Chebyshev polynomial expansion

As demonstrated in Refs. 1 and 2, if the inhomogeneity in Eq. (14) (xT n our case) is energy independent the energy explicitly enters into the Chebyshev polynomial expansion via the energy- (here 9) dependent coefficients ... 

Since this general integral cannot be done in closed form, the most effective way to use it is to expand the integrand as a Taylor series in the small quantities ytmn, and integrate term by term. Once cosines of multiple angles have been removed using the Chebyshev polynomial expansion... 

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. 

We may be able to make some progress in solving the ABC system by appealing to the Newton polynomial expansion, which is a useful method for approximating scalar functions. A special case is the Chebyshev polynomial expansion, which is widely used in commercial algorithms for evaluating special functions [37, 38]. Based on the Chebyshev expansion for u x) = for x real, Tal-Ezer and Kosloff [5]... 

Chebyshev Approximation The well known expansion of exp(— into Chebyshev polynomials T, [23] is one of the most frequently used integration technique in numerical quantum dynamics ... 

The main problem now is to calculate the action of the Green s function onto the initial state xo- The standard strategy is to expand G in a power series of H. For vanishing A, a highly efficient expansion exists [215,235] in terms of Chebyshev polynomials, Tn H), which is similar to the one used in short-time wave packet propagations [166,171]. For finite A, this expansion has to be modified to account for the absorbing potential. As was shown by Mandelshtam and Taylor [221], the analytically continued Chebyshev polynomials, can be used for this purpose. If the initial... 

With Eqs. (32)-(35), one can construct the basis functions (Ej), filtered out of the initial state near the energies Ei. Before applying the modified Chebyshev expansion, one has to rescale the original Hamiltonian H, because Chebyshev polynomials converge only for la < 1 [236]. In fact, H in the equations above has to be substituted with the operator. ffnorm H H))/AH, whose eigenvalues lie between —1 and 1. Then, a basis function can be written as [208]... 

In fact, any analytic function of the Hamiltonian can be expressed as an expansion in terms of the Chebyshev operator (Tj (H) ).[14] It is interesting to note that in both Eqs. (4) and (5), die energy shows up only in the expansion coefficients. In other words, the Chebyshev reeursion yields information at all energies. These advances opened the doors for both time-dependent and time-independent studies of quantum dynamics and spectra using the Chebyshev polynomials. 

The subsequent refinement included profile parameters X, Y, X , Ya, peak asymmetry, sample displacement and transparency shift. Preferred orientation was switched from the March-Dollase to the 8 -order spherical harmonics expansion (6 variables total) and 12 coefficients of the shifted-Chebyshev polynomial background approximation were employed. A reasonably good fit, shown in Figure 7.36, was achieved as a result. 

In order to compute the evolving state i(/)), Tal-Ezer and Kosloff (77) were the first to propose an expansion of the evolution operator in terms of Chebyshev polynomials. They initially developed this method for wavepacket calculations on spatial grids. More recently, this procedure has been adapted and applied to bound systems (20). It involves breaking the total integration time (for instance 2 ps) into smaller time steps At (each about 25 fs), and using a polynomial expansion of the evolution operator (/(Af) over each small time step. This efficient method provides all the transition probabilities Plf(t) from initial state i) in one calculation because it directly provides the evolving state /(r)). 

Chebyshev expansion of a real function /(x). We will first define the Chebyshev polynomial of degree n (n = 0, 1, 2,. ..) denoted T (x),... 

The expansion of the evolution operator in terms of Chebyshev polynomials is then utilized. Tal-Ezer and Kosloff have shown (77) that the degree of the expansion (N) has to be at least 40. How can we now efficiently apply the evolution operator to the state i(t)) to obtain the new state i(t + At)) This latter state is equal to... 

In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary fiow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form... 

As a useful polynomial expansion method we show the Chebyshev expansion scheme below. We consider the general case such as a dissociation process, in which a wavepacket may proceed out of the grid region set in advance. The time propagation operator of the wavepacket x) is expanded in terms of the modified Chebyshev polynomial [255],... 

The Chebyshev procedure consists of expanding the quantum operators in terms of orthogonal Chebyshev polynomials. It is considered to be an efficient and reliable method, since the convergence of the expansion is guaranteed. According to the method, the time-dependent hamiltonian is treated as a constant operator within the time slice dt. Thus, the time propagator is expanded in a series of Chebyshev polynomials for a time t within dt as... 

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