Resonant polynomial

The application of the Chebyshev recursion to complex-symmetric problems is more restricted because Chebyshev polynomials may diverge outside the real axis. Nevertheless, eigenvalues of a complex-symmetric matrix that are close to the real energy axis can be obtained using the FD method based on the damped Chebyshev recursion.155,215 For broad and even overlapping resonances, it has been shown that the use of multiple cross-correlation functions may be beneficial.216... 

Acceleration of Convergence in the Polynomial-Expanded Spectral Density Approach to Bound and Resonance State Calculations. 

Recursion Polynomial Expansion of the Green s Function with Absorbing Boundary Conditions Calculations of Resonances of HCO by Filter Diagonalization. 

In the three-branch horseshoe, the periodic oibit 0 is hyperbolic with reflection and has a Maslov index equal to no = 3 while the off-diagonal orbits 1 and 2 are hyperbolic without reflection with the Maslov index n = 2 [10]. Fitting of numerical actions, stability eigenvalues, and rotation numbers to polynomial functions in E can then be used to reproduce the analytical dependence on E. The resonance spectrum is obtained in terms of the zeros of (4.16) in the complex energy surface. 

Fig. 6.8. A Principle of frequency-multiplexed CARS microspectroscopy A narrow-bandwidth pump pulse determines the inherent spectral resolution, while a broad-bandwidth Stokes pulse allows simultaneous detection over a wide range of Raman shifts. The multiplex CARS spectra shown originate from a 70 mM solution of cholesterol in CCI4 (solid line) and the nonresonant background of coverglass (dashed line) at a Raman shift centered at 2900 cm-1. B Energy level diagram for a multiplex CARS process. C Schematic of the multiplex CARS microscope (P polarizer HWP/QWP half/quarter-wave plate BC dichroic beam combiner Obj objective lens F filter A analyzer FM flip mirror L lens D detector S sample). D Measured normalized CARS spectrum of the cholesterol solution. E Maximum entropy method (MEM) phase spectrum (solid line) retrieved from (D) and the error background phase (dashed line) determined by a polynomial fit to those spectral regions without vibrational resonances. F Retrieved Raman response (solid line) calculated from the spectra shown in (E), directly reproducing the independently measured spontaneous Raman response (dashed line) of the same cholesterol sample...

Therefore, if the total phase shift is known as a function of the energy, the positions of the resonances can be determined by fitting it with a linear combination of arctangent functions plus a smooth polynomial background. See, for example, Ref. [74] and references therein. 

Coefficients of the energy-level polynomials are calculated from the theoretical values of Hg ( ) using rho = 0.866117 in isotopic equations. These polynomials are confirmed by six rotation-vibration transitions (v 3, N 2) observed between 1642 and 1869 cm" by an Infrared laser-resonance method (8). Our polynomials predict these transitions within 0.1 cm. We give the polynomial coefficients, especially higher order ones, to many more digits than are justified by their accuracy. The equations are very approximate near but, judging by Hg (1 ), this should have little effect on the thermodynamic... 

Guo, H. (1998) An efficient method to calculate resonance Raman amplitudes via polynomial propagation. Cham. Phys. Lett. 289, 396-402. 

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