E(k,L) level statistics of classically integrable quantum systems based on the Berry-Robnik approach

Theory of the quantal level statistics of classically integrable system, developed by Makino et al. in order to investigate the non-Poissonian behaviors of level-spacing distribution (LSD) and level-number variance (LNV)\cite{MT03,MMT09}, is successfully extended to the study of $E(K,L)$ function which constitutes a fundamental measure to determine most statistical observables of quantal levels in addition to LSD and LNV. In the theory of Makino et al., the eigenenergy level is regarded as a superposition of infinitely many components whose formation is supported by the Berry-Robnik approach in the far semiclassical limit\cite{Robn1998}. We derive the limiting $E(K,L)$ function in the limit of infinitely many components and elucidates its properties when energy levels show deviations from the Poisson statistics.

(K − j + 1)E(j, L). (1) The LSD P (K, L) is introduced as a distribution function that denotes a probability density to find two adjacent levels of spacing L containing K levels in between. The nearest-neighbor LSD(NNLSD), P (0, L), which is frequently used to analyze the short-range spectral fluctuation, is a special case of K = 0. In a similar way, the LNV Σ 2 (L), skewness γ 1 (L) and excess kurtosis γ 2 (L), which are respectively, the two, three and four-point correlation functions, are calculated as Σ 2 (L) = C 2 (L), γ 1 (L) = C 3 (L)/C 2 (L) 3/2 and γ 2 (L) = C 4 (L)/C 2 (L) 2 − 3, respectively, where C n (L) is nth moment of the level number fluctuation around its average value L, obtained from E(K, L) as correlation instead of Σ 2 (L), is also calculated from E(K, L) as [13] In this way, it is quite important to determine the E(K, L) function, which provides a basis for the energy level statistics. For the Poissonian level sequence of the unfolded scale, E(K, L) can be characterized by the Poisson distribution: which obviously leads to the results from the Poisson statistics: P (K, L) = L K e −L /K!, Σ 2 (L) = L, γ 1 (L) = L −1/2 , γ 2 (L) = 1/L, and ∆ 3 (L) = L/15. Many works have examined the Berry-Tabor conjecture [1,2,8,[14][15][16][17][18][19][20][21][22][23][24][25][26][27], and the statistical property of eigenenergy levels that the Poisson statistics can characterize is now widely accepted as a universal property of generic integrable quantum systems. However, the mechanism supporting this conjecture is still unclear, and deviation from the Poisson statistics is observed in some classically integrable systems that have a spatial or time-reversal symmetry.
One possible mechanism underlying the deviation from Poisson statistics has been proposed by Makino et al. [1], on the basis of the Berry-Robnik approach [3,4,11]. We briefly review the outline as follows. For an integrable system, individual orbits are confined in each inherent torus whose surface is defined by holding its action variable constant, and the whole region of the phase space is densely covered with infinitely many invariant tori, which have infinitesimal volumes in the Liouville measure. Because of the suppression of quantum tunneling in the semiclassical limit → 0, the Wigner function of each quantal eigenstate is expected to be localized in the phase space region explored by a typical trajectory, and to form independent components [4,33]. For a classically integrable quantum system, the Wigner function localizes on the infinitesimal region in → 0 and tends to a δ function on a torus [34]. Then, the eigenenergy levels can be represented as a statistically independent superposition of infinitely many components, each of which contributes infinitesimally to the level statistics. Therefore, if the individual spectral components are sparse enough, one would expect Poisson statistics to be observed as a result of the law of small numbers [35].
The statistical independence of spectral components is assumed to be justified by the principle of uniform semiclassical condensation of eigenstates in the phase space and by the lack of their mutual overlap, and thus can be expected only in the semiclassical limit [4]. This mechanism was initially introduced as a basis for the Berry-Robnik approach to investigate the energy level statistics of the generic mixed quantum system, and its validity is confirmed by numerical computations in the extremely deep semiclassical region which is called the Berry-Robnik regime [32].
On the basis of this view, Makino and Tasaki investigated the NNLSD of systems with infinitely many components [1]. They derived the cumulative function of NNLSD, M (L) = L 0 P (0, S)dS, which is characterized by a single monotonically increasing functionμ(0, S) ∈ [0, 1] of the nearest level spacing S as The functionμ(0, S) classifies M (L) into three cases: Case 1, Poisson distribution M (L) = 1 − e −L for all L ≥ 0 ifμ(0, +∞) = 0; case 2, asymptotic Poisson distribution, which converges to the Poisson distribution for L → +∞, but possibly not for small spacing L if 0 <μ(0, +∞) < 1; case 3, sub-Poisson distribution, which deviates from the Poisson distribution for ∀L in such a way that M (L) converges to 1 for L → +∞ more slowly than does the Poisson distribution ifμ(0, +∞) = 1. This argument is extended later to the study of LNV [2], whose properties are evaluated for cases 1-3 as follows: Case 1, the LNV is the Poissonian Σ 2 (L) = L; case 2, the LNV deviates from the Poissonian in such a way that the slope is greater than 1 for L > 0 and approaches a number ≥ 1 + 2μ(0, +∞) as L → +∞; case 3, the LNV deviates from the Poissonian in such a way that the slope is greater than 1 for L > 0 and approaches a number ≥ 3 as L → +∞. Therefore, the Berry-Robnik approach, when applied to classically integrable systems, allows the NNLSD and LND to deviate from the Poisson statistics.
In this paper, extending the above arguments of Makino et al. [1,2], we investigate the E(K, L) function of a quantum system whose energy level consists of infinitely many independent components, and elucidate its property when the NNLSD of eigenenergy levels shows cases 2 and 3. This paper suggests the possibility of a new statistical law to be observed in the E(K, L) level statistics of classically integrable quantum systems.
The limiting E(K, L) function is derived as follows: We consider a system whose phase space is decomposed into N disjoint regions that give distinct spectral components. The Liouville measures of these regions are denoted by ρ n (n = 1, 2, 3, · · · , N ), which satisfy the normalization N n=1 ρ n = 1. In the Berry-Robnik approach, these quantities are equivalent to the statistical weights of individual spectral components.
When the entire sequence of energy levels is a product of statistically independent superpositions of N subsequences, E(K, L) is decomposed into the E(K, L) function of subsequences, e n (k, L), as where e n satisfies the normalizations +∞ k=0 e n (k, L) = 1 and +∞ k=0 ke n (k, L) = L. In terms of the normalized level-spacing distribution p n (k, S) of the subsequence, e n (k, L) is described as where p n (j < 0, S) = 0, and p n satisties the normalization conditions Sp n (k, S)dS = (k + 1)/ρ n . Eq. (7) is known as the formula in the theory of point process, which is derived as corollaries of the Palm-Khintchine theorem [36].
In addition to Eq.(6), we introduce two assumptions that were introduced in Refs. [1,2]: Assumption (i). The statistical weights of individual components vanish uniformly in the limit of infinitely many components: max n ρ n → 0 as N → +∞.
Assumption (ii). The weighted mean of the cumulative level-spacing distribution of spec- where the convergence is uniform on each closed interval: S ∈ [0, L]. It is noted that µ(k, S) is monotonically decreasing for increasing k.
In the Berry-Robnik approach, Eq.(6) relates the level statistics in the semiclassical limit with the phase space geometry.
Under assumptions (i) and (ii), Eq.(6) leads to the following new expression in the limit where the factor α K (L) and exponent β(L) of the distribution function are described by the parameter functionμ(k, L). When the lowest-order moment of this function showsμ(0, L) = 0 for all L, one has α K (L) = 1 for all K and β(L) = 0, and the limiting functionĒ(K, L) of the whole energy sequence reduces to the Poisson distribution(4). As shown in Ref. [1], this condition is expected to arise when the individual spectral components are sparse enough. In general, one may expectμ(0, L) > 0, which corresponds to a certain accumulation of levels of individual components. In this case, the limiting functionĒ(K, L) deviates from the Poisson distribution.
The organization of this paper is as follows. In Section 2, the limiting functionĒ(K, L) is derived from Eq.(6) and assumptions (i) and (ii). In Section 3, the property of the limiting functionĒ(K, L) is analyzed for cases 1-3, where the possibilities of deviation from the Poisson statistics are discussed. In Section 4, the numerical investigation of the E(K, L) function is carried out for the rectangular billiard, whose numerical results for the NNLSD have been shown to deviate from the Poisson distribution [1,2]. In Section 5, we discuss some relations between our results and those of related works.  6) is factorized using ratios e n (κ m , ρ n L) ≡ e n (κ m , ρ n L)/e n (0, ρ n L) as with where we have used properties e n (0, ρ n L) −1 = 1 + O(ρ n ) and e n (k > 0, ρ n L) = O(ρ n )(see also Eq. (7)). Since e n (k > 0, ρ n L) = e n (k, ρ n L) + O(ρ 2 n ) and Eq. (7), N n=1 e n (κ m , ρ n L) and N n=1 ln e n (0, ρ n L) are described by the weighted mean µ(k, and N n=1 ln e n (0, Here, N n=1 O ρ 2 n in the equations (9), (11) and (12) shows the convergence, which results from assumption (i). Therefore, by applying assumption (ii), we have the limiting formula in the limit of N → +∞, where α 0 (L) = 1, and For K = 1 − 3, the factors α K (L) are specified as Sinceμ(k, S) monotonically increases for S ≥ 0 and 0 ≤μ(k, S) ≤ 1, α K (L) and β(L) in the limit L → +∞ show and β(L) −→μ(0, +∞).
It should be noted that Case 1 and Case 3 are extreme cases where all factors α K (L) of K > 0 in the limit L → +∞ converge to 1 in Case 1 and to 0 in Case 3.
As is also shown in Ref. [1], one observes Case 1 if the scaled NNLSD of individual components f n (0, ρ n S) = p n (0, S)/ρ n , which satisfy  In general, one may observe Case 2 or Case 3, each of which corresponds to strong accumulation of energy levels, leading to a singular NNLSD of the individual components.
In the next section, we numerically analyze the E(K, L) function for the rectangular billiard system whose NNLSD of the eigenenergy levels has been shown to obey Cases 2 and 3 [2].

Numerical studies of rectangular billiard
We analyze the property of E(K, L) for a rectangular quantal billiard whose eigen-energy levels are given by n,m = n 2 + γm 2 , where n and m are positive integers and γ is the square ratio of two sides, a and b, denoted as γ = a 2 /b 2 . The unfolding transformation { m,n } → {¯ m,n } is carried out by using the leading Weyl term of the integrated density of states, N ( ), as¯ m,n = N ( m,n ) = π m,n /4 √ γ. Berry and Tabor observed that the NNLSD of this system agrees with the Poisson distribution (Case 1) when γ is far from rational, while it deviates from the Poisson distribution when γ is rational [8]. The deviation from the Poisson distribution was precisely analyzed for γ = 1 by Connors and Keating [20]. Working on the basis of Landau's number-theoretical result [31], they proved that the mean degeneracy of the eigen-energy levels increases logarithmically as the energy becomes higher. This property has been confirmed numerically by Robnik and Veble in Ref. [21], where the NNLSD P (0, L) converges to the delta function in the high-energy(semiclassical) limit → +∞.
In this section, we evaluate numerically the behaviors of quantities that converge to α K (L) and β(L) in the semiclassical limit. We carry out a numerical study for an irrational case in addition to a rational case (γ = 1), which is described by a finite continued fraction of the golden mean number ( √ 5 + 1)/2, with an irrational truncation parameter δ ∈ [0, 1). the Poisson distribution (4). Our analysis is valid for K << L max as shown in Refs. [11,21], where L max is determined by the shortest period of classical periodic orbit [29], and it is calculated for the rectangular billiard as L max = √ π¯ n,m γ −1/4 . We used eigen-energy levels n,m ∈ [100 × 10 10 , 101 × 10 10 ] corresponding to L max ∼ 1.8 × 10 6 , which is sufficiently large for our numerical study. The numerical computation in this paper was carried out using the double-precision real number operation. When the continued fraction is close to the golden with odd K is very small. Figure 2 shows numerical plots ofα K (L) for K = 1 − 5 and K = 10, which are obtained by E(K, L) asα In each figure, we show three results for γ corresponding to plots (a)-(c) in figure 1. Note that function (29) is equivalent to α K (L) in the semiclassical limit → +∞. In case γ is close to the golden mean and E(K, L) is well approximated by the Poisson distribution,α K (L) agrees with 1 very well [plot (a)]. On the other hand, in case γ is far from the golden mean and E(K, L) deviates from the Poisson distribution,α K (L) approaches a numberα K (+∞) such that 0 <α K (+∞) < 1[plot (b)], and this result corresponds to Case 2. In case γ = 1, α K (L) quickly converges to 0 as L → +∞, and this result corresponds to Case 3. It is quite interesting thatα K (L) of the 4th approximation, whose result corresponds to Case 2, obviously shows relation (25) of monotonically decreasing as K increases as shown in Figure   3.  [20], and additional argument of Makino et. al. [2], M (+0) of the square billiard is described as M (+0) 1 − 4c/π √ ln , and thus we also havẽ with c 0.764. This indicates an extremely slow convergence ofβ(+0) to 1 as the energy becomes higher and β(+0) ≤ β(+∞) = 1 (Case 3) observed in the semiclassical limit ( → +∞). We finally confirm the approximate expression (30). Figure 5 shows 1 −β(0) vs 4c/π √ ln for various energy ranges. The solid line represents the theoretical curve (30), which is valid in the semiclassical (high energy) region. Although we are not yet far enough in the high-energy region where 1 −β(0) << 1, the agreement between them is very good.

Summary and Conclusion
The basic ideas of our study were to apply the Berry-Robnik approach to a classically integrable quantum system, whose phase space consists of infinitely many fine regions, and to discuss the possibility of deviations from the Poisson statistics. In this paper, we successfully applied these ideas to the study of the E(K, L) function which is one of the most fundamental observables in the research field of energy-level statistics.
In the Berry-Robnik approach, the quantal eigenfunctions, localizing on different phasespace regions in the neighborhood of the semiclassical(high-energy) limit, form mutually independent spectral components, where the statistical weight of each component corresponds to the volume ratio (Liouville measure) of the phase-space region. Therefore, we considered a situation where the system consists of infinitely many components and each of them contributes infinitesimally to the spectral statistics. Then, starting from the superposition formula (6) and assumptions (i) and (ii), the limiting distribution functionĒ(K, L) is derived, which is described by the monotonically increasing functionsμ(K, L), K = 0, 1, 2, · · · of the level-spacing L. when there is a strong accumulation of levels, which leads to a singular level spacing distribution of individual components, and such accumulation is expected to arise for a system that has a symmetry, e.g. spatial symmetry or time-reversal symmetry.
As was shown in the numerical studies of Refs. [1,2], the rectangular billiard is one possible example by which to show case 2 in addition to case 1 when the aspect parameter of the system is irrational, and case 3 when the aspect parameter is rational. In this paper, the numerical study of the rectangular billiard is extended to the analysis of the E(K, L) function, where the theoretical arguments ofĒ(K, L) for cases 1-3 are well reproduced.
Similar results are expected also in the torus billiard [21,25], equilateral-triangular billiard [23,24], and integrable Morse oscillator [26], where the deviations from Poisson statistics are reported to be associated with a spatial symmetry.
The limiting functionĒ(K, L) obtained in the present paper, gives a basis on which to investigate the non-Poissonian behaviors of the other statistical observables. For example, the n-point correlation function for a system with infinitely many components is calculated from the moment function,C and this function for n = 2 is described usingμ(K, L) in the following simple form [2]: For 0 <μ(0, +∞) ≤ 1, C 2 (L) provides the non-poissonian limiting LNV,Σ 2 (L) =C 2 (L), whose slope is larger than that of the Poissonian LNV Σ 2 (L) = L. In a similar way, it is also possible to calculate the skewnessγ 1 (L) and excessγ 2 (L) fromC 3 (L) andC 4 (L) respectively, the LSD P (K, L) of K > 0 from Eq.(1), and ∆ 3 (L) from Eq. (3), which, forμ(0, L) = 0, show the non-Poissonian behaviors. This paper reveals the three different classes of E(K, L) statistics possibly observed for the eigenenergy levels consisting of infinitely many components, and also suggests a possibility of new statistical laws (cases 2 and 3) to be observed in the classically integrable quantum systems that have spatial or time-reversal symmetry. Further case study for individual physical systems will be shown elsewhere.    Numerical test of approximate expression (30) for the square billiard system(γ = 1). The solid line represents the theoretical prediction, witch is valid in the semiclassical limit → +∞. For each plots, we used 4 × 10 7 unfolded energy levels.