Explicit, analytical solution of scaling quantum graphs. Abstract


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1 Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT , USA (Januay 6, 2003) Abstact Based on ealie wok on egula quantum gaphs we show that a lage class of scaling quantum gaphs with abitay topology ae explicitly analytically solvable. This is supising since quantum gaphs ae excellent models of quantum chaos and quantum chaotic systems ae not usually explicitly analytically solvable Mt,03.65.Sq 1 Typeset using REVTEX
2 Integable systems ae ae. Many have become textbook classics, such as the hamonic oscillato and the hydogen atom. It is theefoe always a notewothy event when anothe integable system is discoveed. The pupose of this Lette is to add an impotant class of quantum systems to this set: quantum gaphs. Quantum gaphs have a long histoy in mathematics, chemisty and physics (see, e.g. the excellent eview by P. Kuchment [1]). They have ecently come to the attention of many eseaches as models of quantum chaos [2 5]. It may theefoe be supising that a lage and impotant subset of quantum gaphs, scaling quantum gaphs, is explicitly integable in the fom E n = f(n), whee n is an intege, E n ae the enegy levels of the gaph and f is a function that can be constucted explicitly. Because of the quantum chaos connection it aleady caused quite a sti when about a yea ago explicit solutions of a subclass of quantum gaphs, egula quantum gaphs, wee discoveed [6,7]. This is so, because fo any given n the enegy level E n is individually computable, without knowledge of the othe quantum levels. Thus the counting index n is a quantum numbe which is geneally believed not to exist in a quantum system such as quantum gaphs that show so many featues of quantum chaos [8,9]. While in the case of egula quantum gaphs one might shug off the existence of explicit solutions by pointing to the special, nongeneic natue of egula quantum gaphs (only simply connected, linea and cicula gaphs have so fa been identified as membes of the set of egula quantum gaphs), this is now no longe possible. Thus the esults epoted in this Lette ae a substantial, qualitative step fowad with pofound implications fo quantum gaph theoy and quantum chaos. Focussing on scaling quantum gaphs [6,7] excluding gaphs with tunneling and bound states fo convenience (tunneling and bound states ae discussed in [10]), we will pove below that these quantum gaphs, no matte how complex thei topology, ae explicitly solvable analytically. We emphasize that scaling quantum gaphs [6,7] epesent a huge class of gaphs, fa lage than and including the standad, undessed gaphs pimaily studied in the liteatue [1 5]. The class of quantum gaphs fo which we pove the existence of explicit solutions compise quantum gaphs with abitay topology, abitay scaling potentials on thei bonds and abitay scaling δfunctions on thei vetices. 2
3 It has been shown befoe [6,7] that the spectal function of scaling quantum gaphs is given by N g (0) (k) =cos(s 0 k + ϕ 0 ) a j cos(s j k + ϕ j ), (1) j=1 whee S 0 is the action length of the gaph, S j <S 0 ae cetain combinations of the gaph actions, N is the numbe of diffeent gaph action combinations in (1), ϕ 0, ϕ j ae constant phases and a j ae constant amplitudes. The oots k (0) n = E n of the spectal equation g (0) (k n (0) )=0 (2) ae the wave numbes of the eigenenegies E n of the gaph. It was shown in [7] and put on a solid mathematical foundation in [11] that (2) is explicitly solvable in the fom k (0) n =... fo egula quantum gaphs [6,7,11] which satisfy N a j < 1. (3) j=1 The vast majoity of quantum gaphs does not satisfy the egulaity condition (3). Up to now this was believed to be the hedge that makes geneal quantum gaphs esilient against explicit solution and thus peseves one of the cental tenets of quantum chaos theoy, i.e. the loss of quantum numbes [8,9]. Howeve, thee is a way to solve all scaling quantum gaphs, even if they don t fulfill (3), by constucting a chain of spectal functions teminating with a function that does satisfy (3), and then making full use of (3) and the associated mathematical machiney developed fo egula quantum gaphs [11]. Thus the theoy of egula quantum gaphs povides us with a poweful point of depatue fo solving the moe geneal poblem of all scaling quantum gaphs. Taking the m th deivative of (1) and dividing by S m 0 we obtain whee b (m) j g (m) (k) =cos(s 0 k + ϕ 0 + mπ/2) N j=1 b (m) j cos(s j k + ϕ j + mπ/2), (4) = a j (S j /S 0 ) m. Since S j <S 0, thee always exists an M such that b (M) j < 1, i.e. fo m = M (4) satisfies the egulaity condition (3) and the oots k (M) q 3 of g (M) (k) =0ae
4 explicitly computable via explicit analytical, convegent peiodicobit expansions as shown in [6,7] and poved igoously in [11]. Having bootstapped the oots of (1) at the level M by taking the M th deivative of (1), we now have to go backwads to the levels M 1, M 2,..., until we each level 0 and possess k (0) n explicitly. Retacing ou steps is indeed possible by making use of socalled ootsepaatos ˆk n, which play an impotant ole in the theoy of egula quantum gaphs [11]. A sepaato ˆk n sepaates oots numbe n 1andnumben fom each othe. Because the theoy of egula quantum gaphs applies to g (M) (k), we know the sepaatos go one level up to M 1, we need to know the sepaatos ˆk (M 1) (M) ˆk q.to which sepaate oots numbe 1andnumbe fom each othe. It tuns out [10] that the extema of g (M 1) (k) ae oot sepaatos on the level M 1. But the location of the extema of g (M 1) (k) ae the zeos of g (M) (k), which we know. With the knowledge of the oot sepaatos compute the zeos k (M 1) of g (M 1) (k): ˆk (M 1) we now ˆk(M 1) k (M 1) = ˆk (M 1) 1 k g (M) (k) δ(g (M 1) (k)) dk. (5) It is impotant to ealize that (5) is not just a fomal solution, but yields k (M 1) explicitly, analytically, even numeically, if we wish, to any desied accuacy. This is so since both g (M 1) (k) and the oot sepaatos ˆk (M 1) ae known explicitly. It is futhemoe impotant to ealize that the tick (5) does not geneally wok fo othe quantum systems, even if thei spectal equation g is known. This is so because the oot sepaatos ˆk n,knownand explicitly constuctable in the case of quantum gaphs, ae not in geneal known fo othe quantum systems. So fo moe geneal quantum (chaotic) systems the missing sepaatos is indeed a hedge that, so fa, potects them against explicit solution. Ou etacing pocess fom levels M to M 1 can now be continued until we each level 0. This completes ou poof that all scaling quantum gaphs can be solved analytically in closed fom. 4
5 I. ACKNOWLEDGMENTS The authos acknowledge financial suppot by the National Science Foundation unde Gant No
6 REFERENCES [1] P. Kuchment, Waves in Rand. Media 12(4), R1 (2002). [2] T. Kottos and U. Smilansky, Phys. Rev. Lett. 79, 4794 (1997); Ann. Phys. (N.Y.) 274, 76 (1999). [3] H. Schanz and U. Smilansky, Phys. Rev. Lett. 84, 1427 (2000). [4] T. Kottos and H. Schanz, Physica E 9, 523 (2001). [5] U. Smilansky, in Mesoscopic Quantum Physics, Les Houches, Session LXI, 1994, edited by E. Akkemans, G. Montambaux, J.L. Pichad and J. ZinnJustin (Elsevie Science Publishes, Amstedam, 1995), pp [6] Yu. Dabaghian, R. V. Jensen, and R. Blümel, JETP Lett. 74, 235 (2001); JETP 94, 1201 (2002). [7] R. Blümel, Yu. Dabaghian, and R. V. Jensen, Phys. Rev. Lett. 88, (2002); Phys. Rev. E 65, (2002). [8] M. Gutzwille, Chaos in Classical and Quantum Mechanics (Spinge, New Yok, 1990). [9] H.J. Stöckmann, Quantum Chaos (Cambidge Univesity Pess, Cambidge, 1999). [10] Yu. Dabaghian and R. Blümel, in pepaation fo submittal to Phys. Rev. E (2003). [11] R. Blümel, Yu. Dabaghian, and R. V. Jensen, Mathematical foundations of egula quantum gaphs, to be published (2003). 6
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