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1 was This coherent approach semiclassical p opularized as principle of correspondence It seems quite obvious that the main 3 For a detailed discussion of Schrodingers one studies the b ehaviour of quantum me matrix as Plancks constant tends to zero S In the long run Bohr and Heisenberg seemed to have won It was Fritz ReichePlancks assistant at the University of where he discovered what was later semiclassical limit In the Lenz from Octob er to August see ref p on Jan pap er croigratdtoSchrodingerwanted illustrate by the example of Plancks linear oscil in Hamburg was able to calculate rigorously the energy levels of the his the fties states lator that it is alwaysthe p ossible form to of nd solutions welllocalized of wavechange his packets of whose undulatory mechanics shap center e in of with gravitydescrib es the oscillates the without p erio d classical of tra jectory theorbit of corresp onding a of classical p oint the particle motion electron like and in for thus the example Hatom the Kepler pap er and the roleit playedple in see Heisenbergs discovery On of OctobBohr er the had uncertainty a invited Schrodingergave princi talk him inb etween together Cop enhagen Bohr with where and Heisenberg b een Schrodingerhave vividlyautobiography The describ ed almost by fanatic Heisenberg discussions in his the battle and there istum no mechanics doubt is that one theconsidered of Cop enhagen out the interpretation of main of date reasons quan whyShortly Einsteins after the torus discovery quantization of was the a Schrodingerequation was devised known asBrioullin the WKBmetho d named after Wentzel Kramers and Quantum Chaos talk which was published withoutno delay by inuence the at German all PhysicalIn So ciety on Bohrs had the famous development of pap er physics during in the which next the fourty years exp osed one nds no reference tofor Einsteins integrable talk systems was Einsteins rediscovered torus byin quantization the mathematician Joseph Keller only Berlin from toOne who drew can Kellers only attention sp eculatecal to why Einsteins phase Einsteins talk space deep andhas insight his into b een the recognition ignored structure ofreason for of the lies such classi latters in a the imp ortancewith long development for of Heisenbergs time quantization quantum matrix mechanics mechanicstion a in few in summer years spring later spring starting Schrodingerswave equa and Heisenbergs Already derivation inPauli of the b efore uncertainty the principlehydrogen discovery atom in of from the Heisenbergs Schrodingerequation a quantum great mechanics success which and was decisive testburg considered of was as the wissenschaftlicher Hilfsarb eiter new in theoryf theoretischer urTheoretische Paulis Physik Physik rst at the p osition the rst Institut in holder Ham Wilhelm of the chairuary for theoretical physics Pauli was receivedmathematician the Erich venia Hecke legendi Whereas Physik f urtheoretische from Heisenbergclassical had the completely orbits eliminated from the his theory very Schrodingerwas mechanics much inuenced and by the classical analogy b etweenand the named wellknown after mechanical Hamilton principle and dueIn the to wellknown optical principle of Fermat chanical quantities like energyabilities levels decay wave rates functions or barrier the p enetration prob l ie The Ergodicity dynamical Frank Steiner all in involution conditions such that Ub ergange von einem Der Quantensatz soll almost in phase space initial motion which is unpredictable that is See Arnold and Lichtenberg Today our knowledge of classical on the It is now commonly recognized that Trakte gespalten zu denken die langs chaotic 2 in the sense that even the longtime b e Indeed in integrable systems tra jectories with eindeutige und auch b eim are not typical regular i p in the sense that there exist no constants of motion dimensional tori which in turn causes the motion of l in complex systems lahn uamnnederart da zusammenhangen in demFlachen so entstehen rationellen Phasenraum b ezeichnen under control chaos are assumed to b e smo oth enough and to b e nonintegrable k well R Here and in the following we are only considering Hamiltonian sys the tra jectories dep end sensitively dimensional energysurface densely clo ckwork predictability dimensionaler l l und es versagtder die hier SOMMERFELDEPSTEINsche gegeb enen Quantenbedingung etwas auch erweitertenAt Form the in time when Einsteinphysicist gave He his talk was he athe probably professor was obligation at the the to most University famous teachWissenschaften of living and a Berlin shortly member afterwards with Octob er of thethe the right newly founded Preuische but he Koniglich institute Forschung of not Akademie b ecame f urphysikalische the director der Gesellschaft KaiserWilhelm zur of Wissenschaften Forderungder to day the institute isPlanckInstitut called f urPhysik It Max is thus a remarkable historical fact that Einsteins neighbouring tra jectories in phase spaceEinstein separate at made an the crucial exp onentiallation remark rate of that the the quantum absence conditionsto of the tori and situation furthermore excludes encountered that the in this formu classicalthe applies statistical motion precisely mechanics of where colliding one atoms describdie es mikrokanonische or Gesamtheit der molecules auf in e i aheit n In gas System aquivalent his sich denn Nachtrag b eziehenden nur zur Zeitgesamt in Korrektur to diesem he Poincarein Falle connection referred ist with as the an example threeb o dy problem and he concluded that is their Poisson brackets withand each Lieb erman other vanish forsystems further wind details round As these a result the tra jectories of integrable constants integrable systems to b e very tems that is motion governed by Newtons equation without dissipation systems haviour is neighbouring initial conditions separateEinstein only was as the some rst p ower physicistvariant of who tori time in realized the phase imp ortant spaceden by roleplayed which Phasenraum the he jeweilen in called in Trakte eine He Anzahl said Man hat sich den Gebilde interpretiert die generic systems execute a very irregular Trakt zum anderen stetigetion Funktionen wollen sind wir diese als geometrische Hilfskonstruk sich auf alle Liniensind b eziehen die imHowever rationellen the Ko ordinatenraume integrable geschlossene systemstheir forming the standard textb o oksystems systems with are b esides the energyimplies and that almost therefore all no tra jectories llin invariant the tori absence of invariant torithe whole dynamics is very richimp ortance of and most natural scientists b egin to appreciate the by his Ev The are in and k r illustrated ergodic systems molecule as the dimensions how can classical l is only metastable treated by Pauli was and of course dimensional tori and that each l r and the ground state energy whether such a n r

k ion which 5 n k n and There is however Einsteins second imp or H r n k n I E The question was of a tra jectoryis determined by the top ology of The whole theory is based on Einsteins observation that the exists which was not yet known at that time Pauli calculated are integer quantum numbers and the integers Morse index k n l k the Lagrangian manifold infollows phase from space with that the resp ectis semiclassical to explicitly approximation conguration to given space the by quantal It energy levels the Maslov indices Theand motion takes the place Maslov on ap oints indexwhich socalled or can Lagrangian the manifold b e understo o d as the number of conjugate where the PhD thesis in Quantum Chaos singly ionized H the ground state energy byof quantizing the the molecule radial For motion the with ground state resp ect he to chose the axis the example of the hydrogen molecule obtained a p ositive energy and thus he concluded that H The correct values turn out to b e The EBKformula gives the exact leading asymptotic term as Naturally Einstein could not knowtheory ab out was the not Morse yet index developedfor theorem since the Morse That physical the Maslov prop erties indices of are very atomic imp ortant and molecular systems is Notice that Einsteins conditionwhich is in is general only not the correct case if Eg all for the Maslov harmonic indices oscillator one vanish has is negative For further detailsSo see far we have seenand how rened Einsteins quantum leading conditiontegrable to systems the has b een general rediscovered phase EBKquantization space rules of integrable systems and is foliated into for in orbit moves on antant invariant observation which torus he mentioned in his talk namely that p ossess no invariant toriapplied and that In his fact it quantizationcarries metho is d two known can mutually therefore that not transverse the foliations b e phase each space with of strongly leaves chaotic of systems distance b etween two neighbouring tra jectoriesunstable increases manifold exp onentially along and the decreases exp onentiallyously along the the EBKconstruction stable based on manifoldand actionangle there Obvi variables remained is the no dicultfor task more chaotic to p ossible systems nd It aup to ok semiclassical the another quantization royal metho decade d road untilmechanics towards Martin give an Gutzwiller us answer any to op ened the hints Einsteins classical ab out system question is the ergo diclem quantummechanical energy of Einsteins levels as questionor when Einsteins Iof prob have our phrased mo dern it studies in of the quantum title chaos in of this complex sectionis systems the Before starting I p oint shall come ery tra jectory is the intersection of two manifolds one from each foliation s If k I for reads in the to new k p L which has system with l is a function quantization go es to zero are identied Frank Steiner which is equal vary from to H k I l k E this dimensional vec l as w l constants of motion l integrable EBKquantization th irreducible circuit of actions Today k angles is precisely equal to zero the The is the denes a closed lo op k b ehaviour of quantum mechanics Then each orbit of the dynamical indices EBKquantization condition These problems can b e overcome L In the following it l It is now p ossible to make a canonical E and their conjugate momenta 4 Maslov q Then the where p q q p p from p I p H H actionangle variables runs from to p systems he was able to give the most general semiclassical H k w dimensional torus l If only called k w I integrable I corrections arising Since the integrals are invariant as noticed by Einstein the is dierent from the classical limit for which and are interpreted as new co ordinates while degrees of freedom is characterized by the existence of with Einsteins lo opjugate variables momenta dened in eq and play the roleof new con are the new constants of motion Moreover the new Hamiltonian transformation from the co ordinates co ordinates original phase space variables the torus the top ology of an tors will b e denoted by l system lies on a submanifold in phase space of dimension Einstein Brioullin and Keller Before we shall havemarize a the closer mo dern version lo ok ofesp ecially on Einsteins on chaos torus the and quantization construction quantum Forcent of further textb chaos o ok details by semiclassical I Gutzwiller briey wave functions sum As I discussed b efore refer an to the re to the classical Hamiltonian in involution where one constant of motion is the total energy condition for integrable systems go es under the name of examples of such aApplication b ehaviour of can the WKBmetho d b e isble found straightforward systems in in if the eqs one caseis ignores of some carried simple subtleties separa out and which if in typically b elow one noncartesian arise starts co ordinates if from theone Feynmans separation tries path however to integral apply treatedencounters the serious in WKBmetho d diculties a to which more consistent were complicated notthe way systems solved discovery of during one the the Schrodingerequation Of rst course decadesthese after many problems p eople did at not all realize systems since b eing they not were awaresystems content of This with the is a imp ortance reectedquantum treatment in of mechanics of the usually more the fact do complex that simplest notapp earance even let mention of mo dern the alone ever textb o oks phenomenon more chaotic on of p owerfulall classical chaos computers or problems Also has can the led b e totherefore solved a no numerically widespread more b and elief worthwhile to painful that As b e analytical already pursued mentioned investigations it are wasmathematical Keller derivation who of discovered inrequires the the a semiclassical fties detailed that knowledge aIn of sound the case the of underlying classical phase space structure b ecause in general quantal functions are nonanalytic in quantization rule which turnedapart out from to b e exactly Einsteins torus quantization limit of the actions is j dq ij even i g which dq Thus he that the ij g is a length in time at a R R ds It follows that the unpredictable F Here Sinai bil liard diverge denotes time In ergo dic theory of a dynamical system R K F F t It app ears that Hadamard q even b een claimed i j j It has geodesic ow on p q 7 Lyapunov exponent ij longtime behaviour g i p deterministic chaos on the given surface is the m which is dened by the line element q ij many directions g and that neighbouring tra jectories are the conjugate momenta E mR p geodesics p H dt j Today this sensitivity on initial data is recognized as the most dq unstable ij versions of Sinais billiard have b een fabricated at semiconductor mg sp eculations in where i mesoscopic systems p t e wild should b e considered asToday the the true discoverer sub ject of of chaos extent chaos by is the very widespread p opularfor unqualied which use is of certainly the caused colourful to word a chaos inviting great discovery of chaos theory constitutestwentieth the century the third rst great two revolutiontheory b eing in Statements the physics of invention this in of sort the While are relativity absurd and it and of as is quantum we truewe have seen that have historically most seen false physicists thatcentury did One Hadamard not may op ened take sp eculate notice thethe what of word do ors would chaos chaos have already already until happ ened in atchaos recently if theory to was the Hadamard for illustrate had end many his decades used almost ofcarrying ndings exclusively the It the a less domain is last of fancy a mathematics The name mere reader of fact may that wonder ergo dic whether theory Hadamards pap eras Einsteins exp erienced the talk same fate Duhem This realized is the however philosophical not implicationsseries the of of case Hadamards articles The discovery and French which physicist indynamical app eared Pierre a system in with p and o etic as inspiration he a In describ ed separate Hadamards b o ok Duhem published whose his German articles translation app eared already in the top ology of a doublesmo oth torus Hamiltonian Hadamards which dynamical is system given is by describ ed by a Quantum Chaos scale which is xed by the constant negative curvature classical orbits are the Hadamards dynamical system is called the rate At rst sight Hadamards system app earsabstract to b to e a b e purely relevant mathematical for mo delthe physics to o BoltzmannGibbs However gas Sinai into a translated study the of problem the of bynow famous though the system isas in governed by his a mo del deterministic law like Newtons equations can b e very sensitive to the initial conditions and therefore striking prop erty of systems with Hadamards great achievement washis that system he are could prove that all tra jectories in in turn he could relateexperimental precisely to Hadamards mo delinterfaces of as arrays Recently smo oth ofeld nanometer of p otential wells and have op ened the new was the rst who could show that the Here Einsteins summation convention is used where the inverse of the metric neg Then the one constant Before we on Although the Frank Steiner In the hands of at the b oundary GauBonnet theo and Gibbs with b oundary g surface and moreover that R at q q A Boltzmann stadium bil liard One then obtains compact Riemann m and on the other hand shows the typi 6 which moves freely ie without external F p His system hasm two degrees of freedom and if one imagines hard walls q For an early account see Having in mind our discussions of quantum chaos in p with area It turns out however that they are extremely complex physicists Maxwell H Eg if the b oundary is a circle an ellipse or a square g the particle moves freely on a frictionless surface having g curvature planar bil liard It turns out that the billiard dynamics dep ends very sensitively of genus F Eg for it led nally to the rst pro of given in that the Boltzmann Gaussian Ergo dicity and Chaos mo dern ergo dic theory rem the next sections itnegative is curvature useful and to without sp ecializesurfaces b oundary already here to surfaces with to a discussion of quantumthe chaos development I of want our to ideas recall on briey a ergo dicity few and historical classical facts chaos on We owe the early recognitionteenth of century chaotic motion to in the nature at the end of the nine hand and toWhereas the Boltzmann mathematicians put Poincareand forward Hadamardthe the on BoltzmannGibbs ergo mo dic del the of hypothesis other aand in hand gas statistical as mechanics a leading prototype Poincarewasproblem example mainly to of of celestial concerned thermo dynamics mechanics withtem with the sp ecial threeb o dy Clearly attention the to dynamicalof the systems earthsunmoon utmost studied sys imp by ortance Boltzmannand and even Poincareare to day many of theirof prop erties are ergo dic not theory understo o d and For mo theHadamard dern development chaos theory it introduced already wasenough therefore in very to imp ortant b e that acal treated dynamical b ehaviour mathematically system of which irregular isconsists motion of simple a p oint particle of mass Sinai Gibbs mo del of aInstead gas of is choosing ergo dic aative at surface Hadamard considered a surface with one obtains a The tra jectories of thereections particle at consist of segments of straight lines with elastic the system isa integrable strongly while chaotic a system b oundary the of wellknown the shap e of a stadium leads to the motion is conned to a compact domain on the b oundary discuss Hadamards example it isrst worthwhile to consider another class ofLet systems us assume that the p oint particle moves on a the study of the classicalof billiard ball problem was inuential for the development forces on a given twodimensionaltwo surface but The two dimension could is b e the higher smallest than dimension for which chaos can o ccur Hamiltonian of such a planar billiard is not smo oth but rather discontinuous As group These general to show from is symbolic dynamics enabled him of the mo dular H dened on the halved do This For a recent account see is a noncompact triangle of nite This region is symmetric under re Gutzwil ler trace formula F can b e dened It to ok more than fty as a billiard 9 x k It can b e shown quite generally that the which represent in the hierarchy of chaotic L There are no invariant tori in phase space path integrals systems since already the basic denition x is quasiergo dic In fact Artins billiard b elongs to it was recently found that the formula can b e chaotic F can b e viewed It is the purp ose of this section to describ e the terms of continued fractions Anosov systems j Let us cite Artin Damit hab en wir ab er die physi strongly chaotic z j j z which can represented by the mo dular domain ie by the fun j which forms a mathematically sound basis for the semiclassical systems Although the original convergent systems whose most general form is given by the EBKquantization H region n as a map in F Anosov systems are ergo dic and p ossess the mixingprop erty chaotic The General Semiclassical Trace Formula trace formula quantization of chaos The general framework isterms of Feynmans his formulation sum of over histories quantum or mechanics in damental surface Quantum Chaos the class of socalled plagued with serious divergencies and thusand cannot b e numerical applied instabilities without ambiguities improved and brought intoabsolutely a general form such that all series and integrals are systems the highest levelAll revealing the most sto chastic b ehaviour ever p ossible that the geo desic ow on and thus no irreducible closed lo ops is meaningless for ergo dic systems systems are called geo desic ows on compact symmetric RiemannThere spaces is are Anosov no systems doubtthe that Artins hydrogen work atom which onresults was ergo dicity in and already science Paulis achieved mentioned solution atfew in the years sect of University of Hamburg b elong already to during the the rst top In sect I haveintegrable discussed in detail Einsteinsconditions semiclassical torus quantization and for thep ointed EBKapproximation out already by to Einstein the incompletely quantal in energy these semiclassical levels the quantization rules case fail of hyperb olic area kalische Realisierung Man auf zeichne der der zunachst Traktrix ein Rotationsache mit dem halb enstem Mo duldreieck latsich kongruentes dann Dreieck als Unser dieDreieck mechanisches kraftefreieBewegung eines interpretieren Sy der Massenpunktes in Punkt sei diesem von den gezwungen auf Dreiecksseiten elastisch der zu reektiert Flache In bleib en wird his der pap er the Artin introduced theory for of theUsing rst dynamical an time systems idea an which which imp ortant go esmotion to approach day back into to is Gau known Artin was as able to formulate the geo desic years until Martin Gutzwiller for did the rst step towards a semiclassical theory with appropriate b oundary identications ection on themetrized imaginary axis system and which thus Artin was led to consider the desym main Z the H y ij z The sys ij g Frank Steiner motion on Verlangernwir Die einen win b d where the length Artin who studied c a y Duhem refers also to Auf einer solchen Flache with the top ology of a Werden die Gestirne des q An innitely long horn as x studied rst the K the mo dular group PSL q Artins billiard is a twodimensional H Artin 8 for having shot and killed Graf St urgkh On Denken wir uns die Stirn eines Stieres mit Man kann die Genauigkeit mit der die prak pap er Duhem describ es Hadamards mo del ie the The b o ok was translated by Friedrich Adler a endowed with the hyperb olic metric after the mathematician Emil g In his y The surface can b e realized on the erhalf Poincareupp j d as follows iy F cz x Artins bil liard z b f has b een put equal to one az H R den von Erhohungen denen die Hornerund Ohren ausgehen Linienkonnendie geo datischen recht verschieden aussehen diese Hornerund Ohren inhab en der wir Art eine da wie Flache sie wir sich sie ins studieren Unendliche wollen ausdehnen so geo desic ow on with a foreword byclose Ernst friend Mach of Einsteinshis When family Einstein in moved to in the Z same urich Adler house he as lived and Adler with Einstein ItHadamards would talked pap er b e ab out interesting contains to a Duhems nd lotso out b of o ok whether Riemannian imp ortant and geometry in which Einsteins Hadamards b ecame later workhis mo del on pacist general since relativity friend Inthe Adler Einstein prime then supp orted minister in of jail Austria den sich um unaufhorlich dasum rechte Horn das rechte die resp linke anderen Ohr um das linke o der auch z op erates via fractional linear transformations ie by plane sphere containing an op endescrib ed end by cusp Duhem at innity Poincareand the threeb o dy problemSonnensystems and unter der asks Annahme da dieb en Lagen und die Geschwindigkeiten dersel gleichen seiendrehen wie Wird heute alle es nicht weiter imdem und sich Gegenteil Schwarm unaufhorlich um seiner geschehen um da Gefahrtentrennt die sich eines in Sonne dieserDiese der Gestirne Frage Unendlichkeit sich bildet zu von verlieren das ProblemLaplace der gelostzu hab en S t glaubte der a mo dessen b dernenBem uhungen i Mathematiker auerordentliche l vor Schwierigkeit allem i ab er t ab er d at die die edartun s des Herrn S Poincare o n n eAt n s the y end s t of estrongly m this s chaotic section das system I whose should like Hamiltonian to has describ e exactly briey the another form example of a tischen Angab en b estimmt sind b eliebigAnfangslage kann erhohenman den des Flecken materiellen der die Punktesdas bildet die verkleinern Richtung man der kann man wird Anfangsgeschwindigkeit das enthaltzusammenschnuren do ch B undel niemals dieHorn Linie dreht geo datische von die ihren sich ungetreuenerst Kameraden ohne eb b enso efreien Unterla wie die erstere um nachdem um sieThis das dasselb e sich rechte is Horn zu gewunden a ins Unendliche b eautiful entfernen illustration of chaos already in tem is called this mo del here innonEuclidean billiard Hamburg whose in billiard ballface table of is a constant noncompact negative Riemannian sur Gaussian curvature scale The co ordinates of the p oint particle are a in b e q for are k l of the M univer p erio dic dep ends n q Let The mE length I am quite sure However since strongly chaotic have lengths quantum chaos E c is even since all orbits are and on the b oundary } osc Moreover all g analogue experiments for the trace of the geometrical k E l p M E M ik dep endence A small p ositive e Tr g electrodynamics Tr k in analogy to 0) traversals of E formula accelerator physics ! of l } ( denotes the sp eed of light In typical denotes the and c E X k trace whose length tends to zero if l where 11 Multiple n X q E wave chaos E The p erio dic orbits are characterized by their to Here to b e discussed in sect can then b e directly i p n where X n q This implies that all Lyapunov exp onents g universal signatures of wave chaos a socalled Ksystem l Gutzwil lers f equations also the TMmo des of a at microwave res of the ppo E E and is assuming that the Maslov index of Then osc classical wave properties electrical engineering g g counts the number of rep etitions of the ppo b hyperbolic H spectrum ie the trace of the Greens function reads Tr f b H denotes the socalled zero length contribution which comes from p erio dic orbit ppo E g k ergo dic mixing monodromy matrix resolvent of is identied with the frequency on the fo cusing of the tra jectories close to the ppo orbits are unstable andprimitive isolated length conditions on imaginary part has to b e added to the energy primitive Natural units are used but keeping explicitly the ie Quantum Chaos the circular membrane by Clebsch inhighly nontrivial However in the problem casesIn turns where fact out to the no b e explicit classicalin formula b ouncing the is ball chaotic known case problem forRecently is the it chaotic energy has levels b eenaccording or realized the to wave Maxwells functions thatonator whose the base Schrodingerequation has the describ es shap e of a billiard domain if For the following discussionchosen I in shall such a assume way that that the the billiard corresp onding domain classical system is has b een the strictly p ositive For details see Moreover let us attach to each ppo where exp eriments carried out so farup this to identication holds GHz for These the exp eriments lowest can frequencies b e considered as case the base ofsal the signatures resonator of has quantumtranslated the into chaos corresp onding shap e ofthat a the chaotic prop erties of billiard wave chaosnear will future nd eg imp ortant in practical applications in the character where direct tra jectories going from direct or inverse The contribution from the p erio dic orbits is given by the formal sum the two dierent physicalseems situations appropriate are to describ ed sp eak by ab the out same mathematics it quantum chaos although they haverather nothing deal to with do the with quantum mechanics but n are q where we get denotes n The sum b mR H R Taking the there exists Notice that Frank Steiner q E should vanish q m Here b oundary n n orbits vibrating membrane E E R limit periodic orbits q q and that the semiclassical limit m E where in the form describing a at the billiard ie all m q R n macroscopic will b e reinserted in order to identify Dirichlet Laplacian of the quantum billiard n For the quantal Hamiltonian mn and E can b e chosen real Moreover it follows This implies the 10 m for the given quantum billiard is equivalent q contributions resp ectively in semiclassical space Gutzwiller made the imp ortant of the to n q In the following I shall use natural units phase space b q q H and thus requires a study of the highly excited d tends to zero it is wellknown that the leading n n in q The hard walls n Helmholtz equation are strictly p ositive identical m coordinate n closed q g E is in n m Indeed several membrane problems corresp onding to the E in general not even conditionally convergent for physical nonperturbative f high energy behaviour Z limit Laplacian closed Schrodingerequation eigenvalue problem and prop erties of this eigenvalue problem are standard Occasionally however dimensionless and indep endent of R billiard case have already b een solved in the last century the rectangu is n m semiclassical Then the perturbative the Euclidean the the the mass of thewith atomic a b ouncing macroscopic ball p oint particle is b ecoming so heavy that one is dealing is incorp orated by demanding that the quantal wave functions from the classical Hamiltonian In the semiclassical limit when corresp onds to the limit from that the eigenvalues scale in an arbitrary but xed length scale states ie of the only a discretewhose sp ectrum energy corresp onding levels to an innite number of b ound states expressions The mathematical problem dened in eqseigenvalue problem and of the is rather old b eing the where The corresp onding wave functions has only a formal meaning bgrowth ecause there in are innitely number many p erio as dicand orbits function whose thus of the theenergies sum p erio d is is exp onentialAs for an chaotic illustration systems ofplanar the billiards semiclassical theory as for introduced chaotic in systems sect I shall consider at with clamp ed edges integrable lar membrane by Poisson in the equilateral triangle by Lamein and The following to the following observation that the trace of theFourier energydependent transform Greens of function the which timeevolution is op erator is the classical given orbits by a which formal are sum over all trace of the timeevolution oporbits erator the which contributions are come from those classical contribution to the path integral comes from the classical p h non Since a constant l where C conditions is given by k l b e any function e b C H k k l and h g as instead of the trace of p k For planar billiards we sucient b H the expansion p l l p h b O H p k h e Thus I call this term the h n C The Weyl asymptotics has b een p ipx X k a contribution since it is determined is the sign of the trace of the mon n p parameterize the energy levels in the p dp e X h Lp j Z 13 p at p p Im has b een replaced by Tr p j p h denotes Weyls improved asymptotic formula p dp p h perturbative contribution to the trace formula Notice that a to c are dh We are now able to formulate the Ap N E x p N g j p p p b j N H dp which is dened as the p erturbative contribution to the N absolutely p Z perturbative and p N denotes the total length of the b oundary Moreover M 0) n j O ! E } spectral function contribution to the trace formula It remains to discuss the integral ( p j is an even function is analytic in the strip n p p Weyls improved asymptotic formula p p in general nonanalytic in p h h h L n h p h c a b X n It is obvious thattion the general p erio dicorbit sum in eq is for a given func perturbative term in eq that I call the where by the function o dromy matrix Then the leading asymptotic form of the trace of have the following Laurent expansion in denotes the Fourier transform of as will b e dened bin elow eq Under the conditions converge a to c all series and the integral which is determined bycorners the if curvature of the the b oundary b oundary consists and of by a the angles nite at number the of smo oth segments the rst term in eq is equivalent to Weyls law is called discussed by many authorsentitled in Can particular one by hear Marc thethe shap Kac e expansion of in a his drum intocorresp onds famous For the to pap a er the integral review term see of Inserting eq it is clear that this term eigenvalue counting function which satises the following three conditions where form for the class of planar billiards introduced b efore Let General Trace Formula In ref generalized p erio dicorbitthe sum trace rules of a have b een rather general derived function by of considering the Hamiltonian Quantum Chaos the resolvent that is Tr is a suitable l with a Weyls from Frank Steiner barrier is an arbitrary but In order to derive a Since the divergence regularized resolvent directly E entropy E p plane It is not dicult to p l where follows in eq diverges In order to are smaller than or equal to p n l This E l turns out that the natural variable l A b It is then clear that the sum has e b H 12 n denotes a certain asymptotic average of n momentum l E l e l b whose lengths lim n E l e and the sum over l they are not just of a formal mathematical nature Here N b H X l The real problems with the original trace formula discussed here it not of trace class a in eq is in general divergent l is n as one stays b eyond the socalled is strictly p ositive which reads for twodimensional planar billiards N but rather the a p of ppo l E for E n N O j of would lead to an explicit semiclassical formula for the energy We therefore see that the p erio dicorbit expression has only a formula b is the most imp ortant global prop erty of a strongly chaotic system H j n E ie if Im plane as long A p topological entropy b area for example the trace of cure this problem one could simply consider the trace of a The rst problem with the trace formulaop erator comes form the fact that the resolvent xed subtraction p oint arise however from the p erio dicorbit sum Due to the exp onential increase asymptotic convergent trace formula we are thus ledthe regularized to trace of study the the resolvent analytic in continuation of the complex see that the regularized p erio dicorbithalf sum is absolutely convergent in the upp er It turns out that Hence serious convergence problems the Lyapunov exp onents dened by of the number the innite sum over problems are a consequence ofof the a exp onential law and thus of the existence is not the energy but rather a direct signatureentropy of classical chaos inwhich quantum mechanics expresses A the p fact ositive tially that the fast information ab outformal the meaning system is In order lostit to exp onen cast is the necessary semiclassical tosum approach replace into which the a is divergent sound absolutely sum theory Before convergent I by shall a comemention generalized to that p erio dicorbit a the discussion divergence of ofconvergence the is general trace reallylevels formula a I which fortunate in circumstance should b ecause general like a The would to rst b e completely imp ortant p wrong ointFor is the to decide planar on billiards the appropriate variable to work with } H H H The n H t n As a trace i can b e Selberg e classical H F g chaos in one has the same geo desic segments g Moreover for b ound For which is describ ed by is given by with the H H and the semiclassical g n phenomenon of of on The reader is referred to the recent for Hadamards mo del that is for the On the upp erhalf plane F Then the energy sp ectrum is discrete R In analogy to the construction of Artins 15 on the unit circle exact H Then the Schrodingerequation has the same z In the case of Artins billiard One has to imp ose p erio dic b oundary conditions y y ij and have seen that the the Selb erg trace formula x we With some mo dications one also obtains an exact Selb erg ij g E y In the rst case the sp ectrum is discrete whereas in the latter F and for all z E As an example I refer to the error term in the famous circle problem I cannot review these works here detailed studies of E Universal Signatures of Quantum Chaos sections unitary and thus its sp ectrum lies z dynamics is a genericof prop erty deterministic of chaos complex is systemsneighbouring the sensitive tra The jectories dep endence most in on striking phase initial prop erty space conditions separate such that at an exp onential rate rst place wheremechanics one seems should to seek b eIt for the turns a longtime out however b p ehaviour ossible that in the chaoticder analogy largetime control b ehaviour due limit to in to the in the quantum quantum classical fundamental mechanics fact case is that well the un timeevolution op erator result the longtime b ehaviourThere of arises a the strongly basic chaotic questionsical system whether this is chaos wellestablished unpredictable phenomenon manifests of itself clas which could in b e the called quantum quantumclassical chaos world dynamical By in system this an which wein mean is analogous the the phenomenon strongly following corresp onding chaotic quantal Given is a system there any which manifestation b etrays its chaotic character is hyperb olic metric The classical Hamiltonian of the geo desic ow on form as in eqsimply reads where is now the LaplaceBeltrami op erator which on Schrodingerequation but with Dirichletsp ectively or on Neumann b oundary conditions re case the sp ectrum ishyperb olic b oth geometry continuous and and discrete the SelbOur erg For a trace formula rst see introduction into chosen as a simply connected region whose b oundaries are formula carried outinto during the the complex prop last erties ofchaotic few quantum systems years whose have classical givenliterature limit us is many strongly insights In Quantum Chaos exact order is ansystems extremely dicult problem inwhich mathematics will even b for e integrable mentionedThe in trace remark formula iii atliards represents the It a end turns of out semiclassical however section that approximation it for is planar bil where is a discrete subgroup of PSL can b e realized as a fundamental domain the classical Hamiltonian since it is then identical with the famous geo desic ow on compact Riemann surfaces of genus trace formula trace formula for Artinsbilliard billiard discussed in section a compact Riemann surface can b e dened as of Fur L and p p l one infers of statistical t since for n Frank Steiner } t k of the classical p p iii It should b e 2 hightemperature e l g b etween the quan 2 k Then we obtain t l k n O f p E e t of eq are identical k O t enter only in the combi C This do es not happ en by 2 l l p C e k and the b oundary length e and partition function and t L satisfying the ab ove conditions A t duality relation p p p X k decreases as A h p h These results can also b e generalized L X p t The p erio dicorbit contribution in N is equivalent to the limit 14 In the recent literature the co ecients of } ii t t p p This do es not imply however that the error i The time n and the length sp ectrum E A of a vibrating membrane g p N one obtains the e n n C p using again natural units but keeping } tends to zero and one obtains the heat kernel asymp X n t t t At present these sum rules provide the only substitute n n L E E p f t e } This is actually the basis of the mo dern approach to deter e thermo dynamics X n t They are top ological invariants of where can b e a general Since the class of functions heat kernel is replaced by in A t Tr expansion of the trace of the heat kernel are sometimes called See Thus the semiclassical limit } See for more details 0) t ! t with the frequencies b } H t ( One observes that the co ecients n e E Tr t In fact it is exp ected that this error term increases but determination of its mechanics and thus thelimit semiclassical limit corresp onds to the p erio dic orbits The general trace formula establishes a striking tal energy sp ectrum vanishes exp onentially as a to c is ratherperiodicorbit large sum the rules general traceappropriate formula for represents an quantum systems innite whose numberEBKquantization classical of rules limit applicable is to stronglyAs integrable chaotic for systems an the example let us consider the function totics thermore if nation Here a few remarks are in order as the trace of the ley co ecients manifold the small mine the terms in the Weyl formula chance but reects a deepWeyl relation asymptotics b etween the heat kernel asymptotics and the to the co ecients in Weyls asymptotic formula to the case ofgies unbounded hornshap ed billiards If one identies the ener from that a p erfect ear can hear the area the membrane that is one can hear the shap e of a drum noticed that the asymptotic expansionsystems and is thus identical the for Weyl integrablep ossible asymptotics and ngerprints do es chaotic of not classical contain chaosWeyl any in information formula quantum ab out mechanics is iv simultaneously Notice an that the asymptotic expansion in app ear only in the combination term in the counting function of non case distribu More in the  without time limit a arithmetic d d the f p k normal distribution or in f higher moments of the se a p such that for every piece and is absolutely continuous Z 2 O O k R f k timereversal or p k  2 A n n do es not dep end on Z tends to innity the following mean value converges p Gaussian n p p e R n k f a n n n n X X p p n as 17 p on p p d p n X  p In particular all systems with n p  and N N p n X A p f for chaotic systems with arithmetical chaos for integrable systems for generic chaotic systems N i i f p Z p p and is given by a lim N systems the central limit theorem is satised that is the p osc osc is strictly p ositive in the sense that N N satises resp ectively n with area h h E lim p p f a universal chaotic log where E random numbers have d exist where the o dd moments vanish and the even moments sat is n g case corresp onding to log E E a n which is a probability distribution on N f f invariance p f d Z mean zero and standard deviation k E E a tion where the variance For strongly with resp ect to Leb esgue measure with a density function over the density arithmetic considered as and is given by the ab ove integral where The normalized uctuations reversal with as wise continuous b ounded function isfy quence hyperb olic billiards amplitude Quantum Chaos all L of n p trace N including Frank Steiner increasing average if the corresp onding quantum chaos has nal ly on the basis of which I shall quantum system See for more details However it is known to day that Weyls asymptotic formula completely for longrange correla which violate universality in energy The issue is still op en and all the Then the arithmetical function O maximal ly random Berry using the semiclassical energy range containing on the average 16 a typical conjectures up to an for This fundamental dierence b etween the clas measures the meansquare uctuations in the uctuates ab out zero with in p L n p in contrast to the logarithmic b ehaviour predicted uctuations of chaotic systems are describ ed by the n L arithmetical chaos p osc N n p I shall present two analyzed by Michael n n of energy levels p I E and in a welldened sense b e the p erturbative contribution to the total number was p p Thus the claim seems to b e justied that N N N This and quantal timeevolution has led to the common b elief that it seems universal levels Moreover it was recently found that there exists a very sp ecial class of number by random matrix theory Let been found The two conjectures are the following Conjecture should saturate for large argue that there areare unique uctuation prop erties in quantum mechanics which classical system is strongly chaotictute the I am quantum mechanical convinced thatmechanics analogue these of prop erties the consti phenomenon of chaos in classical L chaotic systems showing level statistics even inIt the thus shortrange app regime ears thatsignature the of prop erties classical of chaos the inIn sp ectral quantum the mechanics rigidity following provide no universal formula Berrys semiclassical arguments suggestied that one sp ectral of the statistics commonly stud the socalled DysonMehta sp ectral rigidity the predictions of random matrixcorrelations theory agree of only the for shorttions quantal and sp mediumrange ectra but fail universal laws of randommatrix theory state systems the sp ectrum isp erio dic discrete in and the the timeevolution sense isis of therefore in Harald almost Bohrs contrast to theoryop erator classical of If almost systems the p whose erio dic classical timeevolution functions systemvillian is is This has ruled mixing a by and continuous the chaoticis part the Liouville sp on ectrum unpredictable of for the the unit large Liou sical circle times and thusunlikely the timeevolution that there isthe anything chaotic within b ehaviour quantum of mechanics classical topreface compare dynamical of with systems his On b o okpreliminary the Gutzwiller other answers writes hand suggest in thatof the quantum us mechanics had is realized moreIn subtle than it most haserties b een of conjectured by the Bohigas energy et level al that the statistical prop terms of the Laurent expansion in energy levels in the case of twodimensional billiards n A It remains C A have b een found Path Integrals Phys Lett which in turn justies it to consider 19 quantum chaos signatures of quantum chaos B as expressed in the two conjectures Ub ergang von der Mikro zur Makromechanik Die Naturwis universal maximal randomness Ub er das Wasserstospektrum vom Standpunkt der neuen Quantenmechanik Ub er das Mo dell des der WasserstomolekulionsAnnalen Physik Leipzig IV the conjectures from the general trace formula Steiner SpaceTime Transformations in Radial J Milnor Morse TheoryM Princeton Morse Variational Analysis New York C Grosche andYork F Steiner The Path Integral on the Pseudosphere Ann Phys New F Steiner Exact Path Integral Treatment of the Hydrogen Atom ibid F Steiner Path IntegralsmeV in to Polar MeV Co ordinates EdsC from MC Grosche eV Gutzwiller and to et F Gev al In Singap Steiner ore Path Path Integrals Integrals from on Curved Manifolds Z Phys senschaften Pro c Int Workshop heldsary in of Vienna birth E Austria Eds Schrodingers G on Badurek the H See o ccasion Rauch also of and Physica the A th Zeilinger anniver Amsterdam Rev JB Keller andNew SI York Rubinow Asymptotic Solution of Eigenvalue Problems Ann Phys Z Physik New York vidensk og Mathem Afd Raekke IVAnn Phys New York schen Physikalischen Gesellschaft derive E stetige SchrodingerDer E SchrodingerAn Undulatory Theory of the Mechanics of Atoms and Molecules Phys W Pauli UniversitatHamburg Selbstverlag der UniversitatHamburg Hamburg VI Arnold Mathematical Metho ds AJ of Lichtenberg Classical and Mechanics MA New Lieb erman York Regular and N Chaotic Bohr Dynamics On Second the edition Quantum Theory of JB LineSp ectra Keller Kgl Corrected Vidensk BohrSommerfeld Quantum Selsk Conditions Skrifter for Natur Nonseparable Systems A Einstein Zum Quantensatz von Sommerfeld und Epstein Verhandlungen der Deut MC Gutzwiller Energy Sp ectrum According to Classical Mechanics J Math Phys MC Gutzwiller Chaos in M Classical Morse and The Quantum Calculus Mechanics of New Variations York in the Large Providence W Pauli W Heisenberg Der Teil F und das Ganze M unchen F Steiner of SchrodingersDiscovery Coherent States In Matter Wave Interferometry to Acknowledgments I would likeRalf to Aurich thank Frank Scheer Ralf andconjecture Gunther Aurich Steil I for and Financial p erforming Jens numerical supp ortcontract tests Bolte No by of DFGSte for the Deutsche helpful is Forschungsgemeinschaft under gratefully discussions acknowledged and References which moreover are In conclusion I b elieve that clear these phenomena as manifestations of expresses the fact of Quantum Chaos This are  trian one has  integrable First tests Recently a H v For b oth  ii Numerical Frank Steiner and f d denotes the area of A The results obtained so P 2 is usually skew and can b e  where 2 f do es not dep end on A Z Gaussian e on the upp erhalf plane P With the obvious mo dications b oth q tends to innity a limit distribution with 18 d n p q generic hyperb olic o ctagon hyperb olic n a dx dy y i Conjectures I and I I have b een formulated for has as  P q and is given by a for b oth symmetry classes Z n can b e very dierent for dierent systems The higher systems generalizes rigorous results recently obtained by chaos A b e the normalized eigenfunctions of a strongly chaotic f N lim n For hyperb olic billiards may not converge to the moments of the limit distribution and universal g n in eq by II n q integrable is such that for every piecewise continuous b ounded function f q d arithmetical n dynamics is chaotic has b een put forward by Berry P P and Artins billiard the following limit converges R b oth unimo dal and multimodal Conjecture Let the o dd moments may not b e zero such that moments of In contrast to the ab ove universal situation for chaotic systems for systems there is inprole general of no the central density limit theorem for the uctuations and the with mean zero and standard deviation A few remarksplanar are billiards in order on gles quantum system Then to replace and is given by the ab ove integral where conjectures are hypothesized to hold for general chaotic systems tests of conjecture Iincluding have b een p erformed for several strongly chaotic systems For strongly chaotic systems thefunction central limit theorem is satised that is the Gaussian distributions with mean zero and universal standard deviation conjectures it is imp ortantthe that case the of central strongly limit chaotic theorem systems is ie assumed that to the hold functions in far strongly supp ort conjecture Ilished A elsewhere detailed iii numerical investigation That willb ehaviour part of of b e conjecture pub I whichHeathBrown deals with on the nonuniversal the famouslattice circle p oint problem problem inside and awave by shifted Bleher functions circle et should iv al b ehave Theclassical as conjecture on that Gaussian the semiclassical randomwith functions if lowlying wave the functions underlying detailed gave test supp ort to hasand this b een conjecture is carried in out excellent agreement for with highly the excited Gaussian quantum b ehaviour eigenstates density A B A B 21 D for a Family of Strongly Chaotic Systems Phys Rev

Math USSR Izvestiya A Z ; A D D RR D D B A D J Bolte Some Studiessertation on Hamburg Arithmetical Chaos Int in J Classical Mo d and Phys Quantum Mechanics Dis J Bolte GEnergyLevel Statistics Steil Phys and Rev FJ Lett Steiner Bolte Arithmetical Periodic Chaos Orbits in and Arithmetical Violation Chaos of on Universality Hyp erb olic in Surfaces Nonlinearity R Aurich andLett J Bolte QuantizationR Rules Aurich J for Bolte Stronglyof C Chaotic Dynamical Matthies Zeta Systems M Functions Mo Sieb d Physica er and Phys F Steiner Crossing the Entropy Barrier M Sieb er The Hyp erb olaSystems Billiard Dissertation A Hamburg Mo del for DESY the Semiclassical rep ort Quantization DESY of Chaotic Rev Rev Lett System Physica by Arithmetic Groups Phys Rev Lett P Sarnak Arithmetic Quantum Chaos Expanded Version and of Blythe the Lectures Schur Toronto Lectures TelAviv Princeton Preprintand Universality of Level Fluctuation Laws Phys Rev Lett M Sieb er and F Steiner Quantum ChaosM in Sieb er and the F Hyp erb ola Steiner Billiard Quantization of Phys Chaos Lett Phys Rev Lett Phys Rev Lett R Aurich and FPhysica Steiner PeriodicOrbit Sum RulesR for Aurich the HadamardGutzwiller and Mo del FPhysica Steiner EnergyLevel Statistics ofR the HadamardGutzwiller Aurich Ensemble and FPro c Roy Steiner So c From London ClassicalR Aurich Periodic and Orbits F Steiner toChaos Staircase Functions the Sp Phys ectral Rigidity Quantization Rev and a of Rule Chaos for Quantizing ica R Aurich EBOctagon Bogomolny Physica and F Steiner Periodic Orbits on the Regular Hyp erb olic J Bolte andCommun F Math Phys Steiner Determinants of Laplacelike Op erators on Riemann Surfaces R Aurich anderalized F Euler Steiner Constant Asymptotic Distribution of the PseudoOrbits and the Gen J Bolte andCommun Math F Phys Steiner The Selb erg Trace Formula for Bordered Riemann Surfaces Hamburger Beitragezur Mathematik aus dem Mathematischen Seminar Heft and the Eigenvalues of theMo dular LaplaceBeltrami Op erator Group on PSL the Fundamental Domain of the nian Spaces with Application to Dirichlet Series J Indian Math So c of Unbounded Quantum Billiards J Math Phys C Matthies and F Steiner Selb ergs Zeta Function and the R Quantization Aurich of C Chaos Matthies Phys M Sieb er and F Steiner Novel Rule for Quantizing Chaos Phys M Sieb er and F Steiner Classical and Quantum Mechanics of a Strongly Chaotic Billiard E Bogomolny B Georgeot MJ Giannoni and C Schmit Chaotic Billiards Generated O Bohigas MJ Giannoni and C Schmit Characterization of Chaotic Quantum Sp ectra R Aurich M Sieb er and F Steiner Quantum Chaos of the HadamardGutzwiller Mo del R Aurich and F Steiner On the Periodic Orbits of a Strongly Chaotic System Phys R Berndt and F Steiner Hyp erb olische Geometrie und Anwendungen in der Physik F Steiner On Selb ergs Zeta Function for Compact Riemann Surfaces Phys Lett AB Venkov Selb ergs Trace Formula for the Hecke Op erator Generated by an Involution A Selb erg Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Rieman F Steiner and P Trillenberg Rened Asymptotic Expansion for the Partition Function Quantum Chaos No No Frank Steiner Part I I Ub ersetzt von Friedrich Adler No Sov Math Dokl 20 A See also Oeuvres Completesde Jacques Hadamard Ub er die mechanischen Analogien des zweiten Hauptsatzes der Thermo dy Ub er die asymptotische Verteilung der Eigenwerte Nachr KoniglGes Wiss YaG Sinai Dynamical Systems with Elastic Reections Russian Math Surveys I I I Paris Pures et Appl vol Paris Math Phys LA Bunimovich andMath YaG Phys Sinai Markov Partitions for Disp ersedpsala Billiards Commun Statistical Mechanics Dokl Akad Nauk namik Journal f urdie reine und angewandte Mathematik Crelles Journal Go em PB Gilkey The Index Theorem and the Heat Equation Boston MC Gutzwiller Periodic Orbits and Classical Quantization Conditions ibid tures In Solid StateBL Physics Altshuler Vol PA Lee NewYork and York RA WebbAD Eds Stone Mesoscopic Phenomena RASystems in Springer Jalab ert Solids Series New in and SolidState Y Science Vol Alhassid Berlin In and Transport Phenomena in Mesoscopic in Mo dern Physics Zeit GottingenMathphys Klasse H Weyl Das asymptotische Verteilungsgesetztialgleichungen der mit Eigenwerte linearer einer partieller Dieren Annalen Anwendung auf die Theorie der Hohlraumstrahlung Math Leipzig Hamburgischen Universitat Phys Symp osium on Quantum ChaosM Cop enhagen Sieb er andSystems F Phys Lett Steiner Generalized PeriodicOrbit Sum Rules for Strongly Chaotic H nouvelles PoincareLes methodes de la I celesteTome mechanique Paris Tome J Hadamard Les surfaces acourbures opp oseeset leurs lignes Math geodesiquesJ L Boltzmann LA Bunimovich On the Ergo dic Prop erties of Nowhere Disp ersing Billiards Commun GD Birkho On the Periodic Motions of YaG Dynamical Sinai Systems On Acta the Mathematica Foundations Up of the Ergo dic Hyp othesis for a Dynamical System of HP Baltes and E HP R McKean Hilf and Sp ectra IM of Singer Finite Curvature Systems and Mannheim the Eigenvalues of the Laplacian J Di M Kac Can one hear the shap e of a drum Amer Math Monthly VI Arnold and A D Heitmann Avez and Ergo dic JP Kotthaus Problems The of Sp ectroscopy of Quantum Classical Dot Arrays Mechanics Physics New CWJ Today York Beenakker and H van Houten Quantum Transport in Semiconductor Nanostruc P Duhem La theoriephysique son ob jet et P sa Duhem La structure Theoriephysique Son Revue Ob P de jet Duhem Sa Philosophie Ziel Structure IVVI und Paris Struktur der physikalischen Theorien C Grosche and F Steiner Table of Feynman Path Integrals To HJ app Mikrowellenbillards ear in Chaos Stockmann Springer in Tracts der Quantenmechanik Physik in unserer H Weyl E Artin Ein mechanisches System mit quasiergo dischen Bahnen Abh Math Sem d RP Feynman SpaceTime Approach to NonRelativistic Quantum Mechanics Rev Mo d F Mautner Geo desic Flows on Symmetric Riemann Spaces Ann Math F Steiner From Feynmans Path Integral to Quantum Chaology Invited Talk at the Z ; A A Frank Steiner A der UniversitatHamburg I I Institut f urTheoretische Physik Lurup er Chausse D Hamburg 22 D D PM Bleher Z Chengthe FJ Number Dyson of and Lattice JL Points Leb owitz Inside Distribution a of the Shifted Error Circle Term Commun for Math Phys of ShortWave Solutions of the Helmholtz Equation Phys Rev PM Bleher FJSome Dyson Integrable Systems and InstituteHEP JL for August Advanced Leb owitz Studies NonGaussian Princeton Energy Preprint IASSNS Level Statistics for R Aurich and F Steiner ExactSystem Theory Physica for the Quantum Eigenstates of a Strongly Chaotic of a Strongly Chaotic System Physica Divisor Problem Acta Arithmetica LX Convex Ovals Duke Math J Exp eriments and Heuristics Sup ercomputer Inst Researchof Rep UMSI Minnesota University DA Hejhal and B Rackner On the Topography of Maass Waveforms for PSL J Math Phys SW McDonald and AM Kaufmann Wave Chaos in the Stadium Statistical Prop erties MV Berry Regular and Irregular Semiclassical Wavefunctions J Phys R Aurich and F Steiner Statistical Prop erties of Highly Excited Quantum Eigenstates ML Mehta Random Matrices MV Revised and Berry Enlarged Semiclassical Second Edition Theory San of Diego Sp ectral Rigidity Pro c Roy So c London DR HeathBrown The Distribution and Moments PM of Bleher the On Error the Term Distribution in of the the Dirichlet Number of Lattice Points Inside a Family of FJ Dyson and ML Mehta Statistical Theory of Energy Levels of Complex Systems