Nonperturbative theory of power spectrum in complex systems
aa r X i v : . [ m a t h - ph ] J a n Nonperturbative theory of power spectrum incomplex systems
Roman Riser , , Vladimir Al. Osipov , and Eugene Kanzieper ‡ Department of Mathematics, Holon Institute of Technology, Holon 5810201, Israel Department of Mathematics and Research Center for Theoretical Physicsand Astrophysics, University of Haifa, Haifa 3498838, Israel
Abstract.
The power spectrum analysis of spectral fluctuations in complex waveand quantum systems has emerged as a useful tool for studying their internaldynamics. In this paper, we formulate a nonperturbative theory of the powerspectrum for complex systems whose eigenspectra – not necessarily of the random-matrix-theory (RMT) type – possess stationary level spacings. Motivated bypotential applications in quantum chaology, we apply our formalism to calculatethe power spectrum in a tuned circular ensemble of random N × N unitarymatrices. In the limit of infinite-dimensional matrices, the exact solution producesa universal, parameter-free formula for the power spectrum, expressed in terms ofa fifth Painlev´e transcendent. The prediction is expected to hold universally, atnot too low frequencies, for a variety of quantum systems with completely chaoticclassical dynamics and broken time-reversal symmetry. On the mathematical side,our study brings forward a conjecture for a double integral identity involving afifth Painlev´e transcendent.Published in: Annals of Physics , 168065 (2020). ‡ Corresponding author. ontents Contents1 Introduction 3 N . . . . . . . . . . 194.2 Proof of Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Power spectrum in TCUE N as a Fredholm determinant . . . . . . . . . 224.4 Power spectrum in TCUE N as a Toeplitz determinant . . . . . . . . . . 25 N limit 26 Appendices 41A Boundary conditions for Painlev´e VI function ˜ σ N ( t ; ζ ) as t → ∞
41B Generating function Φ N ( ϕ ; ζ ) and discrete Painlev´e V equations ( dP V ) 42 Introduction
1. Introduction
The power spectrum analysis of stochastic spectra [1] had recently emerged as apowerful tool for studying both system-specific and universal properties of complexwave and quantum systems. In the context of quantum systems, it reveals whetherthe corresponding classical dynamics is regular or chaotic, or a mixture of both,and encodes a ‘degree of chaoticity’. In combination with other long- and short-range spectral fluctuation measures, it provides an effective way to identify systemsymmetries, determine a degree of incompleteness of experimentally measured spectra,and get the clues about systems internal dynamics. Yet, the theoretical foundations ofthe power spectrum analysis of stochastic spectra have not been settled. In this paper,a nonperturbative theory of the power spectrum will be presented.To set the stage, we review traditional spectral fluctuation measures (Section 1.1),define the power spectrum (Definition 1.1) and briefly discuss its early theoretical andnumerical studies as well as the recently reported experimental results (Section 1.2).We then argue (Section 1.3), that a form-factor approximation routinely used for thepower spectrum analysis in quantum chaotic systems is not flawless and needs to berevisited.
Spectral fluctuations of quantum systems reflect the nature – regular or chaotic –of their underlying classical dynamics [2, 3, 4]. In case of fully chaotic classicaldynamics, hyperbolicity (exponential sensitivity to initial conditions) and ergodicity (typical classical trajectories fill out available phase space uniformly) make quantumproperties of chaotic systems universal [3]. At sufficiently long times t > T ∗ , the singleparticle dynamics is governed by global symmetries of the system and is accuratelydescribed by the random matrix theory (RMT) [5, 6]. The emergence of universalstatistical laws, anticipated by Bohigas, Giannoni and Schmit [3], has been advocatedwithin a field-theoretic [7, 8] and a semiclassical approach [9] which links correlationsin quantum spectra to correlations between periodic orbits in the associated classicalgeodesics. The time scale T ∗ of compromised spectral universality is set by the period T of the shortest closed orbit and the Heisenberg time T H , such that T ≪ T ∗ ≪ T H .Several statistical measures of level fluctuations have been devised in quantumchaology. Long-range correlations of eigenlevels on the unfolded energy scale [5] can bemeasured by the variance Σ ( L ) = var[ N ( L )] of number of levels N ( L ) in the intervalof length L . The Σ ( L ) statistics probes the two-level correlations only and exhibit [10]a universal RMT behavior provided the interval L is not too long, 1 ≪ L ≪ T H /T .The logarithmic behavior of the number variance,Σ ( L ) = 2 π β ln L + O (1) , (1.1)indicates presence of the long-range repulsion between eigenlevels. Here, β = 1 , L ≫ T H /T , system-specific features show up in Σ ( L ) in the form of quasi-random oscillations withwavelengths being inversely proportional to periods of short closed orbits.Individual features of quantum chaotic systems become less pronounced in spectralmeasures that probe the short-range fluctuations as these are largely determined bythe long periodic orbits [9]. The distribution of level spacing between (unfolded)consecutive eigenlevels, P ( s ) = h δ ( s − E j + E j +1 ) i , is the most commonly used short-range statistics. Here, the angular brackets denote averaging over the position j of the reference eigenlevel or, more generally, averaging over such a narrow energy Introduction s ≪ two-point correlations,showing the phenomenon of symmetry-driven level repulsion, P ( s ) ∝ s β . (In asimple-minded fashion, this result can be read out from the Wigner surmise [5]).As s grows, the spacing distribution becomes increasingly influenced by spectralcorrelation functions of all orders. In the universal regime ( s . T H /T ∗ ), theseare best accounted for by the RMT machinery which produces parameter-free (but β -dependent) representations of level spacing distributions in terms of Fredholmdeterminants/Pfaffians and Painlev´e transcendents. For quantum chaotic systems withbroken time-reversal symmetry ( β = 2) – that will be the focus of our study – the levelspacing distribution is given by the famous Gaudin-Mehta formula, which when writtenin terms of Painlev´e transcendents reads [6, 11] P chaos ( s ) = d ds exp (cid:18) ˆ πs σ ( t ) t dt (cid:19) . (1.2)Here, σ ( t ) is the fifth Painlev´e transcendent defined as the solution to the nonlinearequation ( ν = 0)( tσ ′′ ν ) + ( tσ ′ ν − σ ν ) (cid:0) tσ ′ ν − σ ν + 4( σ ′ ν ) (cid:1) − ν ( σ ′ ν ) = 0 (1.3)subject to the boundary condition σ ( t ) = − t/ π − ( t/ π ) + o ( t ) as t → completely chaotic classical dynamics. Quantum systems whose classical geodesics is completely integrable belong to a different, Berry-Tabor universality class [12], partiallyshared by the Poisson point process. In particular, level spacings in a generic integrablequantum system exhibit statistics of waiting times between consecutive events in aPoisson process. This leads to the radically different fluctuation laws: the numbervariance Σ ( L ) = L is no longer logarithmic while the level spacing distribution P int ( s ) = e − s becomes exponential [2], with no signatures of level repulsion whatsoever.Such a selectivity of short- and long-range spectral statistical measures has long beenused to uncover underlying classical dynamics of quantum systems. (For a large class ofquantum systems with mixed regular-chaotic classical dynamics, the reader is referredto Refs. [13, 14, 15].) To obtain a more accurate characterization of the quantum chaos, it is advantageousto use spectral statistics which probe the correlations between both nearby and distanteigenlevels. Such a statistical indicator – the power spectrum – has been suggested inRef. [16].
Definition 1.1.
Let { ε ≤ . . . ≤ ε N } be a sequence of ordered unfolded eigenlevels, N ∈ N , with the mean level spacing ∆ and let h δε ℓ δε m i be the covariance matrix oflevel displacements δε ℓ = ε ℓ − h ε ℓ i from their mean h ε ℓ i . A Fourier transform of thecovariance matrix S N ( ω ) = 1 N ∆ N X ℓ =1 N X m =1 h δε ℓ δε m i e iω ( ℓ − m ) , ω ∈ R (1.4)is called the power spectrum of the sequence. Here, the angular brackets stand for anaverage over an ensemble of eigenlevel sequences. (cid:4) Since the power spectrum is 2 π -periodic, real and even function in ω , S N ( ω + 2 π ) = S N ( ω ) , S ∗ N ( ω ) = S N ( ω ) , S N ( − ω ) = S N ( ω ) , (1.5) Introduction ≤ ω ≤ ω Ny , where ω Ny = π is the Nyquistfrequency. In the spirit of the discrete Fourier analysis, one may restrict dimensionlessfrequencies ω in Eq. (1.4) to a finite set ω k = 2 πkN (1.6)with k = { , , . . . , N/ } , where N is assumed to be an even integer. We shall see thatresulting analytical expressions for S N ( ω k ) are slightly simpler than those for S N ( ω ). Remark 1.2.
We notice in passing that similar statistics has previously been usedby Odlyzko [17] who analyzed power spectrum of the spacings between zeros of theRiemann zeta function. (cid:4)
Considering Eq. (1.4) through the prism of a semiclassical approach, one readilyrealizes that, at low frequencies ω ≪ T ∗ /T H , the power spectrum is largely affectedby system-specific correlations between very distant eigenlevels (accounted for by shortperiodic orbits). For higher frequencies, ω & T ∗ /T H , the contribution of longer periodicorbits becomes increasingly important and the power spectrum enters the universalregime . Eventually, in the frequency domain T ∗ /T H ≪ ω ≤ ω Ny , long periodic orbitswin over and the power spectrum gets shaped by correlations between the nearby levels.Hence, tuning the frequency ω in S N ( ω ) one may attend to spectral correlation betweeneither adjacent or distant eigenlevels.Numerical simulations [16] have revealed that the average power spectrum S N ( ω k )discriminates sharply between quantum systems with chaotic and integrable classicaldynamics. While this was not completely unexpected, another finding of Ref. [16]came as quite a surprise: numerical data for S N ( ω k ), at not too high frequencies,could be fitted by simple power-law curves, S N ( ω k ) ∼ /ω k and S N ( ω k ) ∼ /ω k ,for quantum systems with chaotic and integrable classical dynamics, respectively. Inquantum systems with mixed classical dynamics, numerical evidence was presented [18]for the power-law of the form S N ( ω k ) ∼ /ω αk with the exponent 1 < α < α = 5 /
3. The power spectrum was also measured in Sinai [20] andperturbed rectangular [21] microwave billiards, microwave networks [22, 23] and three-dimensional microwave cavities [24]. For the power spectrum analysis of Fano-Feshbachresonances in an ultracold gas of Erbium atoms [25], the reader is referred to Ref. [26].For quantum chaotic systems, the universal 1 /ω k law for the average power spectrumin the frequency domain T ∗ /T H . ω k ≪ a k = 1 √ N N X ℓ =1 δε ℓ e iω k ℓ (1.7)of level displacements { δε ℓ } , one observes the relation S N ( ω k ) = var[ a k ] . (1.8)Statistics of the Fourier coefficients { a k } were studied in some detail [27] within theDyson’s Brownian motion model [28]. In particular, it is known that, in the limit k ≪ N , they are independent Gaussian distributed random variables with zero meanand the variance var[ a k ] = N/ (2 π βk ). This immediately implies S N ( ω k ≪ ≈ πβω k (1.9) Introduction k (in particular, for k ∼ N ), fluctuationproperties of the Fourier coefficients { a k } are unknown. In view of the relation Eq. (1.8),a nonperturbative theory of the power spectrum to be developed in this paper setsup a well-defined framework for addressing statistical properties of discrete Fouriercoefficients { a k } introduced in Ref. [27].An attempt to determine S N ( ω k ) for higher frequencies up to ω k = ω Ny wasundertaken in Ref. [29] whose authors claimed to express the large- N power spectrumin the entire domain T ∗ /T H . ω k ≤ ω Ny in terms of the spectral form-factor [5] K N ( τ ) = 1 N D N X ℓ =1 N X m =1 e iπτ ( ε ℓ − ε m ) E − D N X ℓ =1 e iπτε ℓ ED N X m =1 e − iπτε m E! (1.10)of a quantum system, τ ≥
0. Referring interested reader to Eqs. (3), (8) and (10) ofthe original paper Ref. [29], here we only quote a small- ω k reduction of their result:ˆ S N ( ω k ≪ ≈ ω k K N (cid:16) ω k π (cid:17) . (1.11)(Here, the hat-symbol ( ˆ ) is used to indicate that the power spectrum ˆ S N ( ω k ≪ A simple mathematical model of eigenlevel sequences { ε , . . . , ε N } with identicallydistributed, uncorrelated spacings { s , . . . , s N } , where ℓ -th ordered eigenlevel equals ε ℓ = ℓ X j =1 s j , (1.12)provides an excellent playing ground to analyze validity of the form-factor approxima-tion. Defined by the covariance matrix of spacings of the form cov( s i , s j ) = σ δ ij , suchthat h s i i = 1, it allows us to determine exactly both the power spectrum Eq. (1.4) andthe form-factor Eq. (1.10). Power spectrum. —Indeed, realizing that the covariance matrix of ordered eigenlevelsequals h δε ℓ δε m i = σ min( ℓ, m ) , (1.13)we derive an exact expression for the power spectrum ( N ∈ N ) S N ( ω ) = 2 N + 14 N σ sin ( ω/ (cid:18) − N + 1 sin (( N + 1 / ω )sin( ω/ (cid:19) . (1.14)Equation (1.14) stays valid in the entire region of frequencies 0 ≤ ω ≤ π . For a set ofdiscrete frequencies ω k = 2 πk/N , it reduces to S N ( ω k ) = σ ( ω k / , < ω k ≤ π. (1.15) Remark 1.3.
Notice that Eqs. (1.14) and (1.15) for the power spectrum of eigenlevelsequences with uncorrelated level spacings hold universally . Indeed, both expressionsappear to be independent of a particular choice of the level spacings distribution; thelevel spacing variance σ is the only model-specific parameter. (cid:4) Introduction Figure 1.
Power spectrum S N ( ω ) as a function of frequency ω for eigenlevelsequences with uncorrelated level spacings. Solid red line corresponds to thetheoretical curve Eq. (1.14) with σ = 1. Blue crosses represent the average powerspectrum simulated for 10 million sequences of N = 2048 random eigenlevels withuncorrelated, exponentially distributed spacings s i ∼ Exp(1). Inset: a log-log plotfor the same graphs.
For illustration purposes, in Fig. 1, we compare the theoretical power spectrum S N ( ω ), Eq. (1.14), with the average power spectrum simulated for an ensemble ofsequences of random eigenlevels with uncorrelated, exponentially distributed levelspacings s i ∼ Exp(1). Since the unit mean level spacing h s j i = 1 is intrinsic tothe model, the unfolding procedure is redundant. Perfect agreement between thetheoretical and the simulated curves is clearly observed in the entire frequency domain0 < ω ≤ π .For further reference, we need to identify three scaling limits of S N ( ω ) that emergeas N → ∞ . In doing so, the power spectrum will be multiplied by ω to get rid of thesingularity at ω = 0.(i) The first – infrared – regime, refers to extremely small frequencies, ω ∼ N − . Itis described by the double scaling limit S ( − (Ω) = lim N →∞ ω S N ( ω ) (cid:12)(cid:12)(cid:12) ω =Ω /N = 2 σ (cid:18) − sin ΩΩ (cid:19) , (1.16)where Ω = O ( N ). One observes: S ( − (Ω) = (cid:26) O (Ω ) , Ω → σ + o (1) , Ω → ∞ . (1.17)(ii) The second scaling regime describes the power spectrum for intermediately smallfrequencies ω ∼ N − α with 0 < α <
1. In this case, a double scaling limit becomestrivial: S ( − α ) ( ˜Ω) = lim N →∞ ω S N ( ω ) (cid:12)(cid:12)(cid:12) ω =˜Ω /N α = 2 σ , (1.18)where ˜Ω = O ( N ). In the forthcoming discussion of a spectral form-factor [Eq. (1.27)],such a scaling limit will appear with α = 1 / Introduction ω = O ( N ) fixed as N → ∞ . In this case, we derive S (0) ( ω ) = lim N →∞ ω S N ( ω ) = σ ω ( ω/ , (1.19)where ω = O ( N ). One observes: S (0) ( ω ) = (cid:26) σ + O ( ω ) , ω → σ π / , ω = π . (1.20)Equations (1.17), (1.18) and (1.20) imply continuity of S ( ω ) across the three scalingregimes. We shall return to the universal formulae Eqs. (1.16), (1.18) and (1.19) lateron. Spectral form-factor. —For eigenlevel sequences with identically distributed, uncorre-lated level spacings, the form-factor K N ( τ ) defined by Eq. (1.10) can be calculatedexactly, too. Defining the characteristic function of i -th level spacing,Ψ s ( τ ) = h e iπτ s i i = ˆ ∞ ds e iπτs f s i ( s ) , (1.21)where f s i ( s ) is the probability density of the i -th level spacing, we reduce Eq. (1.10) to K N ( τ ) = 1 + 2 N Re (cid:20) Ψ s ( τ )1 − Ψ s ( τ ) (cid:18) N − − Ψ Ns ( τ )1 − Ψ s ( τ ) (cid:19)(cid:21) − N (cid:12)(cid:12)(cid:12)(cid:12) Ψ s ( τ ) 1 − Ψ Ns ( τ )1 − Ψ s ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) . (1.22)In Fig. 2, we compare the theoretical form-factor Eq. (1.22) with the average form-factor simulated for an ensemble of sequences of random eigenlevels with uncorrelated,exponentially distributed level spacings as explained below Remark 1.3. The simulationwas based on Eqs. (1.10) and (1.12), and involved averaging [32] over ten millionrealizations. Referring the reader to a figure caption for detailed explanations, weplainly notice a perfect agreement between the simulations and the theoretical resultEq. (1.22).As N → ∞ , three different scaling regimes can be identified for the spectral form-factor. Two of them, arising in specific double scaling limits, appear to be universal .(i) The first – infrared – regime, refers to extremely short times, τ ∼ N − . Assumingexistence and convergence of the moment-expansion for the characteristic functionΨ s ( τ ), we expand it up to the terms of order N − ,Ψ s ( τ ) (cid:12)(cid:12)(cid:12) τ = T/N = 1 + 2 iπ TN − π ( σ + 1) T N + O ( N − ) (1.23)to derive the infrared double scaling limit for the form factor: K ( − ( T ) = lim N →∞ K N ( τ ) (cid:12)(cid:12)(cid:12) τ = T/N = 2 σ (cid:18) − sin(2 πT )2 πT (cid:19) , (1.24)where T = O ( N ). Notice that this formula holds universally as K ( − ( T ) does notdepend on a particular choice of the level spacings distribution; its variance σ is theonly model-specific parameter. One observes: K ( − ( T ) = (cid:26) O ( T ) , T → σ + o (1) , T → ∞ . (1.25)(ii) The second – intermediate – regime, refers to intermediately short times, τ ∼ N − / . Expanding the characteristic function Ψ s ( τ ) up to the terms of order N − , Ψ s ( τ ) (cid:12)(cid:12)(cid:12) τ = T /N / = 1 + 2 iπ T N / − π ( σ + 1) T N + O ( N − / ) , (1.26) Introduction Figure 2.
Spectral form-factor K N ( τ ) as a function of τ for a model and the dataspecified in the caption to Fig. 1. Solid red line corresponds to the theoreticalcurve Eq. (1.22) with Ψ s ( τ ) = (1 − iπτ ) − . Inset: a close-up view of thesame graphs; additional black curves display limiting form-factor in various scalingregimes. Dashed line: regime (I), Eq. (1.24) with τ = T/N . Solid line: regime (II),Eq. (1.27) with τ = T /N / . Dotted line: regime (III), Eq. (1.29), see discussionthere. Notice that the black dashed curve [(I)] starts to deviate from the red curve(after the fourth blue cross the deviation exceeds 10%; as τ grows further, therelative deviation approaches the factor 2). For larger τ , the black solid curve[(II)] becomes a better fit to the red curve. Finally, the red curve approaches theunity depicted by the black dotted line [(III)]. we discover that, for intermediately short times, the double scaling limit of the formfactor reads K ( − / ( T ) = lim N →∞ K N ( τ ) (cid:12)(cid:12)(cid:12) τ = T /N / = σ − e − π σ T π σ T ! , (1.27)where T = O ( N ). Hence, in the intermediate double scaling limit, the form-factorexhibits the universal behavior too, as K ( − / ( T ) depends on a particular choice ofthe level spacings distribution only through its variance σ . One observes: K ( − / ( T ) = (cid:26) σ + O ( T ) , T → σ + o (1) , T → ∞ . (1.28)(iii) The third scaling regime describes the form-factor for τ = O ( N ) fixed as N → ∞ . Spotting that in this case the characteristic function Ψ Ns ( τ ) vanishesexponentially fast, we derive K (0) ( τ ) = lim N →∞ K N ( τ ) = 1 + 2Re (cid:20) Ψ s ( τ )1 − Ψ s ( τ ) (cid:21) . (1.29)Notably, in the fixed- τ scaling limit, the form-factor is no longer universal as it dependsexplicitly on the particular distribution of level spacings § through its characteristicfunction Ψ s ( τ ). One observes: K (0) ( τ ) = (cid:26) σ + O ( τ ) , τ → o (1) , τ → ∞ . (1.30) § For the exponential distribution of level spacings the form-factor in the third scaling regime equalsunity, K (0) ( τ ) ≡ Introduction Figure 3.
Limiting curves ( N → ∞ ) for the form-factor across the three scalingregimes [(I) – Eq. (1.24), (II) – Eq. (1.27), and (III) – Eq. (1.29)], glued together atvertical dotted lines. The functions K ( − ( T ), K ( − / ( T ) and K (0) ( τ ), describingthe regimes (I), (II) and (III), correspondingly, are plotted vs variables T = Nτ , T = N / τ and τ , each running over the entire real half-line compactified usingthe transformation (0 , ∞ ) = tan((0 , π/ ,
3) (red), inverse Gaussian IG(1 ,
3) (green) and uniform U(0 ,
2) (blue)distributions, exhibiting identical mean and variance. The dashed black line –to be discussed in the main text – displays the limiting curve of the functionlim N →∞ ω S N ( ω ) with 0 ≤ ω = 2 πτ ≤ π (that is, 0 ≤ τ ≤ / ) for all threechoices of the level spacing distribution. In the scaling regimes (I), (II) and (III),the curve is described by Eqs. (1.16), (1.18) with α = 1 / The three scaling regimes for the form-factor as N → ∞ are illustrated in Fig. 3. Thecontinuity of the entire curve is guaranteed by equality of limits lim T →∞ K ( − ( T ) =lim T → K ( − / ( T ) and lim T →∞ K ( − / ( T ) = lim τ → K (0) ( τ ), see Eqs. (1.25), (1.28)and (1.30). To highlight occurrence of both universal and non-universal τ -domainsin the form-factor, the latter is plotted for three different choices of level spacingdistributions, s j ∼ Erlang(3 , ,
3) and U(0 , h s j i = 1 and the variance σ = 1 / f s j ( s ) = Θ( s ) × s exp( − s ) , Erlang(3 , (cid:18) πs (cid:19) / exp (cid:18) − s − s (cid:19) , IG(1 , − s ) , U(0 , non-universal as the three curvesevolve differently depending on a particular choice of the level spacing distribution.Yet, all three curves approach unity at infinity. Implications for the power spectrum. —We now turn to the discussion of a relationbetween the power spectrum Eq. (1.4) and the form-factor Eq. (1.10). To this end, weshall compare the limiting forms, as N → ∞ , of the form-factor, studied both ana-lytically and numerically in the previous subsection, with the limiting behavior of the Main results and discussion ω S N ( ω ) | ω =2 πτ as prompted by the form-factor approximation Eq. (1.11).The latter is plotted in Fig. 3 by the black dashed line.(i) For extremely low frequencies ω = O ( N − ) (equivalently, short times τ = O ( N − ))belonging to the first scaling regime [(I)], the two quantities are seen to coincide (universal) K ( − ( T ) = (universal) S ( − (Ω) | Ω=2 πT , (1.32)see Eqs. (1.16) and (1.24). The universal behavior of both spectral indicators in thedomain (I) is illustrated in Fig. 3 arranged for three different level spacing distributionsspecified by Eq. (1.31).(ii) In the second scaling regime [(II)], characterized by intermediately low frequencies ω = O ( N − / ) (equivalently, τ = O ( N − / )), the limiting curve for the form-factorstarts to deviate from the one for the product ω S N ( ω ) | ω =2 πτ , in concert with theanalytical analysis,(universal) K ( − / ( T ) = (universal) S ( − / ( ˜Ω) | ˜Ω=2 π T = 2 σ , (1.33)compare Eq. (1.18) taken at α = 1 / S ( − / ( ˜Ω)is a constant throughout the entire domain (II), the form-factor is described by theuniversal function Eq. (1.27) irrespective of a particular form of the level spacingdistribution; the relative deviation between the two limiting curves reaches itsmaximum (= 2) at the borderline between the regimes (II) and (III), in concert withthe earlier conclusion of Ref. [33]. How fast this factor of 2 is approached depends onlyon the value of σ , as described by Eq. (1.27). Hence, the relation Eq. (1.11) is clearlyviolated in the second scaling regime, apart from a single point at the border betweenthe regimes (I) and (II) as stated below Eq. (1.30).(iii) In the third scaling regime [(III)] emerging for ω = O ( N ) (equivalently, τ = O ( N )) the two limiting curves depart incurably from each other: while theproduct lim N →∞ ω S N ( ω ), shown by the dashed black line, follows the universal lawEq. (1.19), the form-factor displays a non-universal behavior strongly depending onthe particular form of level spacing distribution as highlighted by solid red, green andblue curves, see also Eq. (1.29),(nonuniversal) K (0) ( τ ) = (universal) S (0) ( ω ) | ω =2 πτ . (1.34)Hence, the two spectral statistics – the form-factor and the power spectrum – cannot be reduced to each other for any finite frequency 0 < ω < π as N → ∞ . Conclusion. —Detailed analytical and numerical analysis, performed for eigenlevel se-quences with uncorrelated, identically distributed level spacings, leads us to concludethat the spectral form-factor and the power spectrum are generically two distinct sta-tistical indicators . This motivates us to revisit the problem of calculating the powerspectrum for a variety of physically relevant eigenlevel sequences beyond the form-factorapproximation. In the rest of the paper, this program, initiated in our previous pub-lication [33], will be pursued for (a) generic eigenlevel sequences possessing stationarylevel spacings and (b) eigenlevel sequences drawn from a variant of the circular unitaryensemble of random matrices. The latter case is of special interest as its N → ∞ limitbelongs to the spectral universality class shared by a large class of quantum systemswith completely chaotic classical dynamics and broken time-reversal symmetry.
2. Main results and discussion
In this Section, we collect and discuss the major concepts and results of our work.Throughout the paper, we shall deal with eigenlevel sequences possessing stationarylevel spacings as defined below.
Main results and discussion Definition 2.1.
Consider an ordered sequence of (unfolded) eigenlevels { ≤ ε ≤· · · ≤ ε N } with N ∈ N . Let { s , · · · , s N } be the sequence of spacings betweenconsecutive eigenlevels such that s ℓ = ε ℓ − ε ℓ − with ℓ = 1 , . . . , N and ε = 0. Thesequence of level spacings is said to be stationary if (i) the average spacing h s ℓ i = ∆ (2.1)is independent of ℓ = 1 , . . . , N and (ii) the covariance matrix of spacings is of theToeplitz type: cov( s ℓ , s m ) = I | ℓ − m | − ∆ (2.2)for all ℓ, m = 1 , . . . , N . Here, I n is a function defined for non-negative integers n . (cid:4) Remark 2.2.
While stationarity of level spacings is believed to emerge after unfoldingprocedure in the limit N → ∞ , see Ref. [34], it is not uncommon to observestationarity even for finite eigenlevel sequences. Two paradigmatic examples of finite - N eigenlevel sequences with stationary spacings include (i) a set of uncorrelatedidentically distributed eigenlevels [35] mimicking quantum systems with integrableclassical dynamics and (ii) eigenlevels drawn from the ‘tuned’ circular ensembles ofrandom matrices appearing in the random matrix theory approach to quantum systemswith completely chaotic classical dynamics, see Section 4. (cid:4) First result. —For generic eigenlevel sequences, the power spectrum Eq. (1.4) isdetermined by both diagonal and off-diagonal elements of the covariance matrix h δε ℓ δε m i . In the important case of eigenlevel sequences with stationary level spacings ,the power spectrum can solely be expressed in terms of diagonal elements h δε ℓ i of thecovariance matrix. The Theorem 2.3 below, establishes an exact relation between thepower spectrum (see Definition 1.1) and a generating function of variances of ordered eigenvalues. Theorem 2.3 (First master formula) . Let N ∈ N and ≤ ω ≤ π . The power spectrumfor an eigenlevel sequence { ≤ ε ≤ · · · ≤ ε N } with stationary spacings equals S N ( ω ) = 1 N ∆ Re (cid:18) z ∂∂z − N − − z − N − z (cid:19) N X ℓ =1 var[ ε ℓ ] z ℓ , (2.3) where z = e iω , ∆ is the mean level spacing, and var[ ε ℓ ] = h δε ℓ i . (2.4)For the proof, the reader is referred to Section 3.2. Second result. —Yet another useful representation – the second master formula –establishes an exact representation of the power spectrum in terms of a generatingfunction of probabilities E N ( ℓ ; ǫ ) to observe exactly ℓ eigenlevels below the energy ε , E N ( ℓ ; ε ) = N ! ℓ !( N − ℓ )! ℓ Y j =1 ˆ ε dǫ j N Y j = ℓ +1 ˆ ∞ ε dǫ j P N ( ǫ , . . . , ǫ N ) . (2.5)Here, P N ( ǫ , . . . , ǫ N ) is the joint probability density (JPDF) of N unordered eigenlevelstaken from a positive definite spectrum; it is assumed to be symmetric under apermutation of its arguments. Such an alternative albeit equivalent representationof the power spectrum will be central to the spectral analysis of quantum chaoticsystems. Main results and discussion Theorem 2.4 (Second master formula) . Let N ∈ N and ≤ ω ≤ π , and let Φ N ( ε ; ζ ) be the generating function Φ N ( ε ; ζ ) = N X ℓ =0 (1 − ζ ) ℓ E N ( ℓ ; ε ) (2.6) of the probabilities defined in Eq. (2.5). The power spectrum, Definition 1.1, for aneigenlevel sequence with stationary spacings equals S N ( ω ) = 2 N ∆ Re (cid:18) z ∂∂z − N − − z − N − z (cid:19) z − z ˆ ∞ dǫ ǫ (cid:2) Φ N ( ǫ ; 1 − z ) − z N (cid:3) − ˜ S N ( ω ) , (2.7) where z = 1 − ζ = e iω , ∆ is the mean level spacing, and ˜ S N ( ω ) = 1 N Re (cid:18) z ∂∂z − N − − z − N − z (cid:19) N X ℓ =1 ℓ z ℓ = 1 N (cid:12)(cid:12)(cid:12)(cid:12) − ( N + 1) z N + N z N +1 (1 − z ) (cid:12)(cid:12)(cid:12)(cid:12) . (2.8)For the proof, the reader is referred to Section 3.3. Remark 2.5.
Notably, representations Eqs. (2.6) and (2.7) suggest that the powerspectrum is determined by spectral correlation functions of all orders . Contrary to thespacing distribution, which is essentially determined by the gap formation probability[5] E N (0; ε ), the power spectrum depends on the entire set of probabilities E N ( ℓ ; ε )with ℓ = 0 , , . . . , N . (cid:4) Third result. —To study the power spectrum in quantum systems with broken time-reversal symmetry and completely chaotic classical dynamics, let us consider the tunedcircular unitary ensemble (TCUE N ). Obtained from the traditional circular unitaryensemble CUE N +1 [5] by conditioning its lowest eigen-angle to stay at zero, the TCUE N is defined by the joint probability density of N eigen-angles { θ , . . . , θ N } of the form P N ( θ , . . . , θ N ) = 1( N + 1)! Y ≤ i Let { θ ≤ · · · ≤ θ N } be fluctuating ordered eigen-angles drawn fromthe TCUE N , N ∈ N , with the mean level spacing ∆ and let h δθ ℓ δθ m i be the covariancematrix of eigen-angle displacements δθ ℓ = θ ℓ − h θ ℓ i from their mean h θ ℓ i . A Fouriertransform of the covariance matrix S N ( ω ) = 1 N ∆ N X ℓ =1 N X m =1 h δθ ℓ δθ m i e iω ( ℓ − m ) , ω ∈ R (2.11)is called the power spectrum of the TCUE N . Here, the angular brackets denote averagewith respect to the JPDF Eq. (2.9). (cid:4) Main results and discussion Theorem 2.7 (Power spectrum in TCUE N ) . Let { θ ≤ · · · ≤ θ N } be fluctuatingordered eigen-angles drawn from the TCUE N . Then, for any N ∈ N and all ≤ ω ≤ π ,the power spectrum admits exact representation S N ( ω ) = ( N + 1) πN Re (cid:18) z ∂∂z − N − − z − N − z (cid:19) z − z ˆ π dϕ π ϕ Φ N ( ϕ ; 1 − z ) − ≈ S N ( ω ) , (2.12) where ≈ S N ( ω ) = 1 N (cid:12)(cid:12)(cid:12)(cid:12) − ( N + 1) z N + N z N +1 (1 − z ) (cid:12)(cid:12)(cid:12)(cid:12) − ( N + 1) N | − z | (2.13) and Φ N ( ϕ ; ζ ) = exp − ˆ ∞ cot( ϕ/ dt t (˜ σ N ( t ; ζ ) + t ) ! . (2.14) Here, z = 1 − ζ = e iω whilst the function ˜ σ N ( t ; ζ ) is a solution to the σ -Painlev´e VIequation (cid:0) (1 + t ) ˜ σ ′′ N (cid:1) + 4˜ σ ′ N (˜ σ N − t ˜ σ ′ N ) + 4(˜ σ ′ N + 1) (cid:0) ˜ σ ′ N + ( N + 1) (cid:1) = 0 (2.15) satisfying the boundary condition ˜ σ N ( t ; ζ ) = − t + N ( N + 1)( N + 2)3 πt ζ + O ( t − ) (2.16) as t → ∞ . For the proof of Theorem 2.7, the reader is referred to Section 4.2. Remark 2.8. Theorem 2.7 provides an exact RMT solution for the power spectrumin the TCUE N . Alternatively, but equivalently, the finite- N power spectrum can beexpressed in terms of a Fredholm determinant (Section 4.3), Toeplitz determinant(Section 4.4) and discrete Painlev´e V (dP V ) equations (Appendix B). While theToeplitz representation is beneficial for performing a large- N analysis of the powerspectrum, the dP V formulation is particularly useful for efficient numerical evaluationof the power spectrum for relatively large values of N . (cid:4) Fourth (main) result. —The most remarkable feature of the random matrix theory isits ability to predict universal statistical behavior of quantum systems. In this context,a large- N limit of the power spectrum in the TCUE N is expected to furnish a universal,parameter-free law, S ∞ ( ω ) = lim N →∞ S N ( ω ), for the power spectrum. Its functionalform is given in the Theorem 2.9 below. Theorem 2.9 (Universal law) . For < ω < π , the limit S ∞ ( ω ) = lim N →∞ S N ( ω ) exists and equals S ∞ ( ω ) = A (˜ ω ) ( Im ˆ ∞ dλ π λ − ω e i ˜ ωλ × (cid:20) exp (cid:18) − ˆ ∞ λ dtt (cid:0) σ ( t ; ˜ ω ) − i ˜ ωt + 2˜ ω (cid:1)(cid:19) − (cid:21) + B (˜ ω ) ) , (2.17) where ˜ ω = ω/ π is a rescaled frequency, and the functions A (˜ ω ) and B (˜ ω ) are definedas A (˜ ω ) = 12 π Q j =1 G ( j + ˜ ω ) G ( j − ˜ ω )sin( π ˜ ω ) , (2.18) B (˜ ω ) = 12 π sin( π ˜ ω ) ˜ ω ω − Γ(2 − ω ) . (2.19) Main results and discussion Figure 4. A graph for the power spectrum as a function of frequency. Red linecorresponds to the power spectrum calculated through the dP V representation(Appendix B) of the exact Painlev´e VI solution for N = 10 , see Theorem 2.7.Blue crosses correspond to the power spectrum calculated for sequences of 256unfolded CUE eigen-angles averaged over 10 realizations. Inset: a log-log plot forthe same graphs. Here, G is the Barnes’ G -function, Γ is the Gamma function, whilst σ ( t ; ˜ ω ) = σ ( t ) isthe Painlev´e V transcendent satisfying Eq. (1.3) with ν = 1 and fulfilling the boundaryconditions σ ( t ) = i ˜ ωt − ω − itγ ( t )1 + γ ( t ) + O ( t − ω ) , t → ∞ , (2.20) σ ( t ) = O ( t ln t ) , t → , (2.21) with γ ( t ) being defined by Eq. (5.6). Remark 2.10. As a by-product of this Theorem, we have formulated a conjecture fora double integral identity involving a fifth Painlev´e transcendent. A mathematically-oriented reader is referred to Conjecture 5.9. (cid:4) Theorem 2.11 (Small- ω expansion) . In the notation of Theorem 2.9, the followingexpansion holds as ω → : S ∞ ( ω ) = 14 π ˜ ω + 12 π ˜ ω ln ˜ ω + ˜ ω 12 + O (˜ ω ) . (2.22)For the proof of Theorems 2.9 and 2.11, the reader is referred to Section 5. In Figs. 4 and 5, the parameter-free prediction Eq. (2.17) for the power spectrum isconfronted with the results of numerical simulations for the large- N circular unitaryensemble CUE N . Two remarks are in order. (i) First, the limiting curve for S ∞ ( ω )was approximated by the exact Painlev´e VI solution computed for sufficiently large N through its dPV representation worked out in detail in Appendix B. We haveverified, by performing numerics for various values of N , that the convergence of dP V representation of S N ( ω ) to S ∞ ( ω ) is very fast, so that the N = 10 curve provides an Main results and discussion Figure 5. Difference between the power spectrum and its singular part 1 / πω asdescribed by Eq. (1.9) at β = 2 (see also the first term in Eq. (2.22)). The singularpart of the power spectrum corresponds to δS ∞ ( ω ) = 0 as represented by a graydotted line. Red solid line: analytical prediction computed as explained in Fig. 4.Blue crosses: simulation for 4 × sequences of 512 unfolded CUE eigenvalues.Inset: magnified portion of the same graph for 0 ≤ ω ≤ π/ 4; additional blackdashed line displays the difference δS ∞ ( ω ) calculated using the small- ω expansionEq. (2.22). excellent approximation to the universal law for S ∞ ( ω ). A good match between the N = 10 curve and the one plotted for a small- ω expansion Eq. (2.22) of S ∞ ( ω ) (seeinset in Fig. 5) lends an independent support to validity of our numerical procedure. (ii)Second, even though the theoretical results used for comparison refer to the TCUE N –rather than the CUE N – ensemble (which differ from each other by the weight functionand the way the two are intrinsically unfolded k ), the agreement between the TCUE N theory and the CUE N numerics is nearly perfect, which can naturally be attributed tothe universality phenomenon emerging as N → ∞ .The universal formula for S ∞ ( ω ), stated in Theorem 2.9, is the central result of thepaper. We expect it to hold universally for a wide class of random matrix modelsbelonging to the β = 2 Dyson’s symmetry class, as the matrix dimension N → ∞ .Expressed in terms of a fifth Painlev´e transcendent, the universal law Eq. (2.17) canbe viewed as a power spectrum analog of the Gaudin-Mehta formula Eq. (1.2) for thelevel spacing distribution.Apart from establishing an explicit form of the universal random-matrix-theory lawfor S ∞ ( ω ), our theory reveals two important general aspects of the power spectrumwhich hold irrespective of a particular model of eigenlevel sequences: (i) similarly to thelevel spacing distribution, the power spectrum is determined by spectral correlationsof all orders ; (ii) in distinction to the level spacing distribution, which can solely beexpressed in terms of the gap formation probability, the power spectrum is contributedby the entire set of probabilities that a spectral interval of a given length contains k The spectra in CUE N and TCUE N ensembles are intrinsically unfolded for any N ∈ N , albeit eachin its own way. Indeed, in the CUE N the mean density is a constant [5, 6], while in the TCUE N the mean level spacing is a constant, see Corollary 4.3. In the limit N → ∞ , the two types of unfoldingare expected to become equivalent. Power spectrum for eigenlevel sequences with stationary spacings ℓ eigenvalues with ℓ ≥ 0. As such, it provides a complementary statisticaldescription of spectral fluctuations in stochastic spectra of various origins.Considered through the prism of Bohigas-Giannoni-Schmit conjecture, the universallaw Eq. (2.17) should hold for a variety of quantum systems with completely chaoticclassical dynamics and broken time-reversal symmetry at not too low frequencies T ∗ /T H . ω ≤ π , when ergodicity and global symmetries – rather than system specificfeatures – are responsible for shaping system’s dynamics.Potential applicability of our results to the non-trivial zeros of the Riemann zetafunction deserves special mention. Indeed, according to the Montgomery-Odlyzko law(see, e.g., Ref. [36]), the zeros of the Riemann zeta function located high enough alongthe critical line are expected to follow statistical properties of the eigenvalues of large U ( N ) matrices. This suggests that the universal law Eq. (2.17) could be detected”experimentally”. Extensive, high-precision data accumulated by A. M. Odlyzko forbillions of Riemann zeros [37] provide a unique opportunity for a meticulous test of thenew universal law. 3. Power spectrum for eigenlevel sequences with stationary spacings In this Section, we provide proofs of two master formulae given by Theorem 2.3 andTheorem 2.4. In view of Definition 2.1, we first establish a necessary and sufficient condition foreigenlevel sequences to possess stationarity of level spacings. Lemma 3.1. For N ∈ N , let { ≤ ε ≤ · · · ≤ ε N } be an ordered sequence of unfoldedeigenlevels such that h ε i = ∆ . An associated sequence of spacings between consecutiveeigenlevels is stationary if and only if h ( ε ℓ − ε m ) q i = h ε qℓ − m i (3.1) for ℓ > m and both q = 1 and q = 2 .Proof. The equivalence of Eq. (2.1) to Eq. (3.1) at q = 1 is self-evident. To prove theequivalence of Eq. (2.2) to Eq. (3.1) at q = 2, we proceed in two steps.First, let the covariance matrix of level spacings be of the form Eq. (2.2). SubstitutingEq. (1.12) into the l.h.s. of Eq. (3.1) taken at q = 2, and making use of Eq. (2.2) twice, h ( ε ℓ − ε m ) i = ℓ X i = m +1 ℓ X j = m +1 h s i s j i = ℓ X i = m +1 ℓ X j = m +1 I | i − j | = ℓ − m X i ′ =1 ℓ − m X j ′ =1 I | i ′ − j ′ | = ℓ − m X i ′ =1 ℓ − m X j ′ =1 h s i ′ s j ′ i = h ε ℓ − m i , we derive the r.h.s. of Eq. (3.1) with q = 2.Second, let the ordered eigenvalues satisfy Eq. (3.1) at q = 2. Substituting s ℓ ( m ) = ε ℓ ( m ) − ε ℓ ( m ) − into the definition of covariance matrix cov( s ℓ , s m ) of levelspacings and making use of Eq. (3.1), we observe that Eq. (2.2) indeed holds with I | ℓ − m | of the form I | ℓ − m | = 12 h ε | ℓ − m | +1 i + 12 h ε | ℓ − m |− i − h ε | ℓ − m | i . (3.2) Power spectrum for eigenlevel sequences with stationary spacings It follows from Eq. (3.1) of Lemma 3.1 written in the form h δε ℓ δε m i = 12 (cid:16) h δε ℓ i + h δε m i − h δε | ℓ − m | i (cid:17) , (3.3)where δε ℓ = ε ℓ − ℓ ∆. Substituting Eq. (3.3) into the definition Eq. (1.4) and reducingthe number of summations therein, we derive Eq. (2.3). (cid:3) Remark 3.2. For discrete frequencies ω k = 2 πk/N the power spectrum representationEq. (2.3) simplifies to S N ( ω k ) = 1 N ∆ Re (cid:18) z k ∂∂z k − N (cid:19) N X ℓ =1 var[ ε ℓ ] z ℓk . (3.4)Here, z k = e iω k and the derivative with respect to z k should be taken as if z k were acontinuous variable. (cid:4) To prove the Theorem 2.4, we need the following Lemma: Lemma 3.3. For N ∈ N , let { ε ≤ · · · ≤ ε N } be an ordered sequence of eigenlevelssupported on the half axis (0 , ∞ ) , and let E N ( ℓ ; ε ) be the probability to find exactly ℓ eigenvalues below the energy ε , given by Eq. (2.5), with ℓ = 0 , , . . . , N . The followingrelation holds: ddε E N ( ℓ ; ε ) = p ℓ ( ε ) − p ℓ +1 ( ε ) . (3.5) Here, p ℓ ( ε ) is the probability density of ℓ -th ordered eigenlevel where p ( ε ) = p N +1 ( ε ) =0 for ε > . Equivalently, p ℓ ( ε ) = − ℓ − X j =0 ddε E N ( j ; ε ) , ℓ = 1 , . . . , N. (3.6) Proof. Differentiating Eq. (2.5) and having in mind that the probability density of ℓ -thordered eigenvalue equals p ℓ ( ε ) = N !( ℓ − N − ℓ )! ℓ − Y j =1 ˆ ε dǫ j N Y j = ℓ +1 ˆ ∞ ε dǫ j × P N ( ǫ , . . . , ǫ ℓ − , ε, ǫ ℓ +1 , . . . , ǫ N ) , (3.7)we derive Eqs. (3.5) and (3.6). Proof of Theorem 2.4. —Equipped with Lemma 3.3, we are ready to prove Theorem2.4. First, we observe that Eqs. (2.6) and Eq. (3.6) induce the relation N X ℓ =1 z ℓ p ℓ ( ε ) = − z − z ddε (cid:2) Φ N ( ε ; 1 − z ) − z N (cid:3) . (3.8)Second, we split the variance Eq. (2.3) into var[ ε ℓ ] = h ε ℓ i − ℓ ∆ . The later termproduces the contribution ˜ S N ( ω ) in Eq. (2.7) while the former brings N X ℓ =1 h ε ℓ i z ℓ = − z − z ˆ ∞ dǫ ǫ ddε (cid:2) Φ N ( ε ; 1 − z ) − z N (cid:3) = 2 z − z ˆ ∞ dǫ ǫ (cid:2) Φ N ( ε ; 1 − z ) − z N (cid:3) . (3.9) Power spectrum in the tuned circular unitary ensemble N N ( ε ) in the tail region ( ε, ∞ ) exhibits sufficiently fast decay N N ( ε ) ∼ ε − (2+ δ ) with δ > ε → ∞ ¶ . Substituting Eq. (3.9) into Eq. (2.3), we derive thefirst term in Eq. (2.7). This ends the proof of Theorem 2.4. (cid:3) Remark 3.4. For discrete frequencies ω k = 2 πk/N the power spectrum representationEq. (2.7) simplifies to S N ( ω k ) = 2 N ∆ Re (cid:18) z k ∂∂z k − N (cid:19) z k − z k ˆ ∞ dǫ ǫ [Φ N ( ǫ ; 1 − z k ) − − N | − z k | . (3.10)Here, z k = e iω k and the derivative with respect to z k should be taken as if z k were acontinuous variable. (cid:4) 4. Power spectrum in the tuned circular unitary ensemble In this Section, a general framework developed in Section 3 and summed up in Theorem2.4 will be utilized to determine the power spectrum in the tuned circular ensemble ofrandom matrices, TCUE N , for any N ∈ N . For the definition of TCUE N , the readeris referred to Eqs. (2.9) and (2.10). TCUE N The main objective of this subsection is to establish stationarity of spacings betweenordered TCUE N eigen-angles. To this end, we prove Lemma 4.1 and Lemma 4.2. Thesought stationarity will then be established in Corollary 4.3. Lemma 4.1 (Circular symmetry) . For q = 0 , , . . . and ℓ = 1 , , . . . , N it holds that h θ qℓ i = h (2 π − θ N − ℓ +1 ) q i . (4.1) Proof. The proof is based on the circular-symmetry identity p ℓ ( ϕ ) = p N − ℓ +1 (2 π − ϕ ) (4.2)between the probability density functions of ℓ -th and ( N − ℓ + 1)-th ordered eigenanglesin the TCUE N . This relation can formally be derived from the representation p ℓ ( ϕ ) = 1( N + 1)! N !( ℓ − N − ℓ )! (cid:12)(cid:12) − e iϕ (cid:12)(cid:12) ¶ Indeed, Eqs. (2.5) and (2.6) imply an integral representationΦ N ( ε ; 1 − z ) = N Y ℓ =1 (cid:18) z ˆ ∞ dǫ ℓ + (1 − z ) ˆ ∞ ε dǫ ℓ (cid:19) P N ( ǫ , . . . , ǫ N ) . Letting ε → ∞ , we generate a large- ε expansion of the formΦ N ( ε ; 1 − z ) = z N + N X ℓ =1 z N − ℓ (1 − z ) ℓ ℓ Y j =1 ˆ ∞ ε dǫ j R ℓ,N ( ǫ , . . . , ǫ ℓ ) , where R ℓ,N ( ǫ , . . . , ǫ ℓ ) = N !( N − ℓ )! N Y j = ℓ +1 ˆ ∞ dǫ j P N ( ǫ , . . . , ǫ N )is the ℓ -point spectral correlation function. To the first order, the expansion brings Φ N ( ε ; 1 − z ) = z N + z N − (1 − z ) N N ( ε ) + . . . , where N N ( ε ) is the mean spectral density R ,N ( ǫ ) integrated over theinterval ( ε, ∞ ). Hence, the required decay of N N ( ε ) at infinity readily follows. Power spectrum in the tuned circular unitary ensemble × ℓ − Y j =1 ˆ ϕ dθ j π N − Y j = ℓ ˆ πϕ dθ j π × Y ≤ i It is advantageous to start with the JPDF of ordered eigenangles in the TCUE N , P (ord) N ( θ , . . . , θ N ) = N ! P N ( θ , . . . , θ N ) ≤ θ ≤···≤ θ N ≤ π = 1 N + 1 Y ≤ i A sequence of spacings between consecutive eigenangles in TCUE N isstationary such that the mean position of the ℓ -th ordered eigen-angle equals h θ ℓ i = ℓ ∆ , (4.13) where ℓ = 1 , . . . , N and ∆ = 2 πN + 1 , (4.14) is the mean spacing.Proof. Indeed, combining Lemma 4.1 taken at q = 1 and Lemma 4.2 taken at q = 1and m = ℓ − 1, one concludes that the mean spacing∆ = h θ ℓ − θ ℓ − i = 2 πN + 1is constant everywhere in the eigenspectrum. Now we apply Lemma 3.1 and Lemma4.2 to complete the proof. + Stationarity of level spacings in the TCUE N established in Corollary 4.3 allows us touse a ‘compactified’ version of Theorem 2.4 in order to claim the representation statedby Eqs. (2.12) and (2.13), whereΦ N ( ϕ ; ζ ) = N X ℓ =0 (1 − ζ ) ℓ E N ( ℓ ; ϕ ) (4.15)is the generating function of the probabilities E N ( ℓ ; ϕ ) = N ! ℓ !( N − ℓ )! ℓ Y j =1 ˆ ϕ dθ j π N Y j = ℓ +1 ˆ πϕ dθ j π P N ( θ , . . . , θ N ) (4.16)to find exactly ℓ eigen-angles in the interval (0 , ϕ ) of the TCUE N spectrum. The JPDF P N ( θ , . . . , θ N ) is defined in Eq. (2.9).Substituting Eqs. (4.16) and (2.9) into Eq. (4.15), one derives a multidimensional-integral representation of the generating function Φ N ( ϕ ; ζ ) in the formΦ N ( ϕ ; ζ ) = 1( N + 1)! N Y j =1 (cid:18) ˆ π − ζ ˆ ϕ (cid:19) dθ j π Y ≤ i For a set of discrete frequencies ω ′ k = 2 πkN + 1the free term in Eq. (2.12) nullifies, ≈ S N ( ω ′ k ) = 0, bringing a somewhat tidier formula S N ( ω ′ k ) = ( N + 1) πN Re (cid:18) z ′ k ∂∂z ′ k − N − (cid:19) z ′ k − z ′ k ˆ π dϕ π ϕ Φ N ( ϕ ; 1 − z ′ k ) , (4.21)where z ′ k = e iω ′ k . This is essentially Eq. (17) previously announced in our paperRef. [33]. (cid:4) TCUE N as a Fredholm determinant To derive a Fredholm determinant representation of the TCUE N power spectrum, adeterminantal structure [5, 6] of spectral correlation functions in the TCUE N shouldbe established. This is summarized in Lemma 4.5 below. Lemma 4.5. For ℓ = 1 , . . . , N , the ℓ -point correlation function [5, 6] R ℓ,N ( θ , . . . , θ ℓ ) = N !( N − ℓ )! N Y j = ℓ +1 ˆ π dθ j π P N ( θ , . . . , θ N ) (4.22) in the TCUE N ensemble, defined by Eqs. (2.9) and (2.10), admits the determinantalrepresentation R ℓ,N ( θ , . . . , θ ℓ ) = det ≤ i,j ≤ ℓ [ κ N ( θ i , θ j )] , (4.23) where the TCUE N scalar kernel κ N ( θ, θ ′ ) = S N +1 ( θ − θ ′ ) − N + 1 S N +1 ( θ ) S N +1 ( θ ′ ) (4.24) is expressed in terms of the sine-kernel S N +1 ( θ ) = sin[( N + 1) θ/ θ/ 2) (4.25) of the CUE N +1 ensemble. Power spectrum in the tuned circular unitary ensemble Proof. While the determinantal form [Eq. (4.23)] of spectral correlation functions is auniversal manifestation of the β = 2 symmetry of the circular ensemble [5, 6], a preciseform of the two-point scalar kernel κ N ( θ, θ ′ ) depends on peculiarities of the TCUE N probability measure encoded in the weight function ( z = e iθ ) W ( z ) = 12 | − z | = 1 − cos θ (4.26)characterizing the TCUE N measure in Eq. (2.9). For aesthetic reasons, it is convenientto compute a scalar kernel κ N ( θ, θ ′ ) in terms of polynomials { ψ j ( z ) } orthonormal onthe unit circle | z | = 112 iπ ˛ | z | =1 dzz W ( z ) ψ ℓ ( z ) ψ m ( z ) = δ ℓm (4.27)with respect to the weight function W ( z ). In such a case, a scalar kernel is given byeither of the two representations ( w = e iθ ′ ): κ N ( θ, θ ′ ) = p W ( z ) W ( w ) N − X ℓ =0 ψ ℓ ( z ) ψ ℓ ( w ) (4.28)= p W ( z ) W ( w ) ψ N ( w ) ψ N ( z ) − ψ ∗ N ( w ) ψ ∗ N ( z )¯ wz − . (4.29)Equation (4.29), containing reciprocal polynomials ψ ∗ ℓ ( z ) = z ℓ ψ ℓ (1 / ¯ z ) , (4.30)follows from Eq. (4.28) by virtue of the Christoffel-Darboux identity [40].Since for the TCUE N weight function Eq. (4.26), the orthonormal polynomials areknown as Szeg¨o-Askey polynomials (see § 18 in Ref. [41]), ψ ℓ ( z ) = s ℓ + 1)( ℓ + 2) F ( − ℓ, − ℓ ; z ) , (4.31)the reciprocal Szeg¨o-Askey polynomials are readily available, too: ψ ∗ ℓ ( z ) = r ℓ + 1) ℓ + 2 F ( − ℓ, − ℓ − z ) . (4.32)Hence, Eqs. (4.29), (4.31) and (4.32) furnish an explicit expression for the TCUE N scalar kernel κ N ( θ, θ ′ ).This being said, we would like to represent the TCUE N scalar kernel in a moresuggestive form. To do so, we notice that Szeg¨o-Askey polynomials Eq. (4.31) admityet another representation ψ ℓ ( z ) = s ℓ + 1)( ℓ + 2) ℓ +1 X j =1 jz j − . (4.33)Substituting it further into Eq. (4.28), one obtains: κ N ( θ, θ ′ ) = 2 iN + 1 e − i ( θ − θ ′ ) / sin[ θ/ 2] sin[ θ ′ / θ − θ ′ ) / N X j =0 N X k =0 ( N − j − k ) z j ¯ w k (4.34)= 2 iN + 1 e − i ( θ − θ ′ ) / sin[ θ/ 2] sin[ θ ′ / θ − θ ′ ) / N X j =0 N X k =0 (cid:18) N − z ∂∂z − ¯ w ∂∂ ¯ w (cid:19) z j ¯ w k . (4.35) Power spectrum in the tuned circular unitary ensemble N sine-kernel S N ( θ ) = sin( N θ/ θ/ 2) = e − i ( N − θ/ N − X j =0 z j , (4.36)the above can further be reduced to κ N ( θ, θ ′ ) = 2 N + 1 e i ( N − θ − θ ′ ) / sin[ θ/ 2] sin[ θ ′ / θ − θ ′ ) / × (cid:18) ∂∂θ ′ − ∂∂θ (cid:19) S N +1 ( θ ) S N +1 ( θ ′ ) . (4.37)Calculating derivatives therein, we derive κ N ( θ, θ ′ ) = e i ( N − θ − θ ′ ) / (cid:18) S N +1 ( θ − θ ′ ) − N + 1 S N +1 ( θ ) S N +1 ( θ ′ ) (cid:19) . (4.38)Spotting that the phase factor in Eq. (4.38) does not contribute to the determinant inEq. (4.23) completes the proof. Remark 4.6. An alternative determinantal representation of spectral correlationfunctions in the TCUE N can be established if one views the JPDF of the TCUE N asthe one of the traditional CUE N +1 ensemble, whose lowest eigenangle is conditionedto stay at zero, as spelt out below. (cid:4) Lemma 4.7. For ℓ = 1 , . . . , N , the ℓ -point correlation function, Eq. (4.22), in the TCUE N ensemble admits the determinantal representation R ℓ,N ( θ , . . . , θ ℓ ) = 1 N + 1 det ≤ i,j ≤ ℓ +1 [ S N +1 ( θ i − θ j )] (cid:12)(cid:12)(cid:12) θ ℓ +1 =0 , (4.39) where S N +1 ( θ ) is the CUE N +1 sine-kernel: S N +1 ( θ ) = sin[( N + 1) θ/ θ/ . (4.40) Proof. Equation (4.39) is self-evident as the determinant therein is the ( ℓ + 1)-pointcorrelation function in the CUE N +1 with one of the eigen-angles conditioned to stayat zero whilst the denominator is the CUE N +1 mean density S N +1 (0) = N + 1. Proposition 4.8. The generating function Φ N ( ϕ ; ζ ) in Eq. (2.12) of Theorem 2.7admits a Fredholm determinant representation Φ N ( ϕ ; ζ ) = det (cid:2) − ζ ˆ κ (0 ,ϕ ) N (cid:3) , (4.41) where ˆ κ (0 ,ϕ ) N is an integral operator defined by (cid:2) ˆ κ (0 ,ϕ ) N f (cid:3) ( θ ) = ˆ ϕ dθ π κ N ( θ , θ ) f ( θ ) , (4.42) whilst κ N is the TCUE N two-point scalar kernel specified in Lemma 4.5.Proof. To derive a Fredholm determinant representation of the power spectrum, weturn to Eq. (4.15) rewriting it as a sumΦ N ( ϕ ; ζ ) = N X ℓ =0 (cid:18) Nℓ (cid:19) (cid:18) − ζ ˆ ϕ (cid:19) ℓ (cid:18) ˆ π (cid:19) N − ℓ N Y j =1 dθ j π P N ( θ , . . . , θ N ) . Power spectrum in the tuned circular unitary ensemble N − ℓ ) integrations, we obtainΦ N ( ϕ ; ζ ) = N X ℓ =0 ( − ζ ) ℓ ℓ ! ℓ Y j =1 ˆ ϕ dθ j π R ℓ,N ( θ , . . . , θ ℓ ) , where R ℓ,N ( θ , . . . , θ ℓ ) is the ℓ -point correlation function in TCUE N given by Eq. (4.22).Its determinant representation Eq. (4.23) yields the expansionΦ N ( ϕ ; ζ ) = N X ℓ =0 ( − ζ ) ℓ ℓ ! ℓ Y j =1 ˆ ϕ dθ j π det ≤ i,j ≤ ℓ [ κ N ( θ i , θ j )] . Here, κ N ( θ, θ ′ ) is the two-point scalar kernel of the TCUE N ensemble, see Lemma 4.5for its explicit form. Further, consulting, e.g., Appendix in Ref. [42], one identifies asought Fredholm determinant representation given by Eqs. (4.41) and (4.42).A Fredholm determinant representation of the power spectrum is particularly usefulfor asymptotic analysis of the power spectrum in the deep ‘infrared’ limit ω ≪ ζ = 1 − z ≪ TCUE N as a Toeplitz determinant To analyse the power spectrum in the limit N → ∞ for 0 < ω < π being kept fixed,it is beneficial to represent the generating function Φ N ( ϕ ; ζ ) [Eq. (4.17)] entering theexact solution Eq. (2.12) with ζ = 1 − z in the form of a Toeplitz determinant withFisher-Hartwig singularities. Proposition 4.9. The generating function Φ N ( ϕ ; ζ ) in Eq. (2.12) of Theorem 2.7admits a Toeplitz determinant representation Φ N ( ϕ ; ζ ) = e iϕ ˜ ωN N + 1 D N [ f ˜ ω ( z ; ϕ )] , (4.43) where ˜ ω = ω/ π , and D N [ f ˜ ω ( z ; ϕ )] = det ≤ j,ℓ ≤ N − iπ ˛ | z | =1 dzz z ℓ − j f ˜ ω ( z ; ϕ ) ! (4.44) is the Toeplitz determinant whose Fisher-Hartwig symbol f ˜ ω ( z ; ϕ ) = | z − z | (cid:18) z z (cid:19) ˜ ω g z , ˜ ω ( z ) g z , − ˜ ω ( z ) (4.45) possesses power-type singularity at z = z = e iϕ/ and jump discontinuities g z j , ± ˜ ω ( z ) = (cid:26) e ± iπ ˜ ω , ≤ arg z < arg z j e ∓ iπ ˜ ω , arg z j ≤ arg z < π (4.46) at z = z , with z = e i (2 π − ϕ/ .Proof. Start with the multiple integral representation Eq. (4.17) and make use ofAndr´eief’s formula [43, 44] N Y j =1 ˆ L dθ j π w ( θ j ) det ≤ j,ℓ ≤ N [ f j − ( θ ℓ )] det ≤ j,ℓ ≤ N [ g j − ( θ ℓ )]= N ! det ≤ j,ℓ ≤ N (cid:18) ˆ L dθ π w ( θ ) f j − ( θ ) g ℓ − ( θ ) (cid:19) (4.47) Power spectrum in quantum chaotic systems: Large- N limit w ( θ ) = (1 − ζ Θ( θ )Θ( ϕ − θ )) | − e iθ | , integrationdomain is chosen to be L = (0 , π ), and f j − ( θ ) = g j − ( θ ) = e i ( j − θ , to deriveΦ N ( ϕ ; ζ ) = 1 N + 1 det ≤ j,ℓ ≤ N − [ M j − ℓ ( ϕ ; ζ )] , (4.48)where M j − ℓ ( ϕ ; ζ ) = (cid:18) ˆ π − ζ ˆ ϕ (cid:19) dθ π | − e iθ | e − i ( j − ℓ ) θ . (4.49)Introduce a new integration variable z = e iθ in Eq. (4.49), adopt the standardterminology and notation of Refs. [45, 46] to figure out equivalence of Eqs. (4.48)and (4.49) to the statement of the proposition. 5. Power spectrum in quantum chaotic systems: Large- N limit In the limit N → ∞ , the exact solution for the TCUE N power spectrum shouldconverge to a universal law. To determine it, we shall perform an asymptotic analysisof the exact solution Eqs. (2.12) and (2.13), stated in Theorem 2.7, with the generatingfunction Φ N ( ϕ ; ζ ) being represented as a Toeplitz determinant specified in Proposition4.9. To perform the integral in Eq. (2.12) in the limit N → ∞ , uniform asymptotics ofthe Toeplitz determinant Eq. (4.44) are required in the subtle regime of two mergingsingularities. In our case, one singularity is of a root type while the other one is ofboth root and jump types. Relevant uniform asymptotics were recently studied in greatdetail by Claeys and Krasovsky [46] who used the Riemann-Hilbert technique.Two different, albeit partially overlapping, asymptotic regimes in ϕ can be identified. Asymptotics at the ‘left edge’. —Defining the left edge as the domain 0 ≤ ϕ < ϕ ,where ϕ is sufficiently small ∗ , the following asymptotic expansion holds uniformly as N → ∞ (see Theorems 1.5 and 1.8 in Ref. [46])ln D N [ f ˜ ω ( z ; ϕ )] = ln N − i ( N − ωϕ − ω ln (cid:18) sin( ϕ/ ϕ/ (cid:19) + ˆ − iNϕ dss σ ( s ) + O ( N − ω ) , (5.1)so thatΦ N ( ϕ ; ζ ) = e i ˜ ωϕ (cid:18) sin( ϕ/ ϕ/ (cid:19) − ω exp ˆ − iNϕ dss σ ( s ) ! (cid:0) O ( N − ω ) (cid:1) . (5.2)Here ˜ ω = ω/ π is a rescaled frequency so that z = 1 − ζ = e iπ ˜ ω . The function σ ( s ) isthe fifth Painlev´e transcendent defined as the solution to the nonlinear equation s ( σ ′′ ) = (cid:0) σ − sσ ′ + 2( σ ′ ) (cid:1) − σ ′ ) (cid:0) ( σ ′ ) − (cid:1) (5.3) ∗ In fact, here ϕ = 2 π − ǫ with ǫ > Power spectrum in quantum chaotic systems: Large- N limit ♯σ ( s ) = − ˜ ωs − ω + sγ ( s )1 + γ ( s ) + O (cid:16) e − i | s | | s | − ω (cid:17) + O (cid:0) | s | − (cid:1) as s → − i ∞ (5.4)and σ ( s ) = O ( | s | ln | s | ) as s → − i + . (5.5)The function γ ( s ) in Eq. (5.4) equals γ ( s ) = 14 (cid:12)(cid:12)(cid:12) s (cid:12)(cid:12)(cid:12) − ω ) e − i | s | e iπ Γ(2 − ˜ ω )Γ(1 − ˜ ω )Γ(1 + ˜ ω )Γ(˜ ω ) . (5.6)The above holds for 0 ≤ ˜ ω < / Remark 5.1. Following Ref. [46], we notice that in Eqs. (5.1) and (5.2) the path ofintegration in the complex s -plane should be chosen to avoid a finite number of poles { s j } of σ ( s ) corresponding to zeros { ϕ j = is j /N } in the asymptotics of the Toeplitzdeterminant D N [ f ˜ ω ( z ; ϕ )]. For the specific Fisher-Hartwig symbol Eq. (4.45) we expect { s j } to be the empty set; numerical analysis of D N [ f ˜ ω ( z ; ϕ )] suggests that its zerosstay away from the real line. (cid:4) Asymptotics in the ‘bulk’. —Defining the ‘bulk’ as the domain Ω( N ) /N ≤ ϕ < ϕ , where ϕ is sufficiently small, and Ω( x ) is a positive smooth function such that Ω( N ) → ∞ whilst Ω( N ) /N → N → ∞ , the following asymptotic expansion holds uniformly (see Theorem 1.11 in Ref. [46]): D N [ f ˜ ω ( z ; ϕ )] = N − ω G ˜ ω e i ˜ ωϕ e − i ˜ ωπ (cid:12)(cid:12)(cid:12) (cid:16) ϕ (cid:17)(cid:12)(cid:12)(cid:12) − ω (cid:0) O (cid:0) Ω( N ) − ω (cid:1)(cid:1) (5.7)so thatΦ N ( ϕ ; ζ ) = N − ω G ˜ ω e i ˜ ωϕ ( N +1) e − i ˜ ωπ (cid:12)(cid:12)(cid:12) (cid:16) ϕ (cid:17)(cid:12)(cid:12)(cid:12) − ω (cid:0) O (cid:0) Ω( N ) − ω (cid:1)(cid:1) . (5.8)Here, G ˜ ω is a known function of ˜ ωG ˜ ω = G (2 + ˜ ω ) G (2 − ˜ ω ) G (1 + ˜ ω ) G (1 − ˜ ω ) (5.9)with G ( · · · ) being the Barnes’ G -function. The above holds for 0 ≤ ˜ ω < / 2. Theleading term in Eqs. (5.7) and (5.8) is due to Ehrhardt [48]. Remark 5.2. Since both asymptotic expansions [Eq. (5.1) and (5.7)] hold uniformlyin the domain Ω( N ) /N ≤ ϕ < ϕ , the following integral identity for σ ( s ) should hold:lim T → + ∞ ˆ − iT dss σ ( s ) − i ˜ ωT + 2˜ ω ln T ! = − iπ ˜ ω + ln G ˜ ω , (5.10)see Eq. (1.26) in Ref. [46]. Had this global condition been derived independently, itwould have provided an alternative route to producing the ‘bulk’ asymptotics out ofthose known in the edge region. Notice that as T → ∞ , the boundary conditionEq. (5.4) implies a stronger statement: ˆ − iT dss σ ( s ) − i ˜ ωT + 2˜ ω ln T = − iπ ˜ ω + ln G ˜ ω + O ( T − ) . (5.11) (cid:4) ♯ Notice that, in distinction to Ref. [46], we kept two reminder terms in Eq. (5.4) – oscillatory andnon-oscillatory, even though the latter term is subleading. The reason for this is that the function σ ( s ) will subsequently appear in the integral Eq. (5.11) which will make the non-oscillatory reminderterm dominant. Power spectrum in quantum chaotic systems: Large- N limit In doing the large- N asymptotic analysis of our exact solution for the power spectrum[Eqs. (2.12) and (2.14)], we shall encounter a set of integrals I N,k ( ζ ) = N ˆ π dϕ π ϕ k Φ N ( ϕ ; ζ ) , (5.12)where k is a non-negative integer and Φ N ( ϕ ; ζ ) is given by Eq. (4.17). We shallspecifically be interested in k = 0 and 1. Lemma 5.3. In the notation of Eq. (5.12), we have: I N, ( ζ ) = NN + 1 1 − (1 − ζ ) N +1 ζ . (5.13) Equation (5.13) is exact for any ζ ∈ C .Proof. To compute the integral Eq. (5.12) at k = 0, we invoke the expansion Eq. (4.15)of Φ N ( ϕ ; ζ ) in terms of probabilities E N ( ℓ ; ϕ ) of observing exactly ℓ eigenangles ofTCUE N in the interval (0 , ϕ ), I N, ( ζ ) = N n X ℓ =0 (1 − ζ ) ℓ ˆ π dϕ π E N ( ℓ ; ϕ ) . (5.14)The integral above can readily be calculated by performing integration by parts: ˆ π dϕ π E N ( ℓ ; ϕ ) = δ ℓ,N − ˆ π dϕ π ϕ ddϕ E N ( ℓ ; ϕ )= δ ℓ,N + 12 π ˆ π dϕ π ϕ ( p ℓ +1 ( ϕ ) − p ℓ ( ϕ )) . (5.15)In the second line, we have used the relation Eq. (3.5) which, in the context of TCUE N ,acquires the multiplicative factor 1 / π in its r.h.s.; there, p ℓ ( ϕ ) is the probabilitydensity of the ℓ -th ordered eigenangle. Further, identifying (see Corollary 4.3) ˆ π dϕ π ϕ p ℓ ( ϕ ) = h θ ℓ i = (cid:26) ℓ ∆ , ℓ = 1 , . . . , N ;0 , ℓ = 0 , N + 1. (5.16)where ∆ = 2 π/ ( N + 1) is the mean spacing, we conclude that ˆ π dϕ π E N ( ℓ ; ϕ ) = δ ℓ,N + h θ ℓ +1 i − h θ ℓ i π = 1 N + 1 (5.17)for all ℓ = 0 , . . . , N . Substitution of Eq. (5.17) into Eq. (5.14) ends the proof. Remark 5.4. The fact that I N, ( ζ ) could be expressed in terms of elementary functionscan be traced back to stationarity of level spacings in the TCUE N . For one, in theCUE N , an analogue of I N, ( ζ ) would have to be expressed in terms of the six Painlev´efunction. (cid:4) The integral I N,k . —Unfortunately, exact calculation of the same ilk is not readilyavailable for I N,k with k = 1. For this reason we would like to gain an insightfrom Eq. (5.13) as N → ∞ , which, eventually, is the limit we are mostly concernedwith. To this end, we extract the leading order behavior of I N, ( ζ ) on the unit circle | z | = | − ζ | = 1, I N, ( ζ ) = 1 ζ + (1 − ζ ) N ζ + O ( N − ) , (5.18) Power spectrum in quantum chaotic systems: Large- N limit − ζ ) N = z N = e iπ ˜ ωN , (1 − ζ ) N ζ are contributed by a vicinity of ϕ = 2 π in the integral Eq. (5.12) with k = 0. (ii) Onthe contrary, such a prefactor is missing in the term coming from a vicinity of ϕ = 0,1 ζ . The contribution from the bulk of the integration domain appears to be negligible dueto strong oscillations e i ˜ ωϕN of the integrand therein, see Eq. (5.8).Equipped with these observations, we shall now proceed with an alternative, large- N ,analysis of I N,k ( ζ ) for k = 0 and k = 1, where terms of the same structure (with andwithout strongly oscillating prefactor) will appear. Aimed at the analysis of the powerspectrum [Eq. (2.12)], whose representation contains a very particular z -operator, weshall only be interested in the leading order contributions to both terms. Notably, eventhough for k = 1 a non-oscillating term is subleading as compared to an oscillatingterm, we shall argue that its contribution should still be kept.To proceed with the large- N analysis of I N,k , we first rewrite the integral Eq. (5.12)as a sum of two I N,k ( ζ ) = I (1) N,k ( ζ ) + I (2) N,k ( ζ ) (5.19)such that I (1) N,k ( ζ ) = N ˆ π dϕ π ϕ k (cid:0) Φ N ( ϕ ; ζ ) − Φ E N ( ϕ ; ζ ) (cid:1) (5.20)and I (2) N,k ( ζ ) = N ˆ π dϕ π ϕ k Φ E N ( ϕ ; ζ ) . (5.21)Here, Φ E N ( ϕ ; ζ ) is an arbitrary integrable function; it will be specified later on.Prompted by the ‘edge’ and ‘bulk’ asymptotic expansions of Φ N ( ϕ ; ζ ) [Eqs. (5.2) and(5.8)], we split the integral in Eq. (5.20) into three pieces I (1) N,k ( ζ ) = L (1) N,k ( ζ ) + C (1) N,k ( ζ ) + R (1) N,k ( ζ ) , (5.22)where L (1) N,k ( ζ ) = N ˆ Ω( N ) /N dϕ π ϕ k (cid:0) Φ N ( ϕ ; ζ ) − Φ E N ( ϕ ; ζ ) (cid:1) , (5.23) C (1) N,k ( ζ ) = N ˆ π − Ω( N ) /N Ω( N ) /N dϕ π ϕ k (cid:0) Φ N ( ϕ ; ζ ) − Φ E N ( ϕ ; ζ ) (cid:1) , (5.24) R (1) N,k ( ζ ) = N ˆ π π − Ω( N ) /N dϕ π ϕ k (cid:0) Φ N ( ϕ ; ζ ) − Φ E N ( ϕ ; ζ ) (cid:1) , (5.25)correspondingly.To facilitate the asymptotic analysis, we would ideally like to choose Φ E N ( ϕ ; ζ ) insuch a way that the contribution of the ‘bulk’ integral C (1) N,k ( ζ ) into I (1) N,k ( ζ ) becomesnegligible. For the time being, let us assume that such a function is given by the leadingterm in Eq. (5.8),Φ E N ( ϕ ; ζ ) = N − ω G ˜ ω e i ˜ ωϕ ( N +1) e − i ˜ ωπ (cid:12)(cid:12)(cid:12) (cid:16) ϕ (cid:17)(cid:12)(cid:12)(cid:12) − ω . (5.26) Power spectrum in quantum chaotic systems: Large- N limit I (1) N,k ( ζ ) will be dominated by the contributions coming from the ‘left-edge’[ L (1) N,k ( ζ )] and the ‘right-edge’ [ R (1) N,k ( ζ )] parts of the integration domain. In fact, thecontributions of the left and the right edges are related to each other; an exact relationbetween the two will be worked out and made explicit later on. The integral I (1) N,k ( ζ ) . —Restricting ourselves to k = 0 and 1, we first consider theleft-edge part L (1) N,k ( ζ ). Substituting Eqs. (5.2) and (5.26) into Eq. (5.23), we find, as N → ∞ : L (1) N,k ( ζ ) = N ˆ Ω( N ) /N dϕ π ϕ k e i ˜ ωϕ " (cid:18) sin( ϕ/ ϕ/ (cid:19) − ω exp ˆ − iNϕ dss σ ( s ) ! × (cid:0) O ( N − ω ) (cid:1) − N − ω (2 sin( ϕ/ − ω e i ˜ ωϕN e − i ˜ ωπ G ˜ ω . (5.27)To get rid of N in the integral over the Painlev´e V transcendent, we make thesubstitution λ = N ϕ to rewrite L (1) N,k ( ζ ) in the form L (1) N,k ( ζ ) = ˆ Ω( N )0 dλ π λ k N k e i ˜ ωλ/N " (cid:18) sin( λ/ (2 N )) λ/ (2 N ) (cid:19) − ω exp ˆ − iλ dss σ ( s ) ! × (cid:0) O ( N − ω ) (cid:1) − N − ω (2 sin( λ/ (2 N ))) − ω e i ˜ ωλ e − i ˜ ωπ G ˜ ω . (5.28)Noting that λ/N = O (Ω( N ) /N ) tends to zero as N → ∞ , we can further approximate L (1) N,k ( ζ ) as L (1) N,k ( ζ ) = 1 N k ˆ Ω( N )0 dλ π λ k − ω e i ˜ ωλ " exp ˆ − iλ dss σ ( s ) − i ˜ ωλ + 2˜ ω ln λ ! − e − i ˜ ωπ G ˜ ω + O (Ω( N ) k +1 N − k − ω ) + O (Ω( N ) k +2 N − k − ) . (5.29)Next, one may use Eq. (5.11) to argue that replacing Ω( N ) with infinity in Eq. (5.29)produces an error term of the order O (Ω( N ) k − − ω N − k ): L (1) N,k ( ζ ) = 1 N k ˆ ∞ dλ π λ k − ω e i ˜ ωλ " exp ˆ − iλ dss σ ( s ) − i ˜ ωλ + 2˜ ω ln λ ! − e − i ˜ ωπ G ˜ ω + O (Ω( N ) k +1 N − k − ω )+ O (Ω( N ) k +2 N − k − ) + O (Ω( N ) k − − ω N − k ) . (5.30)Further, choosing Ω( N ) to be a slowly growing function, Ω( N ) = ln N , one readilyverifies that the third error term in Eq. (5.30) is a dominant one out of the three as0 < ˜ ω < / 2. Yet, it is smaller as compared to the integral in Eq. (5.30) by a factorΩ( N ) k − − ω that tends to zero as N → ∞ . Thus, in the leading order, we derive: L (1) N,k ( ζ ) = 1 N k L (1) k ( ζ ) + o ( N − k ) , (5.31)where L (1) k ( ζ ) = ˆ ∞ dλ π λ k − ω e i ˜ ωλ " exp ˆ − iλ dss σ ( s ) − i ˜ ωλ + 2˜ ω ln λ ! − e − i ˜ ωπ G ˜ ω , (5.32) Power spectrum in quantum chaotic systems: Large- N limit k = 0 and 1.Now, let us turn to the ‘right-edge’ integral R (1) N,k ( ζ ). Due to the symmetry relationEq. (4.18) shared by Φ E N ( ϕ ; ζ ) too, we realize that the contributions of the left and theright edges are related to each other: R (1) N,k ( ζ ) = N (1 − ¯ ζ ) N ˆ Ω( N ) /N dϕ π (2 π − ϕ ) k (cid:0) Φ N ( ϕ ; ζ ) − Φ E N ( ϕ ; ζ ) (cid:1) . (5.33)Considering the integral in the r.h.s. of Eq. (5.33) along the lines of the previousanalysis, we conclude that the following formula holds as N → ∞ : R (1) N,k ( ζ ) = (1 − ζ ) N R (1) k ( ζ ) + o (1) , (5.34)where R (1) k ( ζ ) = (2 π ) k ˆ ∞ dλ π λ − ω e i ˜ ωλ " exp ˆ − iλ dss σ ( s ) − i ˜ ωλ + 2˜ ω ln λ ! − e − i ˜ ωπ G ˜ ω , (5.35)with k = 0 and 1.Combining Eqs. (5.31), (5.32), (5.34) and (5.35), we end up with the asymptoticresult [Eq. (5.22)] I (1) N,k ( ζ ) N k L (1) k ( ζ ) + (1 − ζ ) N R (1) k ( ζ ) . (5.36)The notation was used here to stress that the r.h.s. contains each leading ordercontribution of both terms, the oscillating and the non-oscillating, as discussed in theparagraph prior to Eq. (5.19). The integral I (2) N,k ( ζ ) . —As soon as the function Φ E N ( ϕ ; ζ ) contains a strongly oscillatingfactor e i ˜ ωϕN , the integral I (2) N,k ( ζ ) in Eq. (5.21) can be calculated by the stationaryphase method [49]. Since there are no stationary points within the interval (0 , π ), theintegral is dominated by contributions L (2) N,k ( ζ ) and R (2) N,k ( ζ ), coming from the vicinitiesof ϕ = 0 and ϕ = 2 π , respectively. Lemma 5.5. Let I (2) N,k ( ζ ) be defined by Eqs. (5.21) and (5.26), where k is a fixednon-negative integer. Then, as N → ∞ , it can be represented in the following form: I (2) N,k ( ζ ) = L (2) N,k ( ζ ) + R (2) N,k ( ζ ) (5.37) where L (2) N,k ( ζ ) = 1 N k L (2) k + o ( N − k ) , (5.38) R (2) N,k ( ζ ) = (1 − ζ ) N R (2) k + o (1) , (5.39) and L (2) k ( ζ ) = G ˜ ω π e iπ ( k +1 − ω − ω ) / ˜ ω − k − ω Γ( k + 1 − ω ) , (5.40) R (2) k ( ζ ) = G ˜ ω π e iπ ( − ω +2˜ ω ) / ˜ ω − ω (2 π ) k Γ(1 − ω ) . (5.41) Proof. Apply the stationary phase method [49] to calculate the integral Eq. (5.21). Power spectrum in quantum chaotic systems: Large- N limit I (2) N,k ( ζ ) N k L (2) k ( ζ ) + (1 − ζ ) N R (2) k ( ζ ) , (5.42)compare with Eq. (5.36). The integral I N,k ( ζ ) . —The calculation above implies that the main integral of ourinterest admits an asymptotic representation I N,k ( ζ ) N k L k ( ζ ) + (1 − ζ ) N R k ( ζ ) (5.43)with k = 0 , L k ( ζ ) = L (1) k ( ζ ) + L (2) k ( ζ ) , (5.44) R k ( ζ ) = R (1) k ( ζ ) + R (2) k ( ζ ) . (5.45)We notice that both L k ( ζ ) = O (1) and R k ( ζ ) = O (1) and the factor (1 − ζ ) N = z N = e iπ ˜ ωN in Eq. (5.43) is a strongly oscillating function of ˜ ω as N → ∞ , in concert withthe discussion in the paragraph prior to Eq. (5.19). Remark 5.6. Our derivation of the main result of this Section, Eq. (5.43), was basedon the assumption that the choice of Φ E N ( ϕ ; ζ ) in the form Eq. (5.26) makes thecontribution of the ‘bulk’ integral C (1) N,k ( ζ ) [Eq. (5.24)] into I (1) N,k ( ζ ) negligible. If this is not the case, one should replace Φ E N with some ˜Φ E N by adding to Φ E N the higher-ordercorrections (up to O ( N − )) that can be obtained from the full asymptotic expansionof Φ N ( ϕ ; ζ ), see Remark 1.4 of Ref. [45]. Inclusion of these higher-order correctionswill reduce the contribution of C (1) N,k ( ζ ) to a negligible level as guaranteed by the roughupper-bound estimate | C (1) N,k ( ζ ) | = N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ π − Ω( N ) /N Ω( N ) /N dϕ π ϕ k (cid:0) Φ N ( ϕ ; ζ ) − Φ E N ( ϕ ; ζ ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N ˆ π dϕ π ϕ k (cid:12)(cid:12)(cid:12) Φ N ( ϕ ; ζ ) − ˜Φ E N ( ϕ ; ζ ) (cid:12)(cid:12)(cid:12) = O ( N − ) . (5.46)On the other hand, the proposed modification of Φ N ( ϕ ; ζ ) will produce corrections tothe functions L (1) N, , R (1) N, , L (2) N, and R (2) N, , which will clearly be subleading to thosecalculated in the leading order [see Eqs. (5.31), (5.34), (5.38), (5.39)]. For this reason,they will not affect the large- N analysis of the power spectrum where only O (1) termsare kept. (cid:4) Now we are in position to evaluate the power spectrum as N → ∞ . To proceed, westart with the exact, finite- N , representation S N ( ω ) = ( N + 1) πN Re (cid:26)(cid:18) z ∂∂z − N − − z − N − z (cid:19) z − z I N, ( ζ ) (cid:27) − ≈ S N ( ω ) (5.47)following from Eqs. (2.12) and (5.12). Substituting I N, given by Eq. (5.43) intoEq. (5.47) and taking into account the relation R ( ζ ) = 2 π R ( ζ ), following fromEqs. (5.45), (5.41) and (5.35), we derive, as N → ∞ : S N ( ω ) = − π Re ( z − z L ( ζ ) ) + 2Re ( z − z (cid:18) z N +1 ddz R ( ζ ) + R ( ζ )1 − z (cid:19) ) − ( ( z − z N ) | − z | ) + o (1) . (5.48) Power spectrum in quantum chaotic systems: Large- N limit N expansion of ≈ S N ( ω ) [Eq. (2.13)].Surprisingly, the last two terms in Eq. (5.48) cancel each other. This follows from theidentity L ( ζ ) = R ( ζ ) = 1 ζ (5.49)that can be identified by comparing Eq. (5.18) with Eq. (5.43) taken at k = 0. Thecancellation implies the N → ∞ result S ∞ ( ω ) = − π Re ( z − z L ( ζ ) ) . (5.50)Substituting Eqs. (5.44), (5.32) and (5.40) into Eq. (5.50), we derive S ∞ ( ω ) = 1 π Re ( e iπ ˜ ω e iπ ˜ ω − ˆ ∞ dλ π λ − ω e i ˜ ωλ " exp (cid:16) ˆ − iλ dss σ ( s ) − i ˜ ωλ + 2˜ ω ln λ (cid:17) − e − i ˜ ωπ G ˜ ω − G ˜ ω π e − iπ (˜ ω +˜ ω ) ˜ ω − ω Γ(2 − ω ) !) , (5.51)where ˜ ω = ω/ π is a rescaled frequency, and the function σ ( s ) is the fifth Painlev´etranscendent defined by Eqs. (5.3), (5.4) and (5.5). Equation (5.51) can be simplifieddown to S ∞ ( ω ) = A (˜ ω ) ( Im ˆ ∞ dλ π λ − ω e i ˜ ωλ × " exp ˆ − iλ − i ∞ dss (cid:0) σ ( s ) + s ˜ ω + 2˜ ω (cid:1)! − + B (˜ ω ) ) , (5.52)where the functions A (˜ ω ) and B (˜ ω ) are defined as in Eqs. (2.18) and (2.19). To deriveEq. (5.52) we have used the integral identity Eq. (5.10) to transform the exponentexp ˆ − iλ σ ( s ) s ds − i ˜ ωλ + 2˜ ω ln λ ! = G ˜ ω e − iπ ˜ ω × lim T →∞ exp " ˆ − iλ − iT σ ( s ) s ds + i ˜ ω ( T − λ ) + 2˜ ω ln( λ/T ) = G ˜ ω e − iπ ˜ ω exp " ˆ − iλ − i ∞ dss (cid:0) σ ( s ) + ˜ ωs + 2˜ ω (cid:1) . (5.53)Finally, we notice that σ ( s = − it ) = σ ( t ) satisfies Eq. (1.3) with ν = 1 supplementedby the boundary conditions Eqs. (2.20) and (2.21). With help of this, we recover thestatement of Theorem 2.9 from Eq. (5.52). (cid:3) Remark 5.7. Note that the global condition Eq. (5.11) ensures that the expression inthe square brackets in Eq. (5.51) exhibits O ( λ − ) behavior as λ → ∞ . This guaranteesthat the external λ -integral in Eq. (5.51) converges for any ˜ ω ∈ (0 , / (cid:4) Remark 5.8. Notice that Eq. (5.49) combined with Eqs. (5.45), (5.41) and (5.35)taken at k = 0, motivates the following conjecture. (cid:4) Conjecture 5.9. Let 0 < ˜ ω < / σ ( s ) be the solution of the fifth Painlev´etranscendent satisfying Eq. (5.3) and the boundary conditions Eq. (5.4) – (5.6). Thenthe following double integral relation holds ˆ ∞ dλ π λ − ω e i ˜ ωλ " exp ˆ − iλ dss σ ( s ) − i ˜ ωλ + 2˜ ω ln λ ! − e − i ˜ ωπ G ˜ ω = 11 − e πi ˜ ω − i G ˜ ω π e − iπ (˜ ω +˜ ω ) ˜ ω − ω Γ(1 − ω ) . (5.54) Power spectrum in quantum chaotic systems: Large- N limit (cid:4) Remark 5.10. To extend the proof of Theorem 2.9 for ω = π , one would have to usethe Theorem 1.12 of Ref. [46] instead of Theorems 1.5, 1.8 and 1.11 of the same paper.Since numerical calculations indicate that the power spectrum is continuous at ω = π ,we did not study this case analytically. (cid:4) Below, the universal law S ∞ ( ω ) for the power spectrum will be studied in the vicinityof ω = 0. In the language of S N ( ω ) this corresponds to performing a small- ω expansionafter taking the limit N → ∞ . Equation (2.17) will be the starting point of our analysis. Preliminaries. —Being interested in the small-˜ ω behavior of the power spectrumEq. (2.17), we observe that the functions A (˜ ω ) and B (˜ ω ), defined by Eqs. (2.18) and(2.19), admit the expansions A (˜ ω ) = 12 π ˜ ω + (cid:18) − γπ (cid:19) ˜ ω + O (˜ ω ) , (5.55) B (˜ ω ) = 12 + ˜ ω ln ˜ ω + ( γ − 1) ˜ ω + O (˜ ω ln ˜ ω ) , (5.56)so that the power spectrum, as ˜ ω → 0, can be written as S ∞ ( ω ) = 14 π ˜ ω + (cid:18) − π (cid:19) ˜ ω + 12 π ˜ ω ln ˜ ω + (cid:26) π ˜ ω + (cid:18) − γπ (cid:19) ˜ ω + O (˜ ω ) (cid:27) ˆΛ(˜ ω ) + O (˜ ω ln ˜ ω ) . (5.57)Here, ˆΛ(˜ ω ) denotes a small- ω expansion of the functionΛ(˜ ω ) = Im ˆ ∞ dλ π λ − ω e i ˜ ωλ (cid:20) exp (cid:18) − ˆ ∞ λ dtt (cid:0) σ ( t ) − i ˜ ωt + 2˜ ω (cid:1)(cid:19) − (cid:21) , (5.58)such that Λ(˜ ω ) = ˆΛ(˜ ω ) + O (˜ ω ) , (5.59)see Eqs. (2.17) and (5.57). Notice that convergence of the external λ -integral at infinityis ensured by the oscillating exponent e i ˜ ωλ . Small- ˜ ω ansatz for the fifth Painlev´e transcendent. —To proceed, we postulate the fol-lowing ansatz for a small- ω expansion of the fifth Painlev´e function σ ( t ): σ ( t ) = ˜ ωf ( t ) + ˜ ω f ( t ) + ˜ ω f ( t ) + · · · . (5.60)Here, the functions f k ( t ) with k = 1 , , . . . satisfy the equations t f ′′′ k + tf ′′ k + ( t − f ′ k − tf k ( t ) = F k ( t ) , (5.61)where F ( t ) = 0 , (5.62) F ( t ) = 4 f ( t ) f ′ − t ( f ′ ) , (5.63) F ( t ) = 4 f ( t ) f ′ + 4 f ′ f ( t ) − tf ′ f ′ , (5.64)etc. The above can easily be checked by substituting Eq. (5.60) into Chazy form [50, 51] t σ ′′′ ν + tσ ′′ ν + 6 t ( σ ′ ν ) − σ ν σ ′ ν + ( t − ν ) σ ′ ν − tσ ν = 0 (5.65) Power spectrum in quantum chaotic systems: Large- N limit ν = 1. The boundary conditions aregenerated by Eqs. (2.20) and (2.21): f ( t ) → t → , f ( t ) = it + o ( t ) as t → + ∞ , (5.66) f ( t ) → t → , f ( t ) → − t → + ∞ , (5.67) f ( t ) → t → , f ( t ) → t → + ∞ . (5.68)The third order differential equation (5.61) can be solved to bring f k ( t ) = (cid:18) t − t (cid:19) (cid:18) c ,k + ˆ t dxx F k ( x ) (cid:19) + e it t (cid:18) c ,k + ˆ t dx x e − ix ( − x + 2 ix + 2) F k ( x ) (cid:19) + e − it t (cid:18) c ,k + ˆ t dx x e ix ( − x − ix + 2) F k ( x ) (cid:19) . (5.69)This representation assumes that the integrals are convergent; integration constantshave to be fixed by the boundary conditions Eqs. (5.66), (5.67), (5.68), etc. Inparticular, we derive: f ( t ) = i t + 2 cos t − t , (5.70) f ( t ) = − − t + 2 πt − πt + 2 cos t + 8 cos tt − πt cos t − t ) t + 8 γ sin tt − t ) sin tt + 8 ln t sin tt − t ) t + 2 t Si( t ) + 4 cos t Si( t ) t , (5.71)where γ = lim n →∞ − ln n + n X k =1 k ! ≃ . f ( t ) = (cid:18) t − t (cid:19) ˆ ∞ t dxx F ( x ) − tt ˆ ∞ dxx F ( x )+ i Im (cid:26) e it t ˆ t dxx e − ix ( − x + 2 ix + 2) F ( x ) (cid:27) . (5.73)Here, the function F ( t ) is known explicitly from Eqs. (5.64), (5.70) and (5.71). Wenotice that f ( t ) ∈ i R , f ( t ) ∈ R , f ( t ) ∈ i R , and F ( t ) ∈ R , F ( t ) ∈ i R . Representation of ˆΛ(˜ ω ) as a partial sum .—Having determined the functions f ( t ), f ( t )and f ( t ), we now turn to the small- ω analysis of Λ(˜ ω ) [Eq. (5.58)]. Expanding theexpression in square brackets in small ω , we obtain:exp (cid:18) − ˆ ∞ λ dtt (cid:0) σ ( t ) − i ˜ ωt + 2˜ ω (cid:1)(cid:19) − − ˜ ω G ( λ ) − ˜ ω G ( λ ) − ˜ ω G ( λ ) − · · · . (5.74) Power spectrum in quantum chaotic systems: Large- N limit G k ( λ ) can be evaluated explicitly in terms of integrals containing f k ( λ )defined in Eq. (5.60). For example, G ( λ ) = F ( λ ) , (5.75) G ( λ ) = − F ( λ ) + F ( λ ) , (5.76) G ( λ ) = F ( λ ) − F ( λ ) F ( λ ) + 16 F ( λ ) . (5.77)Here, F ( λ ) = ˆ ∞ λ dtt ( f ( t ) − it ) = − i (cid:18) − cos λλ + π − Si( λ ) (cid:19) , (5.78) F ( λ ) = ˆ ∞ λ dtt ( f ( t ) + 2) , (5.79)and F ( λ ) = ˆ ∞ λ dtt f ( t ) . (5.80)Notice that F ( λ ) ∈ i R , F ( λ ) ∈ R , F ( λ ) ∈ i R and, hence, G ( λ ) ∈ i R , G ( λ ) ∈ R , G ( λ ) ∈ i R . Substituting Eq. (5.74) into Eq. (5.58), we split Λ(˜ ω ) into a partial sumΛ(˜ ω ) = Λ (˜ ω ) + Λ (˜ ω ) + Λ (˜ ω ) + · · · , (5.81)where Λ k (˜ ω ) = − ˜ ω k Im ˆ ∞ dλ π λ − ω e i ˜ ωλ G k ( λ ) . (5.82)A small-˜ ω expansion of Λ k (˜ ω ) is of our immediate interest. Calculation of ˆΛ (˜ ω ) . —Equations (5.82), (5.75) and (5.78) yieldΛ (˜ ω ) = 2˜ ω ˆ ∞ dλ π λ − ω cos(˜ ωλ ) (cid:18) − cos λλ + π − Si( λ ) (cid:19) . (5.83)Performing the integral, we obtain:Λ (˜ ω ) = 1 π ˜ ω Γ( − ω ) sin( π ˜ ω ) ( (1 − ˜ ω ) ω − + (1 + ˜ ω ) ω − − ω ω − − − ω − ˜ ω F (cid:18) − ˜ ω , − ˜ ω , − ˜ ω ; 12 , − ˜ ω ; ˜ ω (cid:19) ) . (5.84)Its small-˜ ω expansion Λ (˜ ω ) = ˜ ω + O (˜ ω ) bringsˆΛ (˜ ω ) = ˜ ω , (5.85)see Eq. (5.59) for the definition of ˆΛ( ω ). Estimate of Λ k (˜ ω ) . —To treat Λ k (˜ ω ) for k ≥ 2, we split it into two partsΛ k (˜ ω ) = A k (˜ ω, T ) + B k (˜ ω, T ) , (5.86)where A k (˜ ω, T ) = − ˜ ω k Im ˆ T dλ π λ − ω e i ˜ ωλ G k ( λ ) , (5.87) B k (˜ ω, T ) = − ˜ ω k Im ˆ ∞ T dλ π λ − ω e i ˜ ωλ G k ( λ ) . (5.88) Power spectrum in quantum chaotic systems: Large- N limit T is an arbitrary positive number to be taken to infinity in the end.Since a small-˜ ω expansion of A k (˜ ω, T ) is well justified for any finite T , see e.g.Eq. (5.92) below, we conclude that A k (˜ ω, T ) = O (˜ ω k ) . (5.89)To estimate B k (˜ ω, T ), we refer to Remark 5.7 which implies that G k ( λ ) = O ( λ − ) as λ → ∞ . Replacing G k ( λ ) with 1 /λ in Eq. (5.88), we perform the integration by partstwice in the resulting integral ˆ ∞ T dλ π e i ˜ ωλ λ ω = − e i ˜ ωT iπ ˜ ω T − ω + 1 π e i ˜ ωT T − − ω − ω ) ˆ ∞ T dλ π e i ˜ ωλ λ ω (5.90)to conclude that it is of order O (˜ ω − ). This entails B k (˜ ω, T ) = O (˜ ω k − ) . (5.91)Since we are interested in calculating Λ(˜ ω ) up to the terms O (˜ ω ), see Eq. (5.59), weneed to consider A k (˜ ω, T ) and B k +1 (˜ ω, T ) for k ≤ Calculation of ˆΛ (˜ ω ) . — A small-˜ ω expansion of A (˜ ω, T ) brings A (˜ ω, T ) = − ˜ ω Im ˆ T dλ π λ (cid:18) i ˜ ωλ − ω ln λ − 12 ˜ ω λ + O (˜ ω ) (cid:19) G ( λ ) . (5.92)Since G ( λ ) ∈ R , we even conclude that A (˜ ω, T ) = O (˜ ω ) . (5.93)For this reason, A (˜ ω, T ) does not contribute to ˆΛ (˜ ω ).Evaluation of B (˜ ω, T ), given by B (˜ ω, T ) = − ˜ ω ˆ ∞ T dλ π λ − ω sin(˜ ωλ ) G ( λ ) , (5.94)is more involved. A simplification comes from the fact that, at some point, we shall let T tend to infinity. For this reason, it suffices to consider a large- λ expansion of G ( λ )in the integrand. Straightforward calculations bring F ( λ ) = − iλ − i sin λλ + O (cid:18) cos λλ (cid:19) , (5.95) F ( λ ) = − λ + 8 cos λ ln λλ + 2(4 γ − 1) cos λλ + O (cid:18) ln λλ (cid:19) . (5.96)Equation (5.95) is furnished by the large- λ expansion of Eq. (5.78). To derive Eq. (5.96),we first calculated the integral Eq. (5.79) replacing an integrand therein with its large- t asymptotics, and then expanded the resulting expression in parameter λ → ∞ . Byvirtue of Eq. (5.76), this yields G ( λ ) = − λ + 8 cos λ ln λλ + 2(4 γ − 1) cos λλ + O (cid:18) ln λλ (cid:19) . (5.97)The expansion Eq. (5.97), being substituted into Eq. (5.94), generates two families ofintegrals: I j (˜ ω, T ) = ˆ ∞ T dλ π sin[(˜ ω + j ) λ ] λ ω (5.98)with j = 0 , ± K j (˜ ω, T ) = ˆ ∞ T dλ π ln λ sin[(˜ ω + j ) λ ] λ ω (5.99) Power spectrum in quantum chaotic systems: Large- N limit j = ± 1, such that B (˜ ω, T ) = ˜ ω n I (˜ ω, T ) − (4 γ − I − (˜ ω, T ) + I (˜ ω, T )] − K − (˜ ω, T ) + K (˜ ω, T )] o . (5.100)To determine a small-˜ ω expansion of B (˜ ω, T ), we shall further concentrate on small-˜ ω expansions of its constituents, I (˜ ω, T ), I ± (˜ ω, T ) and K ± (˜ ω, T ). (a) .—The function I (˜ ω, T ) can be evaluated exactly, I (˜ ω, T ) = 14 π sin( π ˜ ω ) ˜ ω ω − Γ(1 − ω ) − π T − ω ˜ ω − ω F (cid:18) − ˜ ω ; 32 , − ˜ ω ; − T ω (cid:19) . (5.101)Expanding this result around ˜ ω = 0 we derive I (˜ ω, T ) = 14 + O (˜ ω ) . (5.102) (b) .—To analyze a small-˜ ω expansion I j =0 (˜ ω, T ) = α ( j, T ) + ˜ ω α ( j, T ) + O (˜ ω ) , (5.103)we proceed in two steps. First, we determine the coefficient α ( j, T ) directly fromEq. (5.98) α ( j, T ) = I j =0 (0 , T ) = ˆ ∞ T dλ π sin( jλ ) λ < ∞ , ∀ T > , (5.104)to deduce the relation ( j = 0) α ( − j, T ) = − α ( j, T ) . (5.105)Second, to determine a linear term of a small-˜ ω expansion of I j =0 (˜ ω, T ), we performintegration by parts in Eq. (5.98) to derive the representation I j =0 (˜ ω, T ) = T − − ω cos[(˜ ω + j ) T ]2 π (˜ ω + j ) − ω ˜ ω + j ˆ ∞ T dλ π cos[(˜ ω + j ) λ ] λ ω (5.106)whose integral term possesses a better convergence when ˜ ω approaches zero, ascompared to the one given by Eq. (5.98). Differentiating Eq. (5.106) with respectto ˜ ω and setting ˜ ω = 0 we derive: α ( j, T ) = d I j =0 (˜ ω, T ) d ˜ ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ ω =0 = − πj (cid:18) sin( jT ) + cos( jT ) jT (cid:19) + 1 j ˆ ∞ T dλ π sin( jλ ) λ + 1 j ˆ ∞ T dλ π cos( jλ ) λ < ∞ , ∀ T > . (5.107)This implies the relation ( j = 0) α ( − j, T ) = α ( j, T ) . (5.108)As a consequence, we conclude that I − (˜ ω, T ) + I (˜ ω, T ) = O (˜ ω ) . (5.109)(It is this particular combination that appears in Eq. (5.100).) (c) .—To examine a small-˜ ω expansion K j =0 (˜ ω, T ) = κ ( j, T ) + ˜ ω κ ( j, T ) + O (˜ ω ) , (5.110) Power spectrum in quantum chaotic systems: Large- N limit κ ( j, T ) directly fromEq. (5.99) κ ( j, T ) = K j =0 (0 , T ) = ˆ ∞ T dλ π ln λ sin( jλ ) λ < ∞ , ∀ T > , (5.111)to observe the relation ( j = 0) κ ( − j, T ) = − κ ( j, T ) . (5.112)Second, to examine a linear term of a small-˜ ω expansion of K j =0 (˜ ω, T ), we performintegration by parts in Eq. (5.99) in order to improve integral’s convergence: K j =0 (˜ ω, T ) = ln T π (˜ ω + j ) cos[(˜ ω + j ) T ] T ω + 1˜ ω + j ˆ ∞ T dλ π cos[(˜ ω + j ) λ ] λ ω − ω ˜ ω + j ˆ ∞ T dλ π cos[(˜ ω + j ) λ ] λ ω ln λ. (5.113)Differentiating Eq. (5.113) with respect to ˜ ω and setting ˜ ω = 0, we obtain: κ ( j, T ) = d K j =0 (˜ ω, T ) d ˜ ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ ω =0 = − sin( jT )2 πj ln T − cos( jT )2 πj T ln T − j ˆ ∞ T dλ π sin( jλ ) λ (1 − ln λ ) − j ˆ ∞ T dλ π cos( jλ ) λ (1 − ln λ ) < ∞ , ∀ T > . (5.114)This implies the relation ( j = 0) κ ( − j, T ) = κ ( j, T ) . (5.115)As a consequence, we conclude that K − (˜ ω, T ) + K (˜ ω, T ) = O (˜ ω ) . (5.116)(Again, it is this particular combination that appears in Eq. (5.100).)Collecting the results Eqs. (5.93), (5.100), (5.102), (5.109), and (5.116), we observethat Λ (˜ ω ) = ˜ ω + O (˜ ω ); henceˆΛ (˜ ω ) = ˆΛ (˜ ω ) = ˜ ω , (5.117)see Eqs. (5.59) and (5.85). Calculation of ˆΛ (˜ ω ).—Since A (˜ ω, T ) = O (˜ ω ), see Eq. (5.89), we need to deal with B (˜ ω, T ) only: B (˜ ω, T ) = − ˜ ω Im ˆ ∞ T dλ π λ − ω e i ˜ ωλ G ( λ ) , (5.118)large- λ asymptotics of G ( λ ) defined by Eq. (5.77) are required.To proceed, we need to complement the expansions Eqs. (5.95) and (5.96) with theone for F defined by Eq. (5.80). To this end we, first, employ Eqs. (5.64), (5.70),(5.71) to determine a large- t behavior of F ( t ), F ( t ) t = i (cid:26) a t + a cos tt + a cos t ln tt + O (cid:18) sin( ⋆ t ) ln tt (cid:19)(cid:27) , (5.119)where a , a and a are real coefficients whose explicit values are not required for ouranalysis; sin( ⋆ t ) stands to denote sin t and sin(2 t ), both of which are present in the Power spectrum in quantum chaotic systems: Large- N limit t behavior of f ( t ): f ( t ) = i (cid:26) a ′ t + a ′ sin tt + a ′ cos tt + a ′ cos tt ln t + a ′ cos tt ln t (cid:27) + O (cid:18) sin t ln tt (cid:19) . (5.120)Here, the coefficients a ′ j ∈ R are real.Now, a large- λ behavior of F ( λ ) can be read off from Eq. (5.80): F ( λ ) = i (cid:26) a ′′ λ + a ′′ sin λλ + a ′′ cos λλ + a ′′ sin λλ ln λ + a ′′ sin λλ ln λ (cid:27) + O (cid:18) cos λλ ln λ (cid:19) , (5.121)where the coefficients a ′′ j ∈ R are real, again. Inspection of Eqs. (5.77), (5.95), (5.96)and (5.121) shows that a large- λ behavior of G ( λ ) coincides with that of F ( λ ).Having determined a large- λ asymptotics of G ( λ ), we turn to the analysis of thefunction B (˜ ω, T ) as ˜ ω → 0. Since G ( λ ) ∈ i R , a substitution of Eq. (5.121) intoEq. (5.118) generates several integrals (see below), whose small-˜ ω behavior should bestudied in order to figure out if B (˜ ω, T ) contributes to ˆΛ (˜ ω ) as defined by Eqs. (5.59),(5.81) and (5.86). This knowledge is required to complete calculation of the small- ω expansion of the power spectrum S ∞ ( ω ), see Eq. (5.57). (a) .—The first integral, originating from the a ′′ term in Eq. (5.121), admits a small-˜ ω expansion B , (˜ ω, T ) = ˆ ∞ T dλ π cos(˜ ωλ ) λ ω = O (˜ ω ) . (5.122)This result is obtained from the real part of the r.h.s. of Eq. (5.90) evaluated at ˜ ω = 0.Hence, due to Eq. (5.118), the contribution of B , (˜ ω, T ) to B (˜ ω, T ) is of order O (˜ ω ). (b) .—The second integral, originating from the a ′′ term in Eq. (5.121), reads B , (˜ ω, T ) = ˆ ∞ T dλ π sin λ cos(˜ ωλ ) λ ω = O (˜ ω ) (5.123)as can be seen by setting ˜ ω = 0 directly in the integrand. (c) .—All other integrals generated by the remaining terms in Eq. (5.121) can be treatedanalogously.As a consequence, we conclude that B (˜ ω, T ) is of order O (˜ ω ). Taken together withEqs. (5.86) and (5.89), this implies that Λ (˜ ω ) = O (˜ ω ) so thatˆΛ(˜ ω ) = 2˜ ω + O (˜ ω ) . (5.124)Substituting Eq. (5.124) into Eq. (5.57), we derive the sought small-˜ ω expansion of thepower spectrum S ∞ ( ω ) as stated in Theorem 2.11. (cid:3) Acknowledgments Roman Riser thanks Tom Claeys for insightful discussions and kind hospitality atthe Universit´e catholique de Louvain. This work was supported by the Israel ScienceFoundation through the Grants No. 648/18 (E.K. and R.R.) and No. 2040/17 (R.R.).Support from the Simons Center for Geometry and Physics, Stony Brook University,where a part of this work was completed, is gratefully acknowledged (E.K). Boundary conditions for Painlev´e VI function ˜ σ N ( t ; ζ ) as t → ∞ AppendicesA. Boundary conditions for Painlev´e VI function ˜ σ N ( t ; ζ ) as t → ∞ To derive the t → ∞ boundary condition for ˜ σ N ( t ; ζ ) satisfying Eq. (2.15) of theTheorem 2.7, we make use of Eqs. (4.17) and (2.14) to observe the relation˜ σ N ( t ; ζ ) = − t − ddϕ ln Φ N ( ϕ ; ζ ) (cid:12)(cid:12)(cid:12) ϕ =2 arctan(1 /t ) (A.1)which holds true for t > ≤ ϕ < π/ 2. Since ϕ → t → ∞ , we shall considera small- ϕ expansion of the generating function Φ N ( ϕ ; ζ )Φ N ( ϕ ; ζ ) = N Y j =1 (cid:18) ˆ π − ζ ˆ ϕ (cid:19) dθ j π P N ( θ , . . . , θ N )= 1 + N X ℓ =1 ( − ζ ) ℓ ℓ ! ℓ Y j =1 ˆ ϕ dθ j π R ℓ,N ( θ , . . . , θ ℓ ) , (A.2)where the JPDF P N ( θ , . . . , θ N ) is that of TCUE N [Eq. (2.9)], and R ℓ,N ( θ , . . . , θ ℓ )stands for the ℓ -th correlation function in TCUE N . Due to the Lemma 4.7, theseadmit a determinantal respresentation R ℓ,N ( θ , . . . , θ ℓ ) = 1 N + 1 det ≤ i,j ≤ ℓ +1 [ S N +1 ( θ i − θ j )] (cid:12)(cid:12)(cid:12) θ ℓ +1 =0 , (A.3)where S N +1 ( θ ) is the CUE N +1 sine-kernel: S N +1 ( θ ) = sin[( N + 1) θ/ θ/ . For one, R ,N ( θ ) = 1 N + 1 det (cid:18) S N +1 (0) S N +1 ( θ ) S N +1 ( θ ) S N +1 (0) (cid:19) , (A.4) R ,N ( θ , θ ) = 1 N + 1 det S N +1 (0) S N +1 ( θ − θ ) S N +1 ( θ ) S N +1 ( θ − θ ) S N +1 (0) S N +1 ( θ ) S N +1 ( θ ) S N +1 ( θ ) S N +1 (0) , (A.5)etc.A straightforward calculation produces a small- ϕ expansion of Φ N ( ϕ ; ζ ) whose severalinitial terms read:Φ N ( ϕ ; ζ ) = 1 − ζ π (cid:18) R ,N (0) ϕ + 12! R ′ ,N (0) ϕ + 13! R ′′ ,N (0) ϕ (cid:19) + 12! (cid:18) ζ π (cid:19) (cid:18) R ,N (0 , ϕ + 12 h R [0 , ,N (0 , 0) + R [1 , ,N (0 , i ϕ (cid:19) − (cid:18) ζ π (cid:19) R ,N (0 , , ϕ + o ( ϕ ) . (A.6)Only one, out of six, coefficients in the expansion is nontrivial, R ′′ ,N (0) = N ( N + 1)( N + 2)6 Generating function Φ N ( ϕ ; ζ ) and discrete Painlev´e V equations ( dP V ) N ( ϕ ; ζ ) = 1 − N ( N + 1)( N + 2)72 π ζϕ + o ( ϕ ) . (A.7)By virtue of Eq. (A.1), the boundary condition for ˜ σ N ( t ; ζ ) as t → ∞ reads˜ σ N ( t ; ζ ) = − t + ζ N ( N + 1)( N + 2)3 πt + O ( t − ) . (A.8)Further terms in the 1 /t -expansion Eq. (A.8) can be restored with the help of thePainlev´e VI equation itself [Eq. (2.15)]. Substituting the large- t ansatz˜ σ N ( t ; ζ ) = − t + ∞ X j =2 σ j ( N, ζ ) t j (A.9)therein, we deduce:˜ σ N ( t ; ζ ) = − t + σ ( N, ζ ) t + σ ( N, ζ ) t + σ ( N, ζ ) t + O ( t − ) , (A.10)where σ ( N, ζ ) = ζ N ( N + 1)( N + 2)3 π ,σ ( N, ζ ) = − N + 4 N + 915 σ ( N, ζ ) ,σ ( N, ζ ) = σ ( N, ζ )3 . (A.11) Remark A.1. Since the above procedure is capable of producing the expansioncoefficients σ j ( N, ζ ) of any finite order, it can also be utilized – by virtue of Eq. (2.14)– to generate a small- ϕ expansion of Φ N ( ϕ ; ζ ) up to required accuracy. (cid:4) B. Generating function Φ N ( ϕ ; ζ ) and discrete Painlev´e V equations ( dP V ) To avoid intricacies [52] of a numerical evaluation of the six Painlev´e function ˜ σ N ( t ; ζ )appearing in the generating function Eq. (2.14), we opt for an alternative representationof Φ N ( ϕ ; ζ ) in terms of discrete Painlev´e V equations.To proceed, we follow Ref. [53] (see also Refs. ([54, 55, 56])), to observe that asequence of U ( N ) integrals I N ( ϕ ; ζ ) = 1 N ! N Y j =1 (cid:18) ˆ π − π − ζ ˆ ππ − ϕ (cid:19) dθ j π Y ≤ i The N -recurrence for the reflection coefficients of polynomialsorthogonal on the unit circle | z | = 1 with respect to the weight w ( z ) = t − µ z − µ − ω − iω (1 + z ) ω (1 + tz ) µ (cid:26) , θ / ∈ ( π − φ, π )1 − ζ, θ ∈ ( π − φ, π ) . (B.3) is governed by two systems of coupled first order discrete Painlev´e equations ( dP V ).The first is g N +1 g N = t ( f N + N )( f N + N + µ ) f N ( f N − ω ) ,f N + f N +1 = 2 ω + N − µ + ωg N − t ( N + µ + ¯ ω ) g N − t , (B.4) subject to the initial conditions g = t µ + ω + (1 + µ + ¯ ω ) r µ + ω + (1 + µ + ¯ ω ) tr , f = 0 , r = − w − w . (B.5) The second system is ¯ g N +1 ¯ g N = t − ( ¯ f N + N )( ¯ f N + N + 2 ω )¯ f N ( ¯ f N − µ ) , ¯ f N + ¯ f N +1 = 2 µ + N + µ + ω ¯ g N − N − µ + ¯ ω ) t − ¯ g N − t − , (B.6) subject to the initial conditions ¯ g = µ + ¯ ω + (1 + µ + ω ) t − ¯ r µ + ¯ ω + (1 + µ + ω )¯ r , ¯ f = 0 , ¯ r = − w w . (B.7) Here, ω = ω + iω and ¯ ω = ω − iω . The coefficients w , w ∓ in Eqs. (B.5) and (B.7)are w ℓ = 12 iπ ˛ dzz ℓ +1 w ( z ) . (B.8) The transformations relating the variables { g N , ¯ g N } to the reflection coefficients { r N , ¯ r N } read: r N r N − = 1 − t − g N g N − N − − µ + ωN + µ + ¯ ω (B.9) and ¯ r N ¯ r N − = 1 − ¯ g N ¯ g N − t − N − − µ + ¯ ωN + µ + ω , (B.10) respectively. The Proposition yields a sought dP V representation of the generating functionΦ N ( ϕ ; ζ ), see Eq. (4.17). Indeed, setting ω = ω + iω = 1 and µ = 0, one observes therelation Φ N ( ϕ ; ζ ) = I N ( ϕ ; ζ ) N + 1so that Eq. (B.2) translates toΦ N +1 Φ N − Φ N = ( N + 1) N ( N + 2) (1 − r N ¯ r N ) , (B.11) Generating function Φ N ( ϕ ; ζ ) and discrete Painlev´e V equations ( dP V ) { r N , ¯ r N } are determined by equations r N r N − = 1 − t − g N g N − NN + 1 (B.12)and ¯ r N ¯ r N − = 1 − ¯ g N ¯ g N − t − NN + 1 , (B.13)considered in conjunction with two systems of coupled first order discrete Painlev´eequations (dP V ): g N +1 g N = t ( f N + N ) f N ( f N − ,f N + f N +1 = 2 + Ng N − t ( N + 1) g N − t (B.14)and ¯ g N +1 ¯ g N = t − ( ¯ f N + N )( ¯ f N + N + 2)¯ f N , ¯ f N + ¯ f N +1 = N + 1¯ g N − Nt ¯ g N − . (B.15)The initial conditions readΦ = 1 , Φ = 1 − ζ π ( ϕ − sin ϕ ) , (B.16) g = t w − w − w − tw − , f = 0 (B.17)and ¯ g = w − t − w w − w , ¯ f = 0 , (B.18)respectively. By virtue of Eq. (B.8), a set of parameters { w , w ± } can be calculatedexplicitly: w = 2 − ζiπ (cid:18) − t t + ln t (cid:19) , (B.19) w ± = 1 ∓ ζiπ (cid:18) 14 ( t ± − t ± − 3) + 12 ln( t ± ) (cid:19) . (B.20)Equations (B.11) – (B.20) provide the dP V representation of the generating functionΦ N ( ϕ ; ζ ). Remark B.2. To avoid numerical z –differentiation of Φ N ( ϕ ; 1 − z ) appearing in theformula Eq. (2.12), it is beneficial to produce a similar system of coupled recurrenceequations for ( ∂/∂z )Φ N ( ϕ ; 1 − z ). Since the resulting recurrences are too cumbersometo state them here, we leave their (straightforward) derivation to the inquisitive reader. (cid:4) Remark B.3. Away from the endpoints ϕ = 0 and ϕ = 2 π , the dP V representationopens a way for effective numerical evaluation of both Φ N ( ϕ ; ζ ) and ( ∂/∂z )Φ N ( ϕ ; 1 − z )for finite N . Since the recurrence procedure tends to accumulate numerical errors, wehave used quadruple precision numbers to achieve sufficient precision for very large N (e.g., for N = 10 , see Figs. 4 and 5). (cid:4) Generating function Φ N ( ϕ ; ζ ) and discrete Painlev´e V equations ( dP V ) Remark B.4. In the vicinity of the endpoints ϕ = 0 and ϕ = 2 π , numerical precisionof the above recurrence procedure worsens drastically since the recurrence equationsstart to exhibit a singular behavior. To circumvent this drawback at ϕ = 0, we haveused a small- ϕ expansion of Φ N ( ϕ ; ζ ) as described in the Remark A.1. In the vicinityof ϕ = 2 π , the symmetry relation Eq. (4.18) combined with a small- ϕ expansion makesthe job. (cid:4) eferences 46 References [1] J. M. G. G´omez, K. Kar, V. K. B. Kota, R. A. Molina, A. Rela˜no, and J. Retamosa: Many-bodyquantum chaos: Recent developments and applications to nuclei. Phys. Rep. , 103 (2011).[2] M. V. Berry: Quantum chaology. Proc. R. Soc. A , 183 (1987).[3] O. Bohigas, M. J. Giannoni, and C. Schmit: Characterization of chaotic quantum spectra anduniversality of level fluctuation laws. Phys. Rev. Lett. , 1 (1984).[4] K. Richter: Semiclassical Theory of Mesoscopic Quantum Systems (Springer, Berlin, 2000).[5] M. L. Mehta: Random Matrices (Elsevier, Amsterdam, 2004).[6] P. J. Forrester: Log-Gases and Random Matrices (Princeton University Press, Princeton NJ, 2010).[7] A. V. Andreev, O. Agam, B. D. Simons, and B. L. Altshuler: Quantum chaos, irreversible classicaldynamics, and random matrix theory. Phys. Rev. Lett. , 3947 (1996)[8] O. Agam, A. V. Andreev, and B. D. Simons: Quantum chaos: A field theory approach. Chaos,Solitons & Fractals , 1099 (1997)[9] K. Richter and M. Sieber: Semiclassical theory of chaotic quantum transport. Phys. Rev. Lett. , 206801 (2002); S. M¨uller, S. Heusler, P. Braun, F. Haake, and A. Altland: Semiclassicalfoundation of universality in quantum chaos. Phys. Rev. Lett. , 014103 (2004); S. Heusler,S. M¨uller, A. Altland, P. Braun, and F. Haake: Periodic-orbit theory of level correlations.Phys. Rev. Lett. , 044103 (2007); S. M¨uller, S. Heusler, A. Altland, P. Braun, and F. Haake:Periodic-orbit theory of universal level correlations in quantum chaos. New J. Phys. , 103025(2009).[10] M. V. Berry: Semiclassical theory of spectral rigidity. Proc. R. Soc. A , 229 (1985);Semiclassical formula for the number variance of the Riemann zeros. Nonlinearity , 399 (1988).[11] M. Jimbo, T. Miwa, Y. Mˆori, and M. Sato: Density matrix of an impenetrable Bose gas and thefifth Painlev´e transcendent. Physica D , 80 (1980).[12] M. V. Berry and M. Tabor: Level clustering in the regular spectrum. Proc. R. Soc. A , 375(1977)[13] M. Tabor: Chaos and Integrability in Nonlinear Dynamics: An Introduction (Wiley, London,1989); G. M. Zaslavsky: The Physics of Chaos in Hamiltonian Systems (Imperial CollegePress, London, 2007)[14] B. Eckhardt: Quantum mechanics of classically non-integrable systems. Phys. Rep. , 205(1988); O. Bohigas, S. Tomsovic and D. Ullmo: Manifestations of classical phase space structuresin quantum mechanics. Phys. Rep. , 43 (1993); S. Tomsovic and D. Ullmo: Chaos-assistedtunneling. Phys. Rev. E , 145 (1994)[15] M. Robnik: Quantising a generic family of billiards with analytic boundaries. J. Phys. A: Math.Gen. , 1049 (1984); T. Prosen and M. Robnik: Semicalssical energy level statistics in thetransition region between integrability and chaos: Transition from Brody-like to Berry-Robnikbehaviour. J. Phys. A: Math. Gen. , 8059 (1994); M. V. Berry, J. P. Keating, and S. D. Prado:Orbit bifurcations and spectral statsitics. J. Phys. A: Math. Gen. , L245 (1998); B. Dietz,T. Friedrich, M. Miski-Oglu, A. Richter, and F. Sch¨afer: Spectral properties of Bunimovichmushroom billiards. Phys. Rev. E , R-035203 (2007)[16] A. Rela˜no, J. M. G. G´omez, R. A. Molina, J. Retamosa, and E. Faleiro: Quantum chaos and 1 /f noise. Phys. Rev. Lett. , 244102 (2002)[17] A. M. Odlyzko: On the distribution of spacings between zeros of the zeta function. Math. Comput. , 273 (1987)[18] J. M. G. G´omez, A. Rela˜no, J. Retamosa, E. Faleiro, L. Salasnich, M. Vraniˇcar, and M. Robnik:1 /f α noise in spectral fluctuations of quantum systems. Phys. Rev. Lett. , 084101 (2005);A. Rela˜no: Chaos-assisted tunneling and 1 /f α spectral fluctuations in the order-chaostransition. Phys. Rev. Lett. , 224101 (2008)[19] K. A. Takeuchi: 1 /f α power spectrum in the Kardar-Parisi-Zhang universality class. J. Phys. A:Math. Theor. , 264006 (2017)[20] E. Faleiro, U. Kuhl, R. A. Molina, L. Mu˜noz, A. Rela˜no, and J. Retamosa: Power spectrumanalysis of experimental Sinai quantum billiards. Phys. Lett. A , 251 (2006)[21] M. Bia lous, V. Yunko, S. Bauch, M. Lawniczak, B. Dietz, and L. Sirko: Long-range correlationsin rectangular cavities containing point-like perturbations. Phys. Rev. E , 042211 (2016)[22] M. Bia lous, V. Yunko, S. Bauch, M. Lawniczak, B. Dietz, and L. Sirko: Power spectrum analysisand missing level statistics of microwave graphs with violated time reversal invariance. Phys.Rev. Lett. , 144101 (2016)[23] B. Dietz, V. Yunko, M. Bia lous, S. Bauch, M. Lawniczak, and L. Sirko: Nonuniversality in thespectral properties of time-reversal-invariant microwave networks and quantum graphs. Phys.Rev. E , 052202 (2017)[24] M. Lawniczak, M. Bia lous, V. Yunko, S. Bauch, and L. Sirko: Missing-level statistics and analysisof the power spectrum of level fluctuations of three-dimensional chaotic microwave cavities.Phys. Rev. E , 012206 (2018)[25] A. Frisch, M. Mark, K. Aikawa, F. Ferlaino, J. L. Bohn, C. Makrides, A. Petrov, and eferences 47 S. Kotochigova: Quantum chaos in ultracold collisions of gas-phase erbium atoms. Nature ,475 (2014)[26] J. Mur-Petit and R. A. Molina: Spectral statistics of molecular resonances in erbium isotopes:How chaotic are they? Phys. Rev. E , 042906 (2015)[27] M. Wilkinson: Random matrix theory in semiclassical quantum mechanics of chaotic systems. J.Phys. A , 1173 (1988); B. Mehlig and M. Wilkinson: Spectral correlations: Understandingoscillatory contributions. Phys. Rev. E , 045203 (2001)[28] F. J. Dyson: A Brownian motion model for the eigenvalues of a random matrix. J. Math. Phys. , 1191 (1962)[29] E. Faleiro, J. M. G. G´omez, R. A. Molina, L. Mu˜noz, A. Rela˜no, and J. Retamosa: Theoreticalderivation of 1 /f noise in quantum chaos. Phys. Rev. Lett. , 244101 (2004)[30] A. M. Garc´ıa-Garc´ıa: Power spectrum characterization of the Anderson transition. Phys. Rev. E , 026213 (2006)[31] A. Rela˜no, L. Mu˜noz, J. Retamosa, E. Faleiro, and R. A. Molina: Power-spectrumcharacterization of the continuous Gaussian ensemble. Phys. Rev. E , 031103 (2008)[32] R. E. Prange: The spectral form factor is not self-averaging. Phys. Rev. Lett. , 2280 (1997)[33] R. Riser, V. Al. Osipov, and E. Kanzieper: Power spectrum of long eigenlevel sequences inquantum chaotic systems. Phys. Rev. Lett. , 204101 (2017)[34] O. Bohigas, P. Leboeuf, and M. J. S´anchez: Spectral spacing correlations for chaotic anddisordered systems. Found. Physics , 489 (2001)[35] R. Riser and E. Kanzieper: unpublished (2019)[36] N. M. Katz and P. Sarnak: Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. ,1 (1999)[37] A. M. Odlyzko: The 10 , 139(2001)[38] P. J. Forrester and N. S. Witte: Application of the τ -function theory of painlev´e equations torandom matrices: P VI , the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. , 29(2004)[39] K. Okamoto: Studies on the Painlev´e equations: I. Sixth Painlev´e equation P VI . Ann. Mat. PuraAppl. (4), , 337 (1987)[40] M. E. H. Ismail: Classical and Quantum Orthogonal Polynomials in One Variable (CambridgeUniversity Press, 2005)[41] NIST Handbook of Mathematical Functions, edited by F. W. J. Olver, D. W. Lozier,R. F. Boisvert and C. W. Clark. (Cambridge University Press, 2010)[42] A. Borodin and E. Kanzieper: A note on the Pfaffian integration theorem. J. Phys. A , F849(2007)[43] C. Andr´eief: Note sur une relation les int´egrales d´efinies des produits des fonctions. M´em. Soc.Sci., Bordeaux , 1 (1886)[44] N. G. de Bruijn: On some multiple integrals involving determinants. J. Indian Math. Soc. ,133 (1955)[45] P. Deift, A. Its, and I. Krasovsky: On the asymptotics of a Toeplitz determinant with singularities,in: Random Matrix Theory, Interacting Particle Systems, and Integrable Systems (CambridgeUniversity Press, New York, 2014), p. 93[46] T. Claeys and I. Krasovsky: Toeplitz determinants with merging singularities. Duke Math. J. , 2897 (2015)[47] T. Claeys, private communication.[48] T. Ehrhardt: A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities, in: Recent Advances in Operator Theory (Groningen, 1998) , Oper.Theory Adv. Appl. (Birkh¨auser, Basel, 2001), p. 217[49] N. M. Temme, Asymptotic Methods for Integrals (World Scientific Publishing, Sngapore, 2014)[50] J. Chazy: Sur les ´equations diff´erentielles du troisi`eme ordre et d’ordre sup´erieur dont l’int´egraleg´en´erale a ses points critiques fixes. Acta Math. , 317 (1911).[51] C. M. Cosgrove: Chazy classes IXXI of third-order differential equations. Stud. Appl. Math. ,171 (2000)[52] F. Bornemann: On the numerical evaluation of distributions in random matrix theory: A review.Markov Processes Relat. Fields , 803 (2010)[53] P. J. Forrester and N. S. Witte: Discrete Painlev´e equations, orthogonal polynomials on the unitcircle and N -recurrences for averages over U ( N ) – P VI τ -functions. arXiv: math-ph/0308036v1(2003)[54] P. J. Forrester and N. S. Witte: Discrete Painlev´e equations and random matrix averages.Nonlinearity , 1919 (2003)[55] P. J. Forrester and N. S. Witte: Discrete Painlev´e equations for a class of PVI τ -functions givenas U ( N ) averages. Nonlinearity , 2061 (2005)[56] P. J. Forrester and N. S. Witte: Bi-orthogonal polynomials on the unit circle, regular semi-classicalweights ans integrable systems. Constr. Approx. , 201 (2006) eferences 48 [57] G. Szeg¨o, Orthogonal Polynomials , Colloquium Publications, vol.23