Resonance pole

The most well-known and dramatic manifestation of an INR is the appearance of a narrow feature in the integral cross-section (ICS), cr(E) at total energy E = Er of width T. Obviously the resonance peak is closely related to the existence of the resonance pole in the S-matrix. Using the normal body-fixed representation for an A + BC v,j) — AB(v, j ) + C reaction, the ICS is related to the S-matrix by... 

The principal frequency dependence in Eq. (8.4) comes through the c and d coefficients. These coefficients demonstrate resonant poles at which the fields at specific locations within the particle can rise by orders of magnitude over the incident field. The wave vector values at which this occurs can be easily found by examining the form of the coefficients ... 

The word resonance is a very widespread term in the scientific world. Common uses range from being in a or on resonance to resonance poles and peaks. As with many such ubiquitous terms, they evolve with time and tend to take a life of their own acquiring new meaning and connotations as time goes by. This can lead to some confusion and ambiguity when different definitions are evoked. Here, we wish to explore the meaning of this term attributed to unstable states in quantum mechanics. 

Figure 2.5 Integration contour for the analytically continued Nevanlinna function, f[z), displaying the new cut and deformations around bound states and resonance poles.

Figure 2.16 Display of the 2P shape resonance in the e —Be collisional system. The crosses represent the first partial wave of the p-wave S-matrix resonance pole as a function of the cut off parameter. Taken from Ref. [90] with permission of IJQC.

S.A. Rakityansky, N. Elander, Analyzing the contribution of individual resonance poles of the S-matrix to two-channel scattering, Int. J. Quantum Chem. 106 (2006) 1105. 

Resonance poles lying within the widths of each other are called overlapping resonances. When they occur, the rapid rise of the phase shift due to one pole begins before the rise due to an adjacent pole finishes. In this sense, the effects of overlapping resonances on the phase shift and, in particular, on the cross section are difficult to separate, in general. Nonetheless, the overlapping Lorentzian profiles in the time-delay spectrum... 

While the resonant pole of the dilated electron propagator is persistent once uncovered and should be invariant to further changes in the complex... 

J,fc TDA) decoupling. Because of multiple points of quasi-stability for many trajectories, the quasi-stable value of the resonant pole is elicited from the corresponding a trajectory in fig. IS. 

In this section we investigate the factors affecting the formation and decay of shape resonances by examining the radial charge density /120/ plots from the Feynman Dyson amplitudes corresponding to the resonant poles identified earlier /22,25,26,40,41/ in sections 3.1 and 3.2 for different atomic and molecular resonances. 

Figure 31 Time-dependent decay curves for the dissociation of NO2 depicted on linear (left-hand panels) and on logarithmic (right-hand panels) scales. Dots experimental results solid lines quantum mechanical decay curves obtained from the resonance poles dashes statistical decay curves obtained from the SACM rates. Note the different time scales. Reprinted, with permission of the American Chemical Society, from Ref. 35.

Figure 7.3 shows the distribution of the first 15 complex poles in the fourth quadrant of the k plane. One sees clearly that the first three-resonance poles are very close together. 

Figure 7.4 Plot of the transmission coefficient T[E) vs E for a quadruple barrier system with parameters as discussed in the text, around the first three-resonance miniband. The exact numerical calculation (full line) is indistinguishable from the resonance expansion using the three resonant poles (dashed line). The dotted line represents the calculation without the interference resonant terms.

In order to reproduce the transmission coefficient over a larger energy interval, one needs to include more resonance terms, particularly in the region where the resonance poles overlap strongly, i.e., a i — as... 

In the complex energy plane, the residue at a proper resonant pole, i.e. > r , reads [17]... 

S. Canute and O. Goscinski Stationarity of Resonant Pole Trajectories in Complex Scahng Int. J. Quantum Chem. 14, 383 (1978). 

Figure 3.13. White noise input spectrum (top) versus BiQuad filtered output spectrum (bottom). The BiQuad has a resonator pole pair at 3000 Hz with r = 0.99, and a zero pair at 5000 Hz with r = 1.0 (same filter as shown in the z-plane view of Figure 3.12).

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