian H of the Chebychev polynomial expansion of G+(E) [Chem. Phys. Lett. 225, 37 (1994); 206, 96 (1993)]; the present results provide a similar generalization for the Chebychev expansion of e−iHt/ℏ [Tal-Ezer and Kosloff, J. Chem. Phys. 81, 3967 (1984)]. Nonelliptic Lγ lead to other, new polynomial representations having superior convergence properties.