Resonance complex energy

The total width of the resonance is directly given by the resonance complex energy. In the case where many channels of autodetachment are open, the question of partial widths for the decay into individual channels arises. This always requires analysis of the wave fimction. The problem of obtaining partial widths from complex coordinate computation has been discussed by Noro and Taylor (39) and Bcicic and Simons (40), and recently by Moiseyev (10). However, these considerations do not seem to have found a practical application. Interchannel coupling for a real, multichannel, multielectron problem has been solved in a practical way within the CESE method by Nicolaides and Mercouris (41). According to this theory the partial widths, 7, and partial shifts to the real energy, Sj, are computed to all orders via the simple formula... 

Moiseyev N 1998 Quantum theory of resonances calculating energies, widths and cross-sections by complex scaling Rhys. Rep. 302 212... 

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. 

M. Shapiro If the resonances overlap to such an extent that we can no longer break the frequency resolved spectrum to a sum of Fano lines in a unique way, then my analysis would not be unique. However this is an extreme situation and even in this case one can try to fit the spectrum (admittedly in a non-unique way) to a sum of Fano lines (or complex energy poles). 

We can use any of the above mentioned techniques to find the resonance solutions of the TISE for the model potential presented in the previous section. Doing so, we find that there is a resonance solution at the complex energy (in atomic units) ... 

This is the resonance solution with the smallest position, i.e., smallest real part of the complex energy. Comparing this result with the measurements from the time-dependent simulation in Section 2, the agreement is truly remarkable. The corresponding momentum kres,... 

The time-dependent resonance wavefunction ijftes(x,t) decays in time because of the negative imaginary part of the complex energy. 

Feshbach resonances is purely model dependent since trapping well may exist on one type of adiabatic potential, say in hyperspherical coordinates, while only a barrier may exist on another type, say in natural collision coordinates. However, this is not correct since there are fundamental differences between QBS and Feshbach states. First, the pole structure of the S-matrix is intrinsically different in the two cases. A Feshbach resonance corresponds to a single isolated pole of the scattering matrix (S-matrix) below the real axis of the complex energy plane, see the discussion below. On the other hand, the barrier resonance corresponds to an infinite sequence of poles extending into the lower half plane. For a parabolic barrier, it is easy to show that the pole positions are given by... 

Rydberg states, which are Feshbach resonance states. Their complex energies n, for a fixed angular momentum and labeled by the principal quantum number n, satisfy the Rydberg formula with a complex quantum defect l n + j-Yn/... 

Resonances in half and in full collisions have exactly the same origin, namely the temporary excitation of quasi-bound states at short or intermediate distances irrespective of how the complex was created. In full collisions one is essentially interested in the asymptotic behavior of the stationary wavefunction L(.E) in the limit R —> 00, i.e., the scattering matrix S with elements Sif as defined in (2.59). The S-matrix contains all the information necessary to construct scattering cross sections for a transition from state i to state /. In the case of a narrow and isolated resonance with energy Er and width hT the Breit- Wigner expression... 

The other source of a channel phase is the complex continuum wave function at the final energy E. At first it would appear from Eq. (15) that the phase of ESk) should cancel in the cross term. This conclusion is valid if the product continuum is not coupled either to some another continuum (i.e., if it is elastic) or to a resonance at energy E. If the continuum is coupled to some other continuum (i.e., if it is inelastic), the product scattering wave function can be expanded as a linear combination of continuum functions,... 

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