Semiclassical Quantization 696

CHAPTER 38. SEMICLASSICAL QUANTIZATION 696

Figure 38.1: A returning trajectory in the conﬁgura- tion space. The orbit is periodic in the full phase space only if the initial and the ﬁnal momenta of a returning trajectory coincide as well. Chapter 38

Figure 38.2: A romanticized sketch of S p(E) = S (q, q, E) = p(q, E)dq landscape orbit. Unstable periodic orbitsH traverse isolated ridges and saddles of Semiclassical quantization the mountainous landscape of the action S (qk, q⊥, E). Along a periodic orbit S p(E) is constant; in the transverse directions it generically changes quadratically.

(G. Vattay, G. Tanner and P. Cvitanovi´c) so the trace receives contributions only from those long classical trajectories which are periodic in the full phase space.

e derive here the Gutzwiller trace formula and the semiclassical zeta func- For a periodic orbit the natural coordinate system is the intrinsic one, with qk tion, the central results of the semiclassical quantization of classically axis pointing in theq ˙ direction along the orbit, and q⊥, the rest of the coordinates W chaotic systems. In chapter 39 we will rederive these formulas for the transverse toq ˙. The jth periodic orbit contribution to the trace of the semiclassical case of scattering in open systems. Quintessential wave mechanics eﬀects such as Green’s function in the intrinsic coordinates is creeping, diﬀraction and tunneling will be taken up in chapter 42. 1 dqk d−1 j 1/2 i S − iπ m tr G (E) = d q |det D | e ~ j 2 j , j (d−1)/2 ⊥ ⊥ i~(2π~) I j q˙ Z j

38.1 Trace formula where the integration in qk goes from 0 to L j, the geometric length of small tube around the orbit in the configuration space. As always, in the stationary phase approximation we worry only about the fast variations in the phase S j(qk, q⊥, E), Our next task is to evaluate the Green’s function trace (35.15) in the semiclassical and assume that the density varies smoothly and is well approximated by its j approximation. The trace value D⊥(qk, 0, E) on the classical trajectory, q⊥ = 0 . The topological index m j(qk, q⊥, E) is an integer which does not depend on the initial point qk and not D D tr Gsc(E) = d qGsc(q, q, E) = tr G0(E) + d qG j(q, q, E) change in the infinitesimal neighborhood of an isolated periodic orbit, so we set Z Z Xj m j(E) = m j(qk, q⊥, E). receives contributions from “long” classical trajectories labeled by j which start The transverse integration is again carried out by the stationary phase method, and end in q after finite time, and the “zero length” trajectories whose lengths with the phase stationary on the periodic orbit, q = 0. The result of the transverse approach zero as q′ → q. ⊥ integration can depend only on the parallel coordinate

First, we work out the contributions coming from the finite time returning 1/2 dq det D⊥ (qk, 0, E) i iπ classical orbits, i.e., trajectories that originate and end at a given configuration 1 k j S j− m j tr G (E) = e ~ 2 , ′ j i~ q˙ det D′ (q , 0, E) point q. As we are identifying q with q , taking of a trace involves (still another!) I ⊥ j k ′ stationary phase condition in the q → q limit, where the new determinant in the denominator, det D′ = ′ ′ ⊥ j ∂S j(q, q , E) ∂S j(q, q , E) + = 0 , ′ 2 ′ 2 ′ 2 ′ 2 ′ ∂qi q′=q ∂qi q′=q ∂ S (q, q , E) ∂ S (q, q , E) ∂ S (q, q , E) ∂ S (q, q , E) det + + + ,  ∂q ∂q ∂q′ ∂q ∂q ∂q′ ∂q′ ∂q′  meaning that the initial and final momenta (37.40) of contributing trajectories  ⊥i ⊥ j ⊥i ⊥ j ⊥i ⊥ j ⊥i ⊥ j    should coincide   is the determinant of the second derivative matrix coming from the stationary ′ phase integral in transverse directions. pi(q, q, E) − pi (q, q, E) = 0 , q ∈ jth periodic orbit , (38.1)

695 traceSemicl - 2mar2004 ChaosBook.org version15.9, Jun 24 2017 . ) i + /π 698 m 1 , the 2 D − (38.5) (38.3) (38.4) / )). The h 2 q p ( Weyl rule , V ) p − . By now we m (no sum on p π 2 i E ( D by counting the − .15) is given by p limit, so it makes m . The final result, nary part is finite, divided by S E ltiplied with 2 ates the number of i ~ q hat you have surely ven by the ( ns of type E p r ,..., which we derive next. ction in terms of peri- e ic orbit , s the imaginary part of 2 → ChaosBook.org version15.9, Jun 24 2017 2 he quantum case we do , / ′ )) 1 sity of states 1 q | p ) , = p r q i ( ) is an estimate of the spectral M ), H E 1 i ( − p − sc . , i N E q )) ( p δ det (1 , | q q 1 D ( , = ∞ ) r H X pd E ) is infinite in the p , − D E T q d ( , E 0 ) actually is. The rule is this: The topological phase space is the sum of the number of times ( q p ), the contribution coming from the infinitesimal ( G Z E X ) counts the number of changes of sign of the ma- Θ 0 ( E D E p ( q ~ D 1 G ( 1 i 0 h D p m G + Im m for each dual pair ( = pd ) q i i ) D E p q D d ( E ∂ ∂ ( d 0 sc G Z integrated over the configuration space Z N tr D q 1 1 π h = d dE − ) → = ′ E = =

38. SEMICLASSICAL QUANTIZATION ) ( q ) ) E ) is the Heaviside function (35.20). sc ( x E E ( ( ( G sc 0 0 Θ N tr d d The resulting formula has a pretty interpretation; it estim The semiclassical contribution to the density of states (35 The topological index ), the energy shell volume in (38.5) is a sphere of radius Gutzwiller trace formula q ( an expression for the trace of the semiclassical Green’s fun change their signs as one goes once around the curve. 38.1.1 Average density of states We still have to evaluate tr V trajectories. The real part of tr available quantum cells in the phase space. This number is gi precisely the semiclassical result (38.6). For Hamiltonia , as the ratio of the phase space volume bounded by energy no sense to writeand it plays down an explicitly important role here. in the However, density the of states imagi formula, the imaginary part of the Gutzwiller trace formula (38.3) mu volume of a quantum cell, index of a closed curve in a 2 traceSemicl - 2mar2004 the partial derivatives staircase (35.19), so its derivative yields the average den odic orbits, is beautiful in its simplicity and elegance. lost track of what the total have gone through so many stationary phase approximations t trix of second derivatives evaluated along the prime period quantum states that can be accommodated up to the energy derivation takes only three pages of text. Regrettably, in t the not know of an honest derivation that takes less than 30 pages where CHAPTER The contribution coming from the zero(37.48) length for trajectories i k q 697 . (38.2)

, where , p r ) ) can also j ′ ⊥ M riod of the q ) in the full M ⊥ . The remain- − ) − p p it by following ) ′ ⊥ ⊥ t , starting point T romy matrix are ( q q dic orbit is also a ⊥ ˙ q , , q / and the topological ( ⊥ ′ ⊥ k is an integral around q p k itive along the trajec- ( k . formula (21.19) - that = dq

∂ ChaosBook.org version15.9, Jun 24 2017 − k dq dic orbit. It can be given ) e integral in the parallel k dq ) = ˙ ⊥ ) ⊥ q ′ ⊥ dq E x p x E , ( traversal. If you do not see dt , k ′ ⊥ . The monodromy matrix of k ∂ − x r q I q (

) ∂ ⊥ ( j p x p = m ( π H 2 i ∂

H single −

j ′ ⊥ ′ ⊥ = = p S q ) shell. = ) i ) ~ ∂ ∂ p ( E

E r , ( ) ( e −

j p q for a surface of section transverse to the ′ ⊥ 2 ) ( ) / S S ⊥ q ′ ⊥ ′ ⊥ ′ ⊥ 1 H | p M q p q ) − j ∂ ∂ , ) , = ⊥ ⊥ ′ ⊥ ′ ⊥ is (by the chain rule for derivatives) has no marginal eigenvalues. The determinant M + q q p q E , . j ( ( p , 1 , − is here to enforce the periodic boundary condition ) ′ ⊥ p ⊥ ∂ ∂ is evaluated along the orbit, so det (1 ′ ⊥ ′ ⊥ 1 M p j q j

p T q ∂ ∂ ( ′ ⊥ − M = = − ∂ D −

det (1 ⊥ M | dt ( p ⊥ ′ ⊥ ⊥ ⊥ p det ( p p integral q q j T / ∂ 1)-dimensional transverse vector j ∂ ∂ ∂ ∂ k 0 X

det q

⊥ Z − ~ 1 i D = = = D = = = ) k t , the single, shortest traversal of a periodic orbit by ) ( th repeat they simply get multiplied by p

38. SEMICLASSICAL QUANTIZATION ˙ dq ⊥ ′ ⊥ E q ′ ⊥ ⊥ r ( j p D D D D L G 0 det det Z det det tr th repeat of a prime cycle is the prime cycle monodromy matrix. Let us denote the time pe The ratio det First, we take into account the fact that any repeat of a perio The classical periodic orbit action r p M why this is so, rethink the derivation of the classicaltraceSemicl - trace 2mar2004 Here we have assumed that Note that the spatial integral corresponds to a along the orbit, seealso figure independent of 38.2. where The eigenvalues of the monod ing integral can be carried out by change of variables of the monodromy matrix, the action the prime cycle phase space we can express the ratio a loop defined by the periodic orbit, and does not depend on the periodic orbit. The action andtory, the so for topological index are add index are all classical invariantsdirection of we the now periodic do orbit. exactly. Th for the semiclassical Green’s function evaluated ona a perio meaning in terms ofobservations the monodromy matrix of the periodic orb Defining the 2( be taken out of the in terms of the monodromy matrix CHAPTER orbit within the constant energy 700 , see n (38.11) E = antum chaos E ce formula in need them, at further ado by ectral determin- ether the formula , discussed below of linear operators . orm ians. But there is no 2 )) 0. A first guess might / formulas by replacing E ChaosBook.org version15.9, Jun 24 2017 1 ( | p lation, however, opens a ) = 0 approximation, how to p r m π 2 i sc has a zero at M ) − , by replacing the trace with we multiply larger and larger → ) E − sc ) E sc ( ) E . ) ~ p regularization E − ) E , E S | E ˆ i ~ ′ H ( ( − − q r det (1 G ) is the (formal) trace of the Green’s | e ˆ ˆ − H H E tr 1 q = ∞ | r − = X ( . ) poles at the semiclassical eigenenergies, ) p ˆ n n ultraviolet divergence. For the regularized H E T sing E E ( q 1 G − p − reg X → − E , E G ( ) ) ~ er an impressionistic sketch of regularization. ′ 1 i / n q tr E ff n E , + X ′ = ) − q = , sc E E q ) ) ( ( ( 0 E E d G n π − − Y i = ˆ ˆ − H H ) = ). This problem is dealt with by Green’s function ) E = n , E ) ′ E q E − , ( − 38. SEMICLASSICAL QUANTIZATION ln det ( ln det ( q ˆ ( H E reg d d dE dE G reg det ( − − tr G regularized A sensible way to compute energy levels is to construct the sp The logarithmic derivative of det ( The take care of the sumconverges at over all. an infinity of periodic orbits, and wh 38.2 Semiclassical spectral determinant The problem with trace formulas is that they diverge where we the individual energy eigenvalues.literature responds What to the tothe challenge do? delta of functions wrestling in Much the theneed of trace density to the of do states qu this (35.16) - by we Gauss can compute the eigenenergies without any remembering that the smart way to determine the eigenvalues is by determining zeros of their spectral determinants. ant whose zeroes yield the eigenenergies, det ( Now you stand where Gutzwiller stood in 1990. You hold the tra your hands. You have no clue how good is the but this product is not well defined, since for fixed be that the spectral determinant is the Hadamard product of f numbers ( The regularized trace still has 1 in appendix 38.1.2. Here we o function poles which can be generated only if det ( way to derive a convergent version of det ( This quantity, not surprisingly, is divergent again. The re figure 38.3. By integrating and exponentiating we obtain traceSemicl - 2mar2004 is obtained by subtracting the the regularized trace CHAPTER Green’s function the Gutzwiller trace formula is exercise 38.2 exercise 38.4 exercise 38.3 , ) < ) or E ) E ( 699 ( q . sc ( (38.8) (38.6) (38.7) (38.9) osc V N (38.10) d 2 2 z + , = ) hich E d d ( cannot be re- 0 d . In two spatial , for D n support 1 h E = nsity of states and . 2), so the average − ) ) for / 2 2 ll n be extracted from / / γ d E ( D D on for small ( ial dimension + Γ ChaosBook.org version15.9, Jun 24 2017 sc 2 / ))] ) ))] − d 1 d q ~ q | − ′ ( ( 1 2 q d / 2) V V r | − ′ q 2 | / − − π/ q d 2 p 2) / π E − E 1 / ( ( | ) q rm | is 2) m m p r ) − r [2 [2 − V M , ~ q q d / − − ) D D ectively finite dimensional. The average 0 0 (( d E d E ff Γ ( ( , = E E 2 d p m < < ν ν π ) ) m 2 det (1 q q | ~ rS ( ( 2 V V for (ln(2 − Z Z 2 cos( 1, and so is its formal trace (35.15). The short ~ zero length Green’s function (37.48) is ultraviolet ) for m ν π 2) 2) 1 γ 2 − / / q > = ∞ r 2 X + z D D 2 /            p d ( D → + 2) Γ ) T π = , 2 m / ν ′ z ) ( ) 2 D (1 p q , Γ 2 π E E Γ X ) is the Euler constant. The real part of the Green’s function f ( 1 d π , ~ | ~ (ln( ~ 2 ′ E π D 1 − 1 2 π π A ( π ... q D 2 h 2 m ~ T −        = -dimensional sphere of radius = 577 d ) q = = = . ) |

38. SEMICLASSICAL QUANTIZATION ≈ ( ) is the classically allowed area of configuration space for w ) is the period of the periodic orbit of fixed energy 0 ) ) ) E E ) ( ( E E E z E E = ( ( ( ( ( ( sing sc ν 0 0 0 osc T γ A N d Y d d d G The semiclassical density of states is a sum of the average de . E dimensions the average density of states is where follows from the trace formula (38.3). 38.1.2 Regularization of the trace The real part of the divergent in dimensions traceSemicl - 2mar2004 where and distance behavior of thethe real real part part of of (37.48) the by Green’s using function the ca Bessel function expansi Physically this means that at a fixed energy the phase space ca short distance is dominated by the singular part surface of a density of states is given by where distinct eigenfunctions; anything finer than the quantum ce density of states is of a particularly simple form in one spat the oscillation of the density of states around the average, solved, so the quantum phase space is e CHAPTER where = ) and 702 E p ( Bohr- p T (38.17) (38.16) (38.15) has two m p . M        p x ping the mon- isely the antization rule. the quantization rm π 2 i is the only zero of ic case, when the r a given energy E, − 7) between ∗ p ral determinant. The x one. The determinant rS i ~ ChaosBook.org version15.9, Jun 24 2017 e nd its zeros are again the r 1 . 2 , while the average number ) and we get . / t r n ) 1 ) has a unit jump in this point X p 2 − )), where 4 m E − x − i / ( (        ) − 4 f p N / E ( ln(1 S ) ( i δ ~ p | exp E − ) e ( m x p p ( = − ′ − m m r f | / (1 ~ π − 2 r i π / 2 i t ) = ~ r + + E ) π p , defined by the condition . ( , 4 ∗ P 2 p n im p x / 4 S + ) p S / p − ) 4 ~ E i S m ( 2 x E ~ i p 2 ( ( , − − p δ − ) S , as explained in sect. 8.3. Isolated periodic orbits can n 2 sin e n m p = E − h the exact number of states is Λ exp 2 for smooth potential at both turning points, and 2 are the zeroes of the arguments of delta functions in (38.14) − = / = n / n 0 in the interval under consideration. We obtain n = = = 1 E E E = ( = p − sc sc δ 2 since the staircase function dq ) ~ = ) ) ) / m and 1 ∗ n π n E E 1 n x 2 E p ( E X = = / − − f − , ) Λ ) 38. SEMICLASSICAL QUANTIZATION ) = q n ˆ ˆ n ( E H H E ) E ( ( p ( p E sc p ( m 4 for two billiard (infinite potential) walls. These are prec N det ( det ( S d I = The 1-dof spectral determinant follows from (38.12) by drop , and the identity (19.7) p p of states is Summation yields a logarithm by In one dimension, thecondition. average At of states can be expressed from In this way the trace formula recovers the well known 1-dof qu So in one dimension, wherenothing there is is gained only by going one from periodicspectral the orbit determinant trace fo formula is to a the real spect functionBohr-Sommerfeld for quantized real eigenenergies energies, (38.15). a 38.4 Two-dof systems For flows in two configuration dimensions the monodromy matri eigenvalues be elliptic or hyperbolic.eigenvalues are real Here and we their absolute discuss value only is not the equal hyperbol traceSemicl to - 2mar2004 This expression can be simplified by using the relation (37.1 odromy matrix part and using (38.16) where m S where the energies the function CHAPTER Sommerfeld quantized energies E chapter 23 he 701 (38.13) (38.14) (38.12) ). There E ( p iodic orbits, T . , the topological he 1-dof form of p         is only the parallel ase. Let us check or spectra of chap- 2 . S of a periodic orbit, . The beauty of this / quantum mechanics 1 2) n 2) | ike train ) π/ p ChaosBook.org version15.9, Jun 24 2017 r 38.8) p π/ ) m M − E . l with period ) ~ ( . semiclassical zeta function − / p ) p ) kx n n S ( π E rm − ir − repetition sum in (38.13) can be ! e − 4 ) det (1 | ′ r / E ~ ( ) r 1 E / / ( ) E 2 cos(2 1 ( E = p ∞ 1 ( reg r X = p ∞ m k G X p − tr rS . We obtain the + X determinant – is intrinsic, coordinate-choice ′ ~ 1 p − . π dE         p ) cos( dE 2 M = / E ) E ) ( kx E exp 0 E ~ π ( ) Z ( d 2 π p i p E ( e − T = S

sc ( ) N δ r −∞ π ∞ E i = X X ( e exp k n sc + X A sketch of how spectral determinants = = = , and the relation (35.19) between the density of states and t ) ) ) dN ~ ~ sc sc E E n ) ) ( ( π π dE p p ) 2 2 38.3: E E − T T E x ( − − (

38. SEMICLASSICAL QUANTIZATION p = = δ ˆ ˆ H H T ) ) and monodromy matrix type singularities at thetral eigenenergies determinant while goes the smoothly spec- through zeroes. Figure convert poles into zeros: The trace shows 1 E E −∞ = p ∞ ( ( = X ) det ( det ( d n d m E ( In one dimension the average density of states follows from t p We already know from the study of classical evolution operat traceSemicl - 2mar2004 spectral staircase, The classical particle oscillates in a single potential wel Now we can useusing (38.11) the relation and (37.17) integratedS between the the terms action coming and the from period per is no monodromy matrix to evaluate,coordinate, as in and one no dimension transverse there directions. The 38.3 One-dof systems It has been awhether long the trek, result makes a sense stationaryin even phase one in spatial the upon dimension. simplest stationary case, ph for the oscillating density (38.10) and of the average density ( independent property of the cycle index ter 22 that this can beformula evaluated is by that means everything of on cycle the expansions right side – the cycle actio rewritten by using the Fourier series expansion of a delta sp CHAPTER We obtain - 704 quan- own as the The derivation al determinants e), the formulas tions have good ace formula that ortant conceptual holm determinants, appellations such as ased on cycle expan- s is the case for inte- derived by the Gutzwiller less transparent cycle ical and quantum me- matically improved? ons to the exact quan- eriodic orbits. ical estimates of ChaosBook.org version15.9, Jun 24 2017 ion of physical interest. classical formulas; how (38.18)       ellent exposition [A1.13] by student. rivation presented here is self 2 / p k p 2 m Λ / π 2 p i / − m 1 | ~ π i / p p − p Λ | iS S e i ~ e − 2 / 1 1       | 1 p 0 = ∞ Λ k | Y = p For “zeta function” nomenclature, see remark 22.4 on Y p t ) E ( , sc ) N p are the natural objects for spectral calculations, with con t π i e − = (1 sc ) p Y E Gutzwiller quantization of classically chaotic systems. Zeta functions. = − Ruelle or dynamical zeta functions

38. SEMICLASSICAL QUANTIZATION ) ˆ s H ( or multifractal formalism; in quantum mechanics they are kn 38.1 38.2 /ζ α det ( 1 − resonances. For hyperbolic systems the dynamical zeta func f Spectral determinants, Selberg-type zeta functions, Fred Zeros of As we have assumed repeatedly during the derivation of the tr For deterministic dynamical flows and number theory, spectr expansions. The 2-dof semiclassical spectral determinant vergence better than for dynamical zeta functions, but with tum yield, in combination with cycle expansions, the semiclass convergence and are a useful tool for determination of class chanical averages. functional determinants eigenvalue estimates, as they tend to beIn divergent in classical the dynamics reg trace formulasthe hide under a variety of Gutzwiller trace formulas. Commentary Remark is a typical example.sions Most of periodic such orbit determinants. calculations are b the periodic orbits are isolated,grable systems and or do in not KAM formdiscussed tori families so of far (a systems are with valid mixed only phase for spac hyperbolic and elliptic p and zeta functions are exact. The quantum-mechanicalapproach, ones, are at best onlytum the spectral stationary phase determinants, approximati andproblem for arises quantum already at mechanics the level an ofaccurate derivation imp are of the they, semi and can the periodic orbit theory be syste page 407. traceSemicl - 2mar2004 CHAPTER given here and in sects. 37.3Martin and Gutzwiller, 38.1 the follows inventor closely of the the exc contained, trace but formula. refs. The [34.3, de 38.1] might also be of help to the Remark - 703 (38.18) ] on the very ? 38.9) the spectral expanding eigen- tionary phase ap- classical periodic         an be described by ent, practice shows mula, possibly im- ng external fields. 2) est to derive, in cal- oth in classical and c orbit formulas are p kr π/ mation is made, what ies of semi-conductors, iments [ p other than the leading Λ ChaosBook.org version15.9, Jun 24 2017 ional vector spaces. In m 2 assical quantities. The / − raction corrections, etc.) . 1 ~ | ff / p r        p p p S k Λ ( | m r Λ ir π 2 i 2 e / − 1 0 p | = ∞ p S k X i ~ Λ , 1 e | = ∞ r X − p r . 1 Λ p        p / kr X 0 1 Λ − = ∞ 2 k         − Y / ) 1 1 | 1 E p p r , exp Y 2 q Λ / | , E E 2 2 1 | ~ ~ q A A 0 2 2 p ( r m m = ∞ i i k sc X Λ e e | = = = = 2 2 dqG / / in the sense that all quantities used in periodic orbit calcu The trace of the semiclassical Green’s function 1 1 | | ) ) Z sc p r p r ) = M E M ) 1 − − − E 38. SEMICLASSICAL QUANTIZATION ( ˆ classical ´ H e sc G det (1 det (1 | tr det ( | Trace formulas. While various periodic orbit formulas are formally equival With the 2-dof expression for the average density of states ( In practice, most “quantum chaos” calculations take the sta ´ esum culations the trace formulas are inconvenient for anything traceSemicl - 2mar2004 is given by a sum over the periodic orbits (38.11). While easi that some are preferablefrequently to used: others. Three classes of periodi lations - actions, stabilities,problem is geometrical then phases to -orbit understand formulas. are and cl control the convergence of as the point of departure.follows Once is the stationary phase approxi determinant becomes paradigm of Bohr’s atom, the hydrogen atom, this time in stro proximation to quantum mechanicsproved (the by Gutzwiller including trace tunneling for periodic trajectories, di R Spectral determinants and dynamical zeta functions arise b quantum mechanics because in boththe the dynamical action evolution of c linear evolutionquantum operators mechanics the on periodic infinite-dimens orbit theory aroseand from the stud unstable periodic orbits have been measured in exper and its inverse can be expanded as a geometric series CHAPTER appearing in the tracevalue formulas as can be written in terms of the 706 , 3525 Lecture erential (Oxford , ff 9608012. / A 36 ctional meth- stics - chapter Periodic Orbit Scattering The- , tive methods and 2 Selberg zeta func- ids and classical de- ttay, Phys. Rev. ChaosBook.org version15.9, Jun 24 2017 Classical, Semiclassical , 1578 (1993). , 61 (1992). 2 , 4839 (1990). E 47 A 23 , 53 (1992). (Addison-Wesley, Reading, 1980); Ergodic Theory Dynamical Systems 2 CHAOS (Addison Wesley, New York 1990). , 805 (1992). (Penguin, London 1937). 5 J. Phys. , 5 (1992). CHAOS 2 , 1237 (1992). 5 , 105, (1992). (Springer, New York 1997), arXiv:chao-dyn Nonlinearity , 4114 (1974). Statistical Physics , CHAOS (1993). Classical Mechanics Statistical Mechanics, Thermodynamical Formalism 485 Time evolution and correlations in chaotic dynamical sys- D10 Geometrical Methods in the Theory of Ordinary Di Mathematical Thought from Ancient to Modern Times Men of Mathematics , (Springer, New York 1983). Phys. Rev. . PhD thesis (Niels Bohr Institute, 1992). , 805 (1996). terminants,” in H. Friedrich and B.and Eckhardt., Quantum eds., Dynamics in Atoms – in Memory of Dieter Wintgen Notes in Physics 22.7. Univ. 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Sepúlveda CHAOS References [38.30] R. Dashen, B. Hasslacher and A. Neveu , “Nonperturba (38.20) Show that . E is the action of the orbit. < ) dq q ) ( , E ) , U , 2 E ) q ( ~ ~ ( E π π ( p A 2 l 2 S m H 705 = = = ) is the classically allowed area of configura- ) ) is the time period of the 1-dimensional mo- ) ) , 1791 E E E E E ( ( ( ( ( ¯ 11 ¯ N d T S A where tion and show that where tion space for which Average density of states in 2 dimensions. in 2 dimensions the average density(38.9) of states is given by where onio, S. Fantoni and 38.4. orbit approximation in ChaosBook.org version15.9, Jun 24 2017 , 1004 (1969); formulas,” in G. Casati . 10 2 / d equals (Pergamon, London, 1959). (38.19) 2) r / Show that d ( Show that the = Γ , (Cambridge Univ. Press, Cambridge , 77 (1992). , 3154 (1984). 2 , 343 (1971) , 17 (1994). 2) , 1979 (1967); 92 / 12 , 7 (1992). Mechanics 8 d 116 68 + (1 CHAOS Γ , 685 (1988). , pp. 57-64 (Ed. 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