Fano s formalism

This case is shown schematically in Fig. 5c. In Eq. (50), qj. are generalized y-photon asymmetry parameters, defined, by analogy to the single-photon q parameter of Fano s formalism [68], in terms of the ratio of the resonance-mediated and direct transition matrix elements [31], j. is a reduced energy variable, and <7/ y, is proportional to the line strength of the spectroscopic transition. The structure predicted by Eq. (50) was observed in studies of HI and DI ionization in the vicinity of the 5<78 resonance [30, 33], In the case of a... 

Fano s formalism, coherence spectroscopy one- vs. three-photon excitation, 164-166... 

Along fhose lines, Cordes and Altick [80] also implemenfed multichannel Cl theory with a basis set of hyperspherical coordinates for the determination of properfies of fhe He (3,3b) resonance. The first Cl multichannel implementation of Fano s formalism was done by Ramaker and Schrader [81] with application to the He (nt ) S autoionizing states. 

For the case of ionization of discrete states into a single continuum, the method developed by Fano [9] is complete. Application of this method to the case of orthogonal continua has been studied by Mies [10]. We have developed an extension of Fano s formalism to this multichannel case oy generalizing scalar quantities to matrices, and have incorporated the WKB-QD parameters into the model [7]. 

Extension of Fano s formalism to the general case of multiple discrete states embedded in multiple continua was first conducted by Mies [54]. Like Fano, Mies assumed a prediagonalized basis. By imposing the asymptotic condition for the continuum state from the scattering theory, Mies derived the complete solution to the total continuum problem and gave formulas for energies and widths of resonances. 

In Fano s [29] formal theory of resonance states, the energy-dependent wavefunctions are stationary, the energies are real, and the formalism is Hermitian. The observable quantities, such as the photoabsorption cross-section in the presence of a resonance, are energy-dependent and the theory provides them in terms of computable matrix elements involving prediagonalized bound and scattering N-electron basis sets. The serious MEP of how to compute and utilize in a practical way these sets for arbitrary N-electron systems is left open. 

For example, this form is in harmony with the superposition of energy states in Eq. (2), whose coefficients have been obtained formally by Fano [29]. Although, for the solution of particular problems involving unstable states, we have implemented, in conjunction with the methods of the SSA, the real-energy, Hermitian, Cl in the continuum formalism that characterizes Fano s theory, e.g.. Refs. [78, 82-87] and Chapter 6, in this chapter I focused on the theory and the nonperturbative method of solution of the complex eigenvalue Schrodinger equation (CESE), Eq. (27). 

Other variants are due to Fano [76], Anderson [77], Lee [78], and Friedrichs [79] and have been successfully applied to study, for example, autoionization, photon emission, or cavities coupled to waveguides. The dynamics can be solved in several ways, using coupled differential equations for the time-dependent amplitudes and Laplace transforms or finding the eigenstates with Feshbach s (P,Q) projector formalism [80], which allows separation of the inner (discrete) and outer (continuum) spaces and provides explicit expressions ready for exact calculation or phenomenological approaches. For modern treatments with emphasis on decay, see Refs. [31, 81]. Writing the eigenvector as [31, 76]... 

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