The phase-shifted MQDT described so far assumes that resonance structures in the MQDT formulation are easily identified when both reactance matrices

${K}^{oo}$ and diagonal elements or degenerate blocks of

${K}^{cc}$ are made zero by phase renormalization. In [

18] it was shown how this assumption leads to factorization of the physical scattering matrix

**S** into the background and resonance part as

**S** =

**S**_{B}S_{R} just like in conventional theory of resonance where

**S**_{R} is given by

$1+2\pi i\overline{V}{\left(E-\overline{E}-i\pi {\overline{V}}^{\u2020}\overline{V}\right)}^{-1}{\overline{V}}^{\u2020}$ [

19]. Unlike the conventional scattering theory of resonance, MQDT considers the short-range scattering matrix [

6,

20] besides the physical scattering matrix

**S**. The reason why another type of scattering matrix is considered in MQDT is the following. In order to treat the energy-sensitive phenomena, MQDT divides the space into the inner and outer regions and utilizes the fact that the energy sensitivity of the observables only comes from the outer region where processes are decoupled and can be easily handled with the known analytical solutions. Transfers in energy, momentum, angular momentum, spin, or the formation of a transient complex occur in the inner region due to the presence of the strong interaction between the ionizing electron and the core. The complex dynamics occurring in the inner region affect the observables only through the short-range scattering matrix

$S$ defined for the incoming wave as

${\mathsf{\Psi}}_{k}^{(-)}\to {\displaystyle {\sum}_{i}{\mathsf{\Phi}}_{i}\left({e}^{i{k}_{i}r}{r}^{i\zeta}-{e}^{-i{k}_{i}r}{r}^{-i\zeta}{S}_{ik}\right)}$ (see [

6]) in the outer region and determined by the match with the solution in the inner region at the boundary. Channel basis functions for the closed channels are not diminished to zero but have comparable magnitudes to those for open channels in the matching boundary so that:

where

c denotes closed channels and

o open channels. Its relation with the physical scattering matrix is well-known [

20] and given by

$S={S}^{oo}-{S}^{oc}{\left({S}^{cc}-{e}^{i\beta}\right)}^{-1}{S}^{co}$, which is consisted of two terms. The second term is related to resonance because of its singular nature. Then,

${S}^{oo}$ may be identified with the background part

**S**_{B}. However,

${S}^{oo}$ is not unitary, owing to the leakage into closed channels, and thus cannot be identified as

**S**_{B}. It is found that

${\sigma}^{oo}$ defined with reactance matrix

K as

$\left(1-i{K}^{oo}\right){\left(1+i{K}^{oo}\right)}^{-1}$ corresponds to

**S**_{B} [

18,

21,

22].

**S**_{R} is subsequently identified as

$1+2i\xi {\left(\mathrm{tan}\beta +{\kappa}^{cc}\right)}^{-1}{\xi}^{T}$ where

${\xi}^{T}=\left({\xi}_{1},...,{\xi}_{{n}_{c}}\right)$ with

${\xi}_{i}=\left({\xi}_{1i},...{\xi}_{ji},...\right)$(

${\xi}_{ji}$≡

${K}_{ji}$) and

${\xi}_{i}$ denotes the coupling strength vector between the closed channel

i with open channels in space

P,

$\beta $ denoting

$\pi \nu $ where

$\nu $ is the effective quantum number defined by

$I-{Z}^{2}\mathrm{Ry}/{\nu}^{2}$ with ionization energy

$I$,

${\kappa}^{cc}$ being the reactance matrix corresponding to

${S}^{cc}$. Note the similarity of this form with

**S**_{R} in the conventional scattering theory. The singularity in

${S}_{R}$ suggests that it represents the resonance part and its phase, and may show the typical resonance behavior. However, it is well known that each eigenphase shift is not only affected by the resonance but also by the avoided crossing interaction between two eigenphase shift curves when they approach each other [

23,

24]. The avoided crossings can be made cancelled out so that only the pure resonance behavior remains by considering their sum, called the eigenphase sum

${\delta}_{\mathsf{\Sigma}}$ (

$\equiv {\displaystyle {\sum}_{j}{\delta}_{j}}$). The eigenphase sum is obtained from the determinant of the physical scattering matrix as follows:

where

$\mathrm{det}\left(\mathrm{tan}\beta +{\kappa}^{cc}\right)$ ≡

$C\mathrm{exp}\left(i{\delta}_{r}\right)$ and

${\delta}_{\mathsf{\Sigma}}^{0}={\displaystyle {\sum}_{j}{\delta}_{j}^{0}}$ are used [

25]. Equation (2) tells us that

$\mathrm{det}\left(\mathrm{tan}\beta +{\kappa}^{cc}\right)$ contains all the information on the resonance structure for the given system. This statement may be called the resonance theorem. Note the form analogy of (2) with

$\mathrm{det}(S)$ =

$\mathrm{det}({S}_{B})$$\mathrm{det}\left(E-\overline{E}+i\pi {\overline{V}}^{\u2020}\overline{V}\right)$/

$\mathrm{det}\left(E-\overline{E}-i\pi {\overline{V}}^{\u2020}\overline{V}\right)$ in the conventional scattering theory of resonance [

26]. The role of the phase renormalization that makes

${K}^{oo}$ zero and diagonal elements or degenerate blocks of

${K}^{cc}$ zero in finding the resonance structures may be unraveled with the help of the resonance theorem. The representation in which

${K}^{oo}=0$,

${K}_{ii}^{cc}=0$ or null degenerate

${K}^{cc}$ block will be marked with a tilde.

${\tilde{K}}^{oo}=0$ corresponds to making

${\tilde{\delta}}_{\mathsf{\Sigma}}^{0}=0$ or

$\mathrm{det}\left({\tilde{\sigma}}^{oo}\right)=1$ so that the phase shift due to the background process is removed in the tilde representation. Additional process of making

${\left({\tilde{K}}^{cc}\right)}_{ii}=0$(

i = 1,…,

${n}_{c}$) removes the real part of

${\kappa}^{cc}$,

$\Re \left({\tilde{\kappa}}^{cc}\right)=0$, so that the diagonal elements of

$\mathrm{tan}{\beta}_{i}+{\kappa}_{ii}^{cc}$ becomes

$\mathrm{tan}{\tilde{\beta}}_{i}+i\Im \left({\tilde{\kappa}}_{ii}^{cc}\right)$. Its role is, thus, to move the resonance center to the coordinate origin. In the case of a single closed channel system, (2) can be rewritten as:

which shows the periodic nature in

$\beta $ of Rydberg series in the Lu-Fano plot.

The goal to be achieved by this phase renormalization is different depending on whether the series converge to the same ionization threshold (degenerate) or to the different thresholds (nondegenerate). In the degenerate case, the closed channels are degenerate and $\mathrm{tan}\beta +{\kappa}^{cc}$ can be made diagonalized to exploit the periodic nature of inter-series channel coupling and its determinant can be written as a product of single closed channel formulas (see Equation (21)). Then each eigen-channel acts as a single autoionizing Rydberg series. Phase renormalization ${\tilde{K}}_{ii}^{cc}=0$ moves the resonance center of each eigen series i to its respective origin.

If there is more than one nondegenerate closed channel, it is meaningless to move the resonance center of each eigen-series to its own origin because of the local, non-uniform and thus aperiodic nature of inter-series channel coupling (see

Figure 1) [

27]. Instead, its utility is in the simplification of the parameters for resonance structure. Consider the two closed channel case. Let their ionization energies be

${I}_{1}$ and

${I}_{2}$ with

${I}_{1}>{I}_{2}$. Then the autoionizing Rydberg series converging to

${I}_{1}$ acts as an interloper series and the other converging to

${I}_{2}$ plays the role of the principal autoionizing Rydberg series perturbed by the interloper. In this case, it is meaningless to consider the resonance center since the energy variation of resonance structures is local, non-uniform and aperiodic. The resonance structure is obtained as follows:

where

${\tilde{\epsilon}}_{1}$ denoting

$\mathrm{tan}{\tilde{\beta}}_{1}/{\tilde{\xi}}_{1}^{2}$ is the reduced energy parameter which vanishes at each resonance

${\tilde{\nu}}_{1}=n-{\tilde{\mu}}_{1}$ and runs from −∞ to ∞ between two successive resonances;

${\tilde{W}}_{i}={\tilde{\xi}}_{i}^{2}$(

i = 1,2),

${\tilde{W}}_{2\mathrm{eff}}={\tilde{W}}_{2}{w}_{2\mathrm{eff}}$ with the reduced width

${\tilde{w}}_{2\mathrm{eff}}$:

where

$\theta $ is the angle two coupling vectors

${\xi}_{1}$ and

${\xi}_{2}$ of closed channels 1 and 2 make in

P space;

${k}_{12}$ being the ratio of direct to indirect couplings, and the reduced energy

${\tilde{\epsilon}}_{2\mathrm{eff}}$ is shifted as

$\left({\tilde{\epsilon}}_{2}-{s}_{2}\right)/{\tilde{w}}_{2\mathrm{eff}}$ with: