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Amplitude distribution of eigenfunctions in mixed systems

We study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space. For an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and good agreement with numerical computations for the family of limacon billiards is found. The natural extension of our result to more general systems, e.g. with a potential, is also discussed.


Introduction
The semiclassical behavior of the eigenfunctions of a quantum mechanical system strongly depends on the ergodic properties of the underlying classical system. The semiclassical eigenfunction hypotheses [1,2] states that the Wigner function of a semiclassical eigenstate is concentrated on a region in phase space explored by a typical trajectory of the classical system. In integrable systems the phase space is foliated into invariant tori, and the Wigner functions of the quantum mechanical eigenfunctions tend to delta functions on these tori in the semiclassical limit [3]. On the other hand, in an ergodic system almost all trajectories cover the energy shell uniformly, and hence the Wigner functions of the eigenstates are expected to become a delta function on the energy shell. That this actually happens for an ergodic system for almost all eigenstates follows from the quantum ergodicity theorem see [4][5][6], and [7,8] for billiards (the relation of the quantum ergodicity theorem to the semiclassical behaviour of Wigner functions is explicitly derived for Hamiltonian systems in [9]). However, a generic system is neither integrable nor ergodic [10], but has a mixed phase space in which regular regions (e.g. islands around stable periodic orbits) and stochastic regions coexist. Whether these numerically observed stochastic regions are ergodic and of positive measure is an open question, see [11] for a review on the coexistence problem. The eigenfunctions in mixed systems are expected to be separated into regular and irregular eigenfunctions according to an early conjecture by Percival [12] which has been numerically confirmed for several systems, see e.g. [13][14][15][16]. In addition, at finite energies there is a small (semiclassically vanishing) fraction of "hierarchical states" which are of intermediate nature, and localize in regions bounded by cantori [17].
Beside the localization properties of the Wigner function, also the local amplitude fluctuations of the eigenfunctions strongly depend on the classical system, as has been pointed out in [1,18]. The basic idea is that an eigenfunction can be represented locally as a superposition of de Broglie waves with wavelength determined by the energy, and momenta distributed according to the semiclassical limit of the Wigner function. In a chaotic system one therefore expects an isotropic distribution of the momenta. If one additionally assumes that the phases are randomly distributed, one obtains locally a Gaussian amplitude distribution of a typical eigenfunction in a quantum mechanical system with chaotic classical limit. For instance in a chaotic billiard a Gaussian amplitude distribution is expected, and this has been confirmed by several numerical studies, see e.g. [19][20][21][22][23][24]. Furthermore, predictions of the random wave model for the maxima of chaotic eigenfunctions have been derived and numerically tested for several systems in [25]. In contrast, in an integrable system the localization of the Wigner function on the invariant tori enforces a more coherent superposition of the de Broglie waves, leading to a regular structure of the eigenfunction [1].
Our aim is to determine the amplitude distribution for irregular states in systems with a mixed classical dynamics. We assume that the motion on a stochastic region D in phase space is ergodic and that the statistical properties of eigenfunctions can be described by a random wave model restricted to D see the following section for a precise definition. The derivation shows that locally the fluctuations are Gaussian with a position dependent variance which is given by the classical probability density on position space defined by the ergodic density on D. Thus the resulting amplitude distribution may be significantly different from a Gaussian. In section 3 we compare the theoretical prediction of the restricted random wave model with numerical computations.
2 Amplitude distribution for the restricted random wave model In this section we consider a restricted random wave model for two-dimensional Euclidean quantum billiards in order to describe the statistical properties of irregular eigenfunctions in systems with a mixed classical phase space. The quantum mechanical system is defined by the Euclidean Laplacian on a compact domain Ω ⊂ R 2 with suitable boundary conditions on the boundary ∂Ω. (Usually one chooses Dirichlet conditions.) The quantum mechanical eigenvalue problem is given by and we are interested in the behavior of the eigenfunctions ψ n in the semiclassical limit E n → ∞. The corresponding classical system is given by a free particle moving along straight lines inside the billiard, making elastic reflections on the billiard boundary ∂Ω. The phase space is T * Ω = R 2 × Ω, and the Hamiltonian is H(p, q) = |p| 2 . Since the Hamiltonian is scaling we can restrict our attention to the equi-energyshell with energy E = 1, Introducing polar coordinates (r, φ) for the momentum p, we can parametrize S * Ω by (φ, q) ∈ [0, 2π)×Ω where φ is the direction of the momentum. In these coordinates the Liouville measure on S * Ω is given by which is invariant under the Hamiltonian flow on S * Ω. Now let D ⊂ S * Ω be an open domain which is invariant under the classical flow, and on which the flow is chaotic. The existence of such a domain where the flow is, for instance, ergodic, is an open problem. But numerically one observes invariant domains on which the flow is at least irregular in the sense that most orbits are unstable, and regular islands inside this domain are very small. The uncertainty principle implies a finite quantum mechanical resolution of phase space quantities at finite energies. Therefore at finite energies the small islands of such an irregular domain are not resolved by the quantum system.
So we expect, in the spirit of [1], that the statistical properties of irregular eigenfunctions associated with D can be described by those of a superposition of plane waves with wave vectors of the same lengths and directions distributed uniformly on D. If we furthermore assume random phases we arrive for real valued functions at the following restricted random wave model, which is a superposition of plane waves of the form Here χ D (·) is the characteristic function of D, the phases ε n are independent random variables equidistributed on [0, 2π], and the momenta k n ∈ R 2 are independent random variables which are equidistributed on the circle of radius √ E. So the characteristic function χ D (·) ensures the localization on D. Furthermore, it is natural to take N ∼ √ E, the scaling of the number of line segments of a typical Heisenberg-length orbit. By vol(D) the volume of D measured with the Liouville measure (3) is denoted. With this choice of normalization the expectation value of the norm ||ψ RRWM,D || is one.
Let us first consider the value distribution P q (ψ) of ψ RRWM,D (q) at a given point q ∈ Ω. Our restricted random wave model (4) is a sum of identical independent random variables which have zero mean and whose variance is given by where e(φ) := (cos(φ), sin(φ)) denotes the unit vector in φ-direction. So by the central limit theorem we obtain for E → ∞, i.e. N → ∞, a Gaussian distribution of ψ RRWM,D (q) at q, with variance given by (5). If the classical dynamics on D is ergodic, then the variance σ 2 (q) is exactly the probability density of finding the particle in the point q ∈ Ω if it moves on a generic trajectory in D. So σ 2 (q) is the classical probability density in position space. By integrating eq. (6) over Ω we obtain the complete amplitude distribution as a mean over a family of Gaussians with variances given by (5), So the amplitude distribution is completely determined by the classical probabality density (5), and it will be typically non Gaussian if σ 2 (q) is not constant. The moments of the distribution (8) can be computed directly and turn out to be proportional to the moments of the classical probability density σ 2 (q), where the factor ρ 2k = (2k)! k!2 k denotes the 2k'th moment of a Gaussian. The odd moments are of course zero. Note that the second moment is always 1/vol(Ω), due to the normalization of ψ.
Finally, we would like to point out that the main ingredient in the formula (7) is the assumption that the local amplitude distribution of an irregular eigenfunction around a point q in position space is Gaussian, with a variance given by the classical probability density in position space σ 2 (q), defined by the projection of the invariant measure on D to position space. Clearly this assumption is not restricted to billiards, but is expected to be true for arbitrary quantum mechanical systems for which the underlying classical system contains chaotic components in phase space. So the formula (7) is expected to be valid in far more general situations, with σ 2 (q) denoting the classical probability density defined by the ergodic measure on the chaotic component.

Comparison with irregular eigenfunctions
We now compare the predictions of the restricted random wave model with the results for some numerically computed eigenfunctions. As systems to study the amplitude distribution of irregular states in mixed systems we have chosen the family of limaçon billiards introduced by Robnik [26,27] with boundary given in polar coordinates by ρ(ϕ) = 1 + ε cos(ϕ), ϕ ∈ [−π, π], with ε ∈ [0, 1] being the system parameter. We consider the case ε = 0.3, for which the billiard has a phase space of mixed type [26], see fig. 1. In [28] examples of eigenstates far in the semiclassical regime have been studied in this system, and in particular the amplitude distribution has been studied numerically, but no analytical predictions have been made.
First we have to determine the classical position space probability density σ 2 (q) of the ergodic measure on the invariant domain D. The normalized ergodic measure on D is given by so we can express the variance σ 2 (q) as a mean value As the motion on D is assumed to be ergodic, in order to determine σ 2 (q) we could replace the integral over S * Ω by a time average over a typical trajectory of D and the δ function by a smoothened δ function, e.g. a narrow Gaussian. However, as we will see below, the eigenfunctions turn out to be not concentrated on the whole chaotic component, but rather on a subset which is almost invariant in the sense that it is bounded by partial barriers in phase space. Since at finite energies quantum mechanics has only a finite resolution in phase space, these partial barriers appear like real barriers. But since any classical trajectory will pass such a barrier after a certain time, the time average is not suitable for the determination of σ 2 (q) in such a situation. For a more direct approach to determine σ 2 (q) we use the Poincaré section P = {(s, p) ; s ∈ [−4, 4], p ∈ [−1, 1]}, which is parametrized by the (rescaled) arclength coordinate s (corresponding to ϕ ∈ [−π, π]) along the boundary ∂Ω and the projection p of the unit velocity vector on the tangent in the point s after the reflection. Let D ⊂ P be the projection of the region D in the energy shell S * Ω := {(p, q) ∈ R 2 × Ω ; ||p|| = 1} to the Poincaré section. This projection is defined as follows: to a point (e(φ), q) ∈ D we can associate the trajectory which passes through q in direction e(φ), then s(φ, q) is defined as the first intersection with the boundary ∂Ω when traversing the trajectory backwards from q, and p(φ, q) := e(φ)T(s(φ, q)) which is the projection of the unit velocity vector e(φ) to the unit tangent vector T(s(φ, q)) to ∂Ω at s(φ, q).
For a given point q we therefore get a curve parameterized by φ (p(φ, q), s(φ, q)) ∈ P .
Since χ D (e(φ), q) = χ D ((p(φ, q), s(φ, q)), we get and therefore we have to determine the fraction of the angular interval(s) for which the curve (11) is in D. That is, one has to determine the angles φ entry i (q) and φ exit i (q) where the curve (11) enters or leaves the region D, i.e. the intersection points of (11) with the boundary of D.
In terms of these angles we obtain which is proportional to the fraction of directions in the ergodic component visible from the point q.

6
With this classical probability density σ 2 (q) one can compute the corresponding amplitude distribution via eq. 8. If σ 2 (q) = 0 for some region, then the local amplitude distribution (6) becomes a delta function, and it is necessary to consider for a concrete comparison a binned distribution, We now use a Husimi Poincaré section representation of the eigenstate (see e.g. [29,30]) to determine the boundary of the relevant component D by a spline approximation. The Poincaré Husimi representation of an eigenfunction ψ n in a billiard is defined by projecting the normal derivative u n (s) of an eigenfunction ψ n (q) at the boundary onto a coherent state on the boundary. The coherent states, semiclassically centered in (s, p) ∈ P, are defined as with k n = √ E n ; the prefactor ensures the normalization H n (s, p) dp ds = 1. A first example is shown in fig. 2. In a) a high lying eigenfunction (E = 1002754.70 . . . , approximately the 130568 th state of odd symmetry) in the limaçon billiard with ε = 0.3 is shown as density plot (black corresponding to high intensity of |ψ| 2 ). In b) the corresponding Husimi representation on the Poincaré section is shown. The boundary of the irregular region D is described by a cubic spline which is shown as full curve. With these boundary curves we can use (13) to compute σ 2 (q), which is shown in fig. 2c). Finally, in fig. 2d) the comparison of the amplitude distribution of ψ with the prediction of the restricted random wave model is given. Clearly, P (ψ) is non-Gaussian, and the agreement is very good. Table 1 lists the first moments and also very good agreement of the results using (9) and the moments of ψ is found. Both the resulting amplitude distribution P RRWM,D and the moments turn out to be quite robust with respect to small changes of the selection of D. Notice that we have rescaled σ 2 (q) such that the variance of the distributions is one.
Another example is shown in fig. 3. The eigenfunction (E = 1003030.75 . . . , approximately the 130607 th state of odd symmetry) plotted in a) has a quite large region in the center where it is almost vanishing. So from this alone the amplitude distribution is expected to show a very clear deviation from the normal distribution. Using the same procedure as in the previous case we determine D, compute σ 2 (q) and then P RRWM (ψ). The comparison of the prediction with P (ψ) is shown in fig. 3d). The strongest deviation occurs for ψ ≈ 0. The peak of P RRWM (ψ) in the limaçon is shown as grey scale plot (black corresponding to high intensity). In b) the corresponding Husimi function on the Poincaré section is shown together with the boundary (full curves) of the region on which the eigenfunction is concentrated. In c) a density plot of σ 2 (q), computed via eq. (13), is shown. In d) the cumulative amplitude distribution of the eigenfunction is compared with the prediction of the restricted random wave model (RRWM); on this scale no differences are visible. The left inset shows P (ψ) and for the right inset a logarithmic vertical scale is used to emphasize the tails of the distribution. For comparison the normal distribution is shown as grey curve.   at ψ = 0 is due to the fact that σ 2 (q) = 0 for the region in the center of the billiard. The eigenfunction, however, is not exactly zero, but shows a decay into that region and thus still fluctuates there. This causes a broadening of the δ-contribution, which is clearly visible in the plot of P (ψ) in fig. 3d). For |ψ| > 0.25 this region is not relevant anymore, and the agreement of P (ψ) and P RRWM (ψ) is very good. In the right inset of fig. 3d) the distribution is shown with a logarithmic vertical scale to illustrate the agreement of the distributions even in the tails. The moments, computed via eq. (9), are listed in table 1. The agreement of the moments of the eigenfunction with the prediction of the restricted random wave model is quite good. All moments of the two examples are larger than those of a Gaussian, corresponding to the larger tails. Compared to the moments of the restricted random wave model those of the eigenfunctions tend to be smaller, in particular for the larger moments. This is reasonable, as an actual eigenfunction is always bounded, which reduces higher moments compared to the result of eq. (9).
We furthermore have tested our basic assumption (6), that the local value distribution of a sufficiently high lying eigenfunction is Gaussian with a variance given by the local classical probability density associated with D, more directly. To this end we have computed the value distribution of the eigenfunction in fig. 3 for three small regions on which σ 2 (q) is almost invariant, and we therefore expect a Gaussian. The results are shown in fig. 4, and good agreement with the prediction (6) is found. Since many fewer wavelengths are contained in these small domains than in Ω the statistics is of course not as good as for the full system, but the results give strong support for a local Gaussian behavior. The variances for the two domains A and C coincide with the expected classical one σ 2 (q). But for domain B the observed variance is larger than σ 2 (q) = 0. This corresponds to the widening of the delta peak in fig. 3, and is due to the fact that the eigenfunction cannot become exactly zero on some open set at finite energies, but instead fluctuates around zero.

Summary
In this work we have extended the random wave model for eigenfunctions from the case of chaotic systems to the case of irregular eigenfunctions in systems with mixed phase space. Our main result is one particular prediction of this model, namely the amplitude distribution (7) of irregular eigenfunctions. Numerical tests have been performed for two high lying eigenfunctions of the limaçon billiard with ε = 0.3, and impressive agreement, even in the tails of the distribution, with the theoretical prediction was found.
The physical picture underlying our analysis is that the local hyperbolicity in the irregular part of the phase space forces the eigenfunctions localizing on this part of phase space to behave locally like a Gaussian random function with a variance given by the classical probability density in position space defined by the uniform measure on the irregular component. Taking the mean over all these local Gaussians with varying variance gives our result for the global amplitude distribution. We have tested this intuitive picture by computing local amplitude distributions. The agreement of these with the Gaussian prediction is very good, giving further strong support to the picture of local Gaussian fluctuations with variance determined by the underlying classical system. A natural further question relates to the correlations of such eigenfunctions between different points in position space; this topic is addressed in [31].
Although we have restricted our study to Euclidean billiards, the general picture of local Gaussian fluctuations is of course not limited to these special type of systems. We therefore expect our results to be valid for irregular eigenfunctions in arbitrary systems (e.g. systems with potential), with σ 2 (q) defined as the projection of the ergodic measure on the irregular component to the position space.