One series of autoionising resonances

K-matrix theory can be regarded as a method of reducing complicated spectral fluctuations to a minimum number of nearly constant quantities. The same, of course, is true of MQDT, and the relation between the two is fundamental to the present section. Our goal is to preserve analyticity in order to investigate the general behaviour of overlapping poles one may then explore the full range of variation in parameter space and search for remarkable situations (zeros, cancellations, symmetry reversals, etc) with confidence that unitarity is automatically satisfied at all points. 

As above, contact is made between K-matrix and MQDT formulations through the replacement  

This replacement is of key importance it acts like a change of variables from Tra, En, E to X, /x, v. 

We begin from the K-matrix form for one Rydberg series of resonances Kres = tan A = = E (8.95) 

The sum begins with the lowest resonance, and must be taken to infinity at the series limit. In MQDT, the bottom of the channel is not defined, but a sum to infinity does not arise either. The quantity Tra (the linewidth) scales as (n — /x)-3 for an asymptotically Coulombic potential. A zero in the cross section occurs between each resonance. 

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