II. Non-paper: Zeta functions in scattering problems evaluated on the real wave number axis. Andreas Wirzba. Institut fur Kernphysik, TH Darmstadt

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1 Damstadt, Octobe, 995 II. Non-ae: Zeta functions in scatteing oblems evaluated on the eal wave numbe axis Andeas Wizba Institut fu Kenhysik, TH Damstadt Schlogatenst. 9, D-6489 Damstadt, Gemany www : htt :==cunch:ik:hysik:th-damstadt:de=wizba= ABSTRACT Using the two-dimensional -disk system (in the A eesentation) we comae the cluste hase shifts of the exact quantum mechanical oblem with the coesonding cluste hase shift of the following semi-classical zeta functions: (a) the Gutzwille-Voos zeta function, (b) the dynamical zeta function and (c) the quasi-classical deteminant. Futhemoe we show esults fo the squaed moduli of the quantum mechanical and semi-classical deteminants on the eal wave numbe axis. As the cluste hase shifts ae in incile measuable quantities, even exeimentalists can now decide which of the thee choices is the best. Intoduction The question might be aised which of the following thee zeta functions is descibing the quantum mechanical scatteing data the best, (a) the Gutzwille-Voos zeta function Z GV (z; k) = ex 8 < X :? X = z [] t (k)? 9 = Y ; = Y j=!? z[] t j ; (.) (b) the dynamical zeta-function? (z; k) = ex (? X X = z[] t (k) ) = Y? z [] t ; (.) (c) o the quasi-classical zeta-function of ef.[] Z qcl (z; k) = ex 8 >< X >:? X =? z [] t (k)? 9 >= >; ; (.) whee t (k) = e ikl?i= =j j is the th imay cycle, [] its toological length and z is a book-keeing vaiable fo keeing tack of the toological ode #. L is the length of the th The newest vesion of this non-ae can always be found on this www home-age. # The vaious zeta-functions ae Taylo-exanded in z aound u to a given cuvatue ode. Finally, z :=.

2 obit, its Maslov index (lus the gou theoetical weight), and its stability (the leading eigenvalues of the stability matix) see efs.[, ] fo details and denitions. In the ast this evaluation was made by comaing the exact and semi-classical esonances (esecially the sub-leading ones) of the -disk system in the A eesentation, see e.g. the nonae []. As the deviations ae most onounced fo the sub-leading esonances (which ae shielded by the leading ones), one could ague that exeimentally it doesn't matte which of the thee zeta functions ae used to descibe the measued data, as all thee give aoximately the same edicitions fo the leading esonances. Below we will show that even exeimentally one can tell the thee semi-classical zeta functions aat and that in fact the Gutzwille-Voos one is by fa the best. Cluste hase shifts In ef.[] the exact and semi-classical exessions fo the deteminant of S-matix fo nonovelaing n-disk systems have been constucted. Fo the case of the -disk system they ead det l S () (k) = s:c:?! det l S () det l M A (k ) y det l M A (k ) y (ka) det l M A (k) det l M A (k) e?in (k) e Z-disk(l)(k ) ez -disk(l)(k) ez -disk()(k ) ez -disk()(k) det l M E (k ) y (det l M E (k))! e Z A (k ) ez A (k) ez A (k ) ez A (k) ez E (k ) ez E (k) (.) whee the tilde indicates that diactional coections have to be included in geneal. Esecially fo the A eesentation of the -disk system we theefoe have the elation between the quantum mechanical kenels and the Gutzwille-Voos zeta functions det l M A (k ) y det l M A (k) s:c:?! Z A(k ) Z A (k) ; (.) whee we now neglect diactional coections. As agued in ef.[] the l.h.s. and the.h.s. of eq.(.) and theefoe eq.(.) esect unitaity; i.e. if the wave numbe k is eal, the left hand sides and also the ight hand sides of eqs.(.) and (.) can be witten as exfi(k)g with a eal hase shift (k). In fact, we can dene a total hase shift fo the coheent at of the -disk scatteing oblem (hee always undestood in the A eesentation) in the following way: e iqm(k) := det lm(k ) y det l M(k) e i GV(k) := Z GV (k ) Z GV (k) e i dyn(k) e i qcl(k) (.) (.4) :=? (k )? (k) (.5) := Z qcl(k ) Z qcl (k) : (.6)

3 Please comae this hase shift denition with the cluste hase shift given in section 4 of Lloyd and Smith [4]. Fo a seaable system, as e.g. the -disk system (in the angula momentum eesentation), the cluste hase shift coesonds just to the sum (k) = X l=? l (k) ; (.7) as the S-matix in the one-disk system (evaluated with esect to the cente of the disk) eads ()?H l (ka) S ll (k) = H () l (ka) ll = e i l(k) ll ; (.8) such that dets(k) = +Y l=? e i l(k) : (.9) Let us once moe stess: the coheent o cluste hase shift is an exeimentally accessible quantity: one just has to constuct fom the measued coss sections the elastic scatteing amlitude. This leads to the full hase shift of the -disk system including the contibution of the single disks. Howeve, the incoheent at can of couse be subtacted by making efeence exeiments with just single disks at the same osition whee they used to be in the -disk oblem. In this way one can seaate the incoheent hase shifts fom the coheent ones. Finally, one can use Levinson's theoem in ode to secify the oveall fee hase. So, qm (k) is \measuable". Can we now use it in ode to disciminate between the vaious zeta functions? This is done in the next section. The comaison Below, we comae the exact quantum mechanical cluste hase shift qm with (a) the semi-classical cluste hase shift GV (k) of the Gutzwille-Voos zeta function (.), (b) the semi-classical cluste hase shift dyn (k) of the dynamical zeta function (.), (c) and the semi-classical cluste hase shift qcl (k) of the quasi-classical zeta function (.). The zeta functions in the numeato as well as in the denominato of Z(k) =Z(k) have been exanded to cuvatue ode (=toological length). Fo the Gutzwille-Voos zeta function this is ovekill as aleady cuvatue ode 4 should descibe the data below Rek = 95=a #. Howeve, the quasi-classical one has oblems fo lowe cuvatue odes with edicting the (sub-)leading esonances (see ef.[]); theefoe, we wanted to give the quasi-classical zeta function an as fai chance as ossible. # In fact, we have not seen any dieence in the Gutzwille-Voos calculation between the cuvatue ode and esults fo k =a and u to gue accuacy. Cuvatue ode, howeve, gives in the egime =a k =a noticable deviations.

4 We will show the data in the window =a k =a. This window is by no means secial. We could have chosen any othe window and thee would have been no change in the qualitative behavio. The eason why we show such a small window is that we othewise would not see anything because of the aid oscillations. Also k =a is big enough to suess diaction eects. Futhemoe, although we have no hysical inteetation in tems of the S-matix, we also comae the exact quantum mechanical oduct detm(k)detm(k ) y with the oducts (a') Z GV (k)z GV (k ), the squaed modulus of the Gutzwille-Voos zeta function, (b')? (k)? (k ), the squaed modulus of the dynamical zeta function, (c') and Z qcl (k)z qcl (k ), the squaed modulus of the quasi-classical zeta function, with k always eal and fo the -disk system in the A eesentation. Note nally that the zeta function dened by the atio (using the denitions of ef.[] # ) inseted into Z(k) := F +( ; k)f?( 7 ; k) F? ( ; k)f +( 5 ; k) (.) e iat(k) = Z(k ) Z(k) (and also into the oduct Z(k)Z(k) ) woks on the eal wave numbe axis and in the limit n! (.) (whee n is the cuvatue ode) as well as the oiginal Gutzwille-Voos zeta function. So it doesn't matte whethe the Gutzwille-Voos zeta function is diectly exanded in the cuvatue exansion o whethe the the individual deteminants F + ( ; k), F?( 7 ; k), F?( ; k) and F +( 5 ; k) ae each exanded in seaate cuvatue exansions u to the same cuvatue ode and then inseted in the atio (.). Note that the esence o absence of the subleading facto ( + j j?4 ) (given in the footnote below) does not change the esults u to gue accuacy. So hee ae the data: # The quasi-classical zeta functions F+(; k; z) and F?(; k; z) ae dened as follows, whee the subleading facto ( + j j?4 ) of eq.() in ef.[] has been emoved as in eq.() of ef.[]: F +(; k; z) = ex 8 >< X X >:? F?(; k; z) = ex 8 > < >:? X The inut quantities ae dened as below eq.(.). = X =? j j? z [] t (k)?? z [] t (k)?? j j?+ 9 >= >; j j?+ 9 >= >; : 4

5 .8 Coheent cluste hase shifts of the -disk-system with R=6a and in A_ eesentation QM (exact): Gutzwille-Voos:.6.4. \eta(k) (a) k [/a] Figue.: The coheent cluste hase shifts of the -disk scatteing system in the A eesentation with R = 6a. The exact quantum mechanical data ae denoted by diamonds. The semi-classical Gutzwille- Voos edictions ae calculated u to th ode in the cuvatue exansion and ae denoted by cosses. Below the same fo the squaed moduli of the sectal deteminants. Coheent squaed moduli of the sectal deteminant in the -disk system with R=6a and in A_ eesentation QM (exact): Gutzwille-Voos:.5 Z(k) Z(k)*.5.5 (a') k [/a] 5

6 .8 Coheent cluste hase shifts of the -disk-system with R=6a and in A_ eesentation QM (exact): dynamical zeta fct:.6.4. \eta(k) (b) k [/a] Figue.: The coheent cluste hase shifts of the -disk scatteing system in the A eesentation with R = 6a. The exact quantum mechanical data ae denoted by diamonds. The semi-classical data of the dynamical zeta function ae calculated u to th ode in the cuvatue exansion and ae denoted by cosses. Below the same fo the squaed moduli of the sectal deteminants. Coheent squaed moduli of the sectal deteminant in the -disk system with R=6a and in A_ eesentation QM(exact): dynamical zeta fct.:.5 Z(k) Z(k)*.5.5 (b') k [/a] 6

7 .8 Coheent cluste hase shifts of the -disk-system with R=6a and in A_ eesentation QM (exact): Quasi-classical:.6.4. \eta(k) (c) k [/a] Figue.: The coheent cluste hase shifts of the -disk scatteing system in the A eesentation with R = 6a. The exact quantum mechanical data ae denoted by diamonds.. The semi-classical quasi-classical edictions ae calculated u to th ode in the cuvatue exansion and ae denoted by cosses. Below the same fo the squaed moduli of the sectal deteminants. Coheent squaed moduli of the sectal deteminant in the -disk system with R=6a and in A_ eesentation QM(exact): Quasi-classical:.5 Z(k) Z(k)*.5.5 (c') k [/a] 7

8 4 Conclusion Conclusions ae eally not necessay. The data seak fo themselves see esecially the height of the maxima and minima and the \beating node". Let us stess that these ae not \ funny" data as the sub-leading esonances (which ae comletely shielded by the leading esonances), but had data #4 which even exeimentalists can econstuct fom thei measued coss sections. So one of the conclusions of ef.[] has to be modied: even exeimentally one can tell the thee semi-classical zeta functions aat and see which is the best. The esult is that the Gutzwille-Voos whethe used diectly o whethe dened as the atio (.) of fou quasiclassical deteminants as in ef.[] is by fa the best. Refeences [] P. Cvitanovic, P.E. Rosenqvist, G. Vattay and H.H. Rugh, A Fedholm Deteminant fo Semiclassical Quantization, CHAOS (99) [] P. Cvitanovic and G. Vattay, Entie Fedholm Deteminants fo Evaluation of Semiclassical and Themodynamical Secta, Phys. Rev. Lett. 7 (99) [] A. Wizba and M. Hensele, Non-ae about The missing-link between the quantum mechanical and semiclassical detemination of scatteing esonance oles, htt :==www:nbi:dk=edag=qccouse= and htt :==cunch:ik:hysik:th-damstadt:de=wizba= [4] P. Lloyd and P.V. Smith, Multile-scatteing theoy in condensed mateials, Adv. Phys. (97) 69-4 and efeences theein. #4 This holds at least fo the hase shifts. Fo the absolute squaed deteminants thee is no obvious hysical inteetation. 8