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Extra resources for Physics Reports vol.190
In Figs. 19—27, for increasing curvature order, the resonances are plotted as the real part of the wave number (resonance “energy”) versus the imaginary part of the wave number (resonance “width”). Some features of the resonance spectra allow for an immediate interpretation [13,15,17]: The mean spacing of the resonances is approximately 2 /¸M , where ¸M is the average of the geometrical lengths of the shortest periodic orbits, namely the lengths ¸ and ¸ of the two periodic orbits of topologic length one.
43] has been removed as in (12) of Ref. N . When Eq. 13) works on the real wave-number axis and in the limit nPR (where n is the curvature order) as well as the original Gutzwiller—Voros zeta function. 12). Note that the presence or absence of the subleading factor (1#" P "@\) in the definitions of F ( , k; z) and N > F ( , k; z) does not change the results up to figure accuracy. \ Let us stress that phase shifts are not only of theoretical interest, as are the subleading resonances (which are completely shielded by the leading resonances), but hard data which can be extracted, in principle, from measured differential cross sections.
61). 7, these relations can be K\K proven by complete induction. Now, by solely inserting Eqs. 87) into the second and third line of Eq. R ) # GK GK G G G G G G j K\ 2R j 2R j K\ 2R G # GK . K! R j K K ! R j j K\! ! K! K R #2 R # R K K K j¸ ¸ j¸ j¸ K K\K K K\K R ! R R K# K! 89) A. R j K# K\ R ! K R ! 89)" K K ¸ j¸ ¸ j¸ K\K K\K j K\ 2R K\ 2R K\GK ! 1# K\GK " R #¸ 1# K j¸ K\G K\G G G j K\ 2R K\ 2R G !