Resonance states of the three-disk scattering system

For the paradigmatic three-disk scattering system, we confirm a recent conjecture for open chaotic systems, which claims that resonance states are composed of two factors. In particular, we demonstrate that one factor is given by universal exponentially distributed intensity fluctuations. The other factor, supposed to be a classical density depending on the lifetime of the resonance state, is found to be very well described by a classical construction. Furthermore, ray-segment scars, recently observed in dielectric cavities, dominate every resonance state at small wavelengths also in the three-disk scattering system. We introduce a new numerical method for computing resonances, which allows for going much further into the semiclassical limit. As a consequence we are able to confirm the fractal Weyl law over a correspondingly large range.


Introduction
The structure of eigenstates, along with spectral properties, is central for the understanding of quantum systems.In closed chaotic quantum systems, like quantum billiards or quantum maps, their structure is well understood.As stated by the quantum ergodicity theorem, almost all eigenstates are uniformly distributed on the energy shell in the semiclassical limit [1][2][3][4][5][6].The statistical properties of eigenstates are well described by the random wave model [7][8][9][10][11].An exception are states showing enhanced intensities on short unstable periodic orbits, so-called periodic-orbit scars [12][13][14].The number of eigenstates can be obtained from the Weyl law [15,16].
Recently, a factorization conjecture for fully chaotic open systems was introduced, that applies to resonance states with arbitrary decay rate [38,71]: One factor is given by universal exponentially distributed intensity fluctuations corresponding to a complex random wave model.The other factor depends on the decay rate and is given by some classical conditionally-invariant measure that is suitably smoothed.There is no semiclassical theory to derive these measures, so far.Heuristically motivated measures for all decay rates have been suggested for the case of full escape [39] as well as for partial escape in maps [40] and dielectric cavities [38].They show good, but not perfect, agreement.For a Baker map with local randomization the exact measure has been derived from a random vector model [72].The two factors of the factorization conjecture identify which features of a resonance state are universal quantum (wave) phenomena and which are system specific with a classical (ray) origin.In particular, this explains which features neighboring resonance states have in common and which are individual.
A new type of scarring of resonance states along segments of rays was recently observed in dielectric cavities [38].It is unrelated to periodic-orbit scars of closed systems [12][13][14].Ray-segment scars were found in every resonance state at small wavelengths and they were conceptually explained based on the factorization conjecture.

(b).
A semiclassical theory for resonance poles of the three-disk scattering system is based on dynamical zeta functions consisting of periodic orbits and evaluated using the cycle expansion [74,79,81,90].Recently, with these methods it was shown in Refs.[90,91] how one can determine semiclassical resonance states in a Husimi representation that combines left and right resonance states [42].Hence, in principle there exists a semiclassical description for resonance states of the three-disk scattering system.This approach has, however, the following limitations: (i) The convergence of the cycle expansion is in practice too slow for small wavelengths or small distances of the disks.
(ii) To date there is no semiclassical description of the (right) resonance states fulfilling the Schrödinger equation (in contrast to the left-right Husimi representation mentioned above).(iii) Although individual resonance states can be computed from hundred thousands of periodic orbits, no insight about their structure is obtained, e.g. about their dependence on the decay rate or about similarities and differences of neighboring resonance states.This, however, is the virtue of the factorization conjecture and it is desirable to test it for the three-disk scattering system.In this paper, we confirm the factorization conjecture for resonance states in the three-disk scattering system.In particular, we demonstrate that one factor is given by universal exponentially distributed intensity fluctuations.For the other factor, we show that the classical density is very well, but not perfectly, described by extending a construction from maps with full escape to the three-disk scattering system.Furthermore, we observe ray-segment scars in every resonance state at small wavelengths.All these results on resonance states are made possible due to a new numerical method, which allows for going about two orders of magnitude further into the semiclassical limit than before.This allows for confirming the fractal Weyl law over a correspondingly large range.

Classical dynamics
The three-disk scattering system consists of three hard disks of radius a with centers at the corners of an equilateral triangle of side length R, see Fig. 1(a).It is uniquely characterized by the dimensionless ratio R/a.In this paper we focus on the parameter R/a = 2.1 for reasons related to the analysis of resonance states and discussed in Sec.4.1.A point particle moves along straight lines between collisions with the disks and is specularly reflected at their boundaries.The dynamics is chaotic with no stable periodic orbits present.Using the systems C 3v symmetry [92], the dynamics can be reduced to a fundamental domain, see Fig. 1(a).
The phase space of the three-disk scattering system is four-dimensional with dynamics taking place on the three-dimensional energy shell.A further reduction is achieved by a Poincaré surface of section at the disk's boundary, resulting in the two-dimensional boundary phase space.It is parametrized by dimensionless Birkhoff coordinates (s, p), where sa is the arc length along the disk's boundary, see Fig. 1(a), and p is the normalized momentum projected onto the tangent of the boundary.In the fundamental domain one has s ∈ [0, π] and p ∈ [−1, 1].
Of special importance are invariant sets of the scattering system [17,18,93].The backward-trapped set Γ b consists of points that under backward time evolution are trapped in the system.It is shown in Fig. 2(a).This set will be important for the localization of quantum resonance states.The forward-trapped set Γ f is defined analogously, see Fig. 2(b).The chaotic saddle Γ s is given by the intersection Γ b ∩ Γ f of backward-and forward-trapped set, see Fig. 2(c), and thus consists of those points in phase space which under backward and forward time evolution are trapped in the system.These three sets are related, namely Γ b and Γ f are the unstable and stable manifolds of Γ s , respectively.Numerically, they can be obtained by the sprinkler method [17,94].
These sets are known to have a multifractal structure [17,18,26,95] which is well visible in Fig. 2. The generalized dimension D q of the chaotic saddle in the fourdimensional phase space can be described by D q = 2d q + 2 [18].Here, the partial dimension d q is the generalized dimension along one direction of the chaotic saddle in the two-dimensional boundary phase space.We determine the box-counting dimension of the chaotic saddle in the boundary phase space leading to the partial box-counting dimension d 0 = 0.84 for R/a = 2.1.This will be used for the fractal Weyl law of resonance poles, see Sec. 3.
An important property of an open system is its natural measure µ nat [18,[25][26][27][28][29] shown in Fig. 2(d), which will be important for the structure of resonance states, see Sec. 4. It emerges asymptotically from a uniform distribution in phase space under time evolution leading to its concentration on the backward-trapped set Γ b .It is a conditionally-invariant measure with natural decay rate γ nat .This decay rate is also called the classical escape rate [18,75], but we prefer to explicitly link it to the natural measure, as in systems with partial escape there are additional classically motivated rates, see e.g.Ref. [38].The natural decay rate γ nat can be expressed using the time scale τ = a/v, which is the time a particle with velocity v needs to travel the distance a.For R/a = 2.1 we find γ nat τ = 0.436.The natural measure µ nat is uniformly distributed on the backward-trapped set in phase space, while on the boundary phase space one observes a 1 − p 2 dependence, see Fig. 2(d).In position representation it is shown in Fig. 11 (top, second from left).
For the structure of resonance states it was shown in Ref. [39] that the temporal distance t ϵ (x) to an ϵ-surrounding of the chaotic saddle is relevant, see Sec. 5.This temporal distance t ϵ (x), shown in Fig. 2(e), is defined as the time a particle starting at a phase-space point x ∈ Γ b on the backward-trapped set needs to come within a distance ϵ of the chaotic saddle Γ s under backward time evolution.In particular, on the chaotic saddle the temporal distance t ϵ is zero.The temporal distance t ϵ (x) shows a partitioning on the backward trapped set Γ b .This partitioning can be well described using stable and unstable manifolds of the periodic orbit at (s, p) = ( π 6 , 0), which has to be crossed in position space to exit the system, see Fig. 2(f).

Quantization and numerical method
The quantum dynamics of the three-disk scattering system is described by the free Schrödinger equation where the wave function ψ(r) fulfills Dirichlet boundary conditions at the boundary of each disk.We focus on the antisymmetric A 2 -representation of the system's C 3v symmetry, where Dirichlet boundary conditions are also imposed on the symmetry lines [77].The wavenumber k ∈ C is complex with its real part being inversely proportional to the wavelength.Its imaginary part is proportional to the decay rate γ of the resonance state, where the time scale τ = a/v is based on the k-dependent velocity v = ℏ Re k/m of a particle with mass m.Note that, for electromagnetic waves the constant velocity v is given by the speed of light c.
The physically most interesting states of the three-disk scattering system are its resonances which consist of outgoing waves only and occur at discrete values k n ∈ C. The resonance poles k n are the poles of the S-matrix and can be obtained from the zeros of a matrix M (k), explicitly given in Appendix A, such that det(M (k n )) = 0 [77].The left singular vector of M (k n ), which corresponds to the singular value zero, allows for determining the (right) wave function ψ n (r) of this resonance state fulfilling Eq. ( 1) [86].For an exemplary resonance state see Fig. 1(b).
We are able to determine poles and resonance states up to Re ka ≈ 10 5 for R/a = 2.1, which is about two orders of magnitude further in the semiclassical limit compared to previous publications on the three-disk scattering system [52,80,86].For larger R/a, e.g.R/a = 2.5, we even find poles for Re ka ≈ 10 6 (not shown).To the best of our knowledge this goes well beyond existing numerical analysis of any other closed or open billiard system.This is based on three ingredients, see Appendix A and Python code provided as supplementary material: (i) We use that in the three-disk scattering system the application of the matrix M (k) to a vector can be efficiently calculated using fast Fourier transforms, which need O(N log N ) instead of O(N 2 ) operations.This allows for treating matrix dimensions N ≈ Re ka for which the matrix M (k) could not be stored in memory.
(ii) We find all poles near a complex wavenumber k using a Taylor expansion of the matrix M (k), extending an approach for quantum billiards [96,97] to complex k.We increase the accuracy of poles and states to a desired precision by subsequent convergence steps.Previously, this procedure was successfully applied to dielectric cavities [38].For these steps, the matrix-vector multiplication from (i) allows for employing computationally efficient iterative methods for eigensystems and sets of linear equations.
(iii) The calculation of wave functions ψ(r) and Husimi functions H(x) from the normal derivative of the wave function on the disk's boundary is substantially accelerated by using fast Fourier transforms.

Spectrum and fractal Weyl law
The spectrum for R/a = 2.1 is obtained for Im ka ∈ [−1, 0] and selected intervals in Re ka up to ka ≈ 10 5 , see Fig. 3.At the upper end of the spectrum a gap to the real line is observed [76,[83][84][85], except for smaller k.In particular, for Re ka < 60 all poles are very close to the real line as the corresponding wavelength is larger than the opening between the disks.For increasing k the gap increases with the upper end of the spectrum converging towards Im ka = − γnatτ 2 , see Eq. ( 2).This is in agreement with a recent conjecture about the asymptotic spectral gap [98,99].For larger |Im ka| the density decreases as expected from the analogy to truncated random matrices [46,100,101].
This unprecedented large number of resonance poles allows for a comparison with the fractal Weyl law [46,47,[49][50][51][52][53][54][55][56][57][58][59][60][61][62]  as where d H is the partial Hausdorff dimension of the chaotic saddle.As we computed the spectrum in intervals only, see Fig. 3, we cannot determine N (κ).Instead, we compare to the fractal Weyl law for the density of states, dN (κ This is approximated by ∆N ∆κa using ∆κa = 50 in Fig. 4 where we find a power law κ δ with δ = 0.85 up to κa = 10 5 .This is in agreement to the partial box-counting dimension d 0 = 0.84 of the chaotic saddle, see Sec. 2.1, which is used to approximate the partial Hausdorff dimension d H appearing in the fractal Weyl law.

Resonance states
In Fig. 5(a) the intensity |ψ(r)| 2 of four representative resonance states is shown in position space.Due to the large wavenumber Re ka ≈ 2 • 10 4 the wave nature of the resonance states becomes apparent only by zooming into a small region of position space.These resonance states show a strong dependence on Im ka which corresponds to a varying dimensionless decay rate γτ , Eq. ( 2).For the smallest decay rate (left) the resonance state is strong in the inner region between the three disks.For larger decay rates (right) it becomes stronger towards the outer region of the scattering system.
A resonance state can be represented by a Husimi function H(x) on the boundary phase space x = (s, p) [86,102].This is shown in Fig. 5  examples further in the semiclassical limit at Re ka ≈ 10 5 with similar values of Im ka as in Fig. 5(a).They concentrate on the fractal backward-trapped set Γ b [21][22][23], cf.Fig. 2(a).The Husimi function is smooth on a scale of area 2π/(Re ka) corresponding to a Planck cell, which needs a magnification for the studied large value of k.For increasing decay rates (right) one observes more structure along the backward-trapped set.In particular, the intensity on the chaotic saddle is decreased.We observe that resonance states with a similar value of Im ka, i.e. similar dimensionless decay rate γτ , Eq. ( 2), show the same overall structure in position and Husimi representation (with increasingly fine details for increasing Re ka).This observation motivates the factorization of resonance states discussed in Sec.4.2.For convenience we use from here on for the dimensionless decay rate γτ just the symbol γ.In this paper we restrict the investigation to the parameter R/a = 2.1 of the threedisk scattering system.Values of R/a slightly above 2 have the advantage that the backward-trapped set Γ b has a large fractal dimension close to the maximal integer value.As the resonance states concentrate on the backward-trapped set, see Fig. 5(b), they can be much better analyzed for such values of R/a.In particular, there are many more fluctuations in the Husimi function than for the case of larger R/a, where the fractal dimension of the backward-trapped set tends to zero.
Recently, in dielectric cavities it was observed that all resonance states for large wavenumber k show strong scars, which were termed ray-segment scars [38,103].It can be observed just as well for the three-disk scattering system in all resonance states of Fig. 5(a) and the magnification of one of them in Fig. 6.A ray-segment scar is an enhancement of the intensity |ψ(r)| 2 along a segment of a ray.It extends for thousands of wavelengths, often beyond one or two reflections on the boundary.It is not related to the well-known periodic-orbit scars [12][13][14] of eigenstates of closed chaotic billiards, which occur for a small fraction of eigenstates only.In contrast, ray-segment scars are observed in every resonance state at sufficiently large wavenumber k.That ray-segment scars are not on periodic orbits can be seen best in Fig. 5(a, right) where one finds scars along rays leaving the system.Ray-segment scars can be conceptually explained [38] based on the factorization of resonance states discussed in Sec.4.2.Their properties will be discussed in Sec. 6.The observation of ray-segment scars in the three-disk scattering system also establishes this phenomenon in systems with full escape.Together with their observation in dielectric cavities with partial escape [38], this demonstrates the universal relevance of ray-segment scars for quantum chaotic scattering.

Factorization
In Refs.[38,71] the following conjecture was stated: Chaotic resonance states ψ(r) in scattering systems are a product of (i) conditionally invariant measures from classical dynamics with a smoothed spatial density ρ γ (r) depending on the states's decay rate γ and (ii) universal exponentially distributed fluctuations η(r) with mean one.The equality in Eq. ( 5) is understood in the sense that the right hand side has the same statistical properties as the resonance state on the left hand side.This factorization is just as well conjectured to hold in the Husimi representation, in phase space.This was validated for quantum maps [71] and dielectric cavities [38].
Here, we verify it for the three-disk scattering system, an autonomous system with full escape.Averaging Eqs. ( 5) and ( 6) over resonance states with a similar value of Im ka, i.e. similar decay rate γ, leads to relating the classical densities to the corresponding quantum average.Before studying the semiclassical origin of ρ γ (r) and ρ γ (x) in Sec. 5, we present in Sec.4.3 the averages ⟨|ψ(r)| 2 ⟩ γ and ⟨H(x)⟩ γ and discuss their dependence on the decay rate.In order to study the universality of the fluctuations η(r) and η(x) without using any knowledge about ρ γ (r) and ρ γ (x), we replace ρ γ in Eqs. ( 5) and ( 6) by the average, Eq. ( 7).This gives for the fluctuations of the resonance state ψ with decay rate γ, where x has to be chosen on the backward-trapped set Γ b .These fluctuations will be studied in Sec.4.4.
The factorization of an exemplary resonance state in the Husimi representation, H(x) = ⟨H(x)⟩ γ • η(x), is visualized in Fig. 7.The average shows a clear multifractal structure and the fluctuations show a uniform distribution on the backward-trapped set.

Average resonance states
The averages in position representation ⟨|ψ(r)| 2 ⟩ γ and on the boundary phase space ⟨H(x)⟩ γ at four representative value of Im ka are shown in Fig. 8.The averaging is in each case done over 500 resonance states with poles closest to the chosen Im ka.These resonance poles are also highlighted in Fig. 3.
For the average of resonance states their normalization relative to each other is important.While for closed billiards [104][105][106] and dielectric cavities [38] this is welldefined by integrating the absolute square of the wave function over the area of the system, there is no obvious area to integrate over for the three-disk scattering system.Therefore, we choose to normalize the resonance states such that 2π 0 ds |a 2 ϕ n (s)| 2 = |k n a| 2 where ϕ n (s) is the normal derivative of the wave function ψ n (r) on the outer side of the disk's boundary.This is motivated by the normalization of a wave function inside a closed circular boundary [104][105][106] (up to an irrelevant factor of 2).In particular, it is numerically convenient, see Appendix A.3, in contrast to integrating |ψ n (r)| 2 over some region in position space.
In Fig. 8(b) the averaged resonance states in Husimi representation show a strong dependence on Im ka, which is even more pronounced than for individual resonance states, see Fig. 5(b).As expected for Im ka corresponding to the natural decay rate γ nat (Fig. 8(b), second from left) a uniform distribution on the backward-trapped set is observed [30][31][32][33][34][35][36][37][38] corresponding to the natural measure µ nat , see Fig. 2(d).By smoothing µ nat on the scale of a Planck cell, see Fig. 10 (second from left), perfect agreement is achieved.With increasing |Im ka| (Fig. 8(b), to the right) the probability density on the backward-trapped set becomes fractal along the backward-trapped set, in particular it decreases on the chaotic saddle.For resonance states with smallest |Im ka| (Fig. 8(b), left) one observes a small increase on the chaotic saddle, which is noticeable by averaging only.The multifractal structure is qualitatively well described by the partitioning of the backward-trapped set based on the stable and unstable manifold of the periodic orbit at the exit, see Fig. 2(f).
In position representation, shown in Fig. 8(a), the averaged resonance states, just like individual resonance states, become stronger towards the outer region of the scattering system for increasing |Im ka|.Beyond that one finds very weak structure especially in comparison to dielectric cavities in position representation [38].In Sec. 5 we will relate the averages ⟨|ψ(r)| 2 ⟩ γ and ⟨H(x)⟩ γ according to Eq. ( 7) to classical densities ρ γ (r) and ρ γ (x).

Universal fluctuations
In closed systems the distribution of the intensities |ψ(r)| 2 and H(x) of chaotic eigenstates follow the universal distribution of the random wave model [7][8][9][10][11].For the complex (real) random wave model this is a complex (real) Gaussian distribution of the eigenstate amplitudes leading to an exponential (Porter-Thomas) distribution of the intensities.In Fig. 9 In Fig. 9(b) the distributions of the fluctuations η(r) and η(x), defined in Eq. ( 8), are shown.They follow a universal exponential distribution with mean one over more than three orders of magnitude.This shows that the fluctuations η follow a complex random wave model, demonstrating the importance of the factorization for understanding the structure of resonance states.This is in line with previous findings for quantum maps [71] and dielectric cavities [38].
For other decay rates we adopt a conditionally-invariant measure from quantum maps in Sec.5.1, which very well describes the averaged resonance states.We find in Sec.5.2, however, that this is not the correct semiclassical limit measure.Recent improvements are discussed in Sec. 6.

Measure based on temporal distance
In a system with full escape it was conjectured in Ref. [39] that the semiclassical measure is uniformly distributed on sets with the same temporal distance t ϵ to an ϵ-surrounding of the chaotic saddle Γ s .This yields a conditionally-invariant measure with decay rate γ, for any set A in phase space.This measure modifies the natural measure µ nat depending on the temporal distance t ϵ (x) shown in Fig. 2(d, e).The modification becomes stronger with increasing deviation of the desired decay rate γ from γ nat .For the reasoning behind Eq. ( 9), see Ref. [39].The measure µ ϵ γ smoothed on the scale of the corresponding Planck cell is displayed in phase space in Fig. 10 and compared to the averaged Husimi representation ⟨H⟩ γ .On a qualitative level, we observe very good agreement for the three additional decay rates γ ̸ = γ nat .In particular, this holds for the partitioning by the stable and unstable manifold of the periodic orbit at the exit, see Fig. 2(f).However, we observe that the value of the measure shows some deviations which increase with the distance from γ nat .This will be further quantified using the Jensen-Shannon divergence, see Sec. 5.2.
In position space one can define a measure µ ϵ γ (r) in analogy to Eq. ( 9).It is shown in Fig. 11, integrated over the size of each pixel, and compared to the averaged intensities ⟨|ψ(r)| 2 ⟩ γ .We find very good agreement, with the grain of salt that there is not much structure in position space to make a detailed comparison.
Note that in Ref. [39] for the ϵ-surrounding the specific value ϵ = ℏ/2 was used, leading to the h-resolved chaotic saddle.In our case of Reka ≈ 10 5 this would correspond to ϵ = 1/(2 Re ka) ≈ 2 • 10 −3 , which however does not lead to a good agreement in a figure (not shown) equivalent to Fig. 10, where we choose heuristically ϵ = 10 −4 .We explain this in the following way: In order to have a certain resolution of the temporal distance t ϵ along the backward-trapped set (i.e. the unstable direction) one needs to define the ϵ-surrounding of the chaotic saddle on a much finer scale in this direction.As the temporal distance has a sufficiently fine resolution, see Fig. of ϵ = 10 −4 seems sufficiently small.In fact for ϵ = 10 −5 we find no changes.

Jensen-Shannon divergence
In Fig. 10 we find for the averaged Husimi representation perfect agreement for γ nat but observe deviations for γ ̸ = γ nat using the measure µ ϵ γ based on the temporal distance.We now quantify this using the Jensen-Shannon divergence d JS [107], which compares two probability distributions.This was used to compare a measure to resonance states in systems with partial escape in Ref. [40].For computing the Jensen-Shannon divergence one has to choose a partitioning of phase space.For studying the semiclassical limit we choose a fixed scale independent of k given by squares with side length √ h max = 2π/(Re k min a) and k min a = 1000.The Jensen-Shannon divergence is shown in Fig. 12 for six decay rates and Re ka from 10 3 to 10 5 .For γ nat , corresponding to |Im ka| = 0.218, we find convergence in the semiclassical limit confirming the qualitative impression of Fig. 10 and previous findings [30][31][32][33][34][35][36][37][38].For γ < γ nat , corresponding in Fig. 12 to |Im ka| = 0.18, one finds small values of d JS , however the convergence cannot be studied, as there are no resonance poles in this region in the semiclassical limit, see Fig. 3.For γ > γ nat we find no convergence in the semiclassical limit which suggests that the measure µ ϵ γ based on the temporal distance is not the correct semiclassical limit measure.Recent improvements are discussed in Sec. 6.

Discussion and outlook
We demonstrate that the factorization conjecture holds for resonance states in the threedisk scattering system.One factor is described by universal exponentially distributed intensity fluctuations.For the other factor, also describing averaged resonance states, we apply a construction from maps with full escape to the three-disk scattering system.We observe very good, but not perfect, agreement.The factorization conjecture complements the periodic-orbit approach to resonance states of the three-disk scattering system.It allows for obtaining new insights into their structure, in particular about their dependence on the decay rate as well as about similarities and differences of neighboring resonance states.By computing resonance states further in the semiclassical limit than before, we are able to validate the fractal Weyl law over a very large range of wavenumbers.
The recently observed ray-segment scars in dielectric cavities [38] are also found in the three-disk scattering system.This demonstrates the universal relevance of raysegment scars for quantum chaotic scattering.They have been explained [38] based on the factorization conjecture: Whenever the multifractal classical density shows strong intensity enhancements in phase space, then the additional universal fluctuations give rise to some phase-space points with extremely high intensities.In every resonance state this leads to scars along segments of rays, which are to periodic orbits.The most likely directions are determined by the high intensities of the multifractal classical density.The specific direction of the ray segment varies from state to state, as the phasespace points with extreme intensities vary due to the universal fluctuations.Going to the semiclassical limit the multifractal classical density is resolved on finer scales, leading to higher intensities and thus to stronger scars.In the future it is desirable to analyze ray-segment scars quantitatively, in particular their length, width and intensity distributions.
Another future aim is to derive the semiclassical measure which perfectly describes the resonance states in systems with full or partial escape.In particular, it would be desirable to find a common approach, as currently one either needs the natural measure and the temporal distance (full escape [39]) or the natural and the inverse natural measure (partial escape [38,40]).In fact, work in progress shows that such a common approach to derive a semiclassical measure exists, using ideas from systems with local randomization [72], which describes the resonance states even better [108].Another approach for the three-disk scattering system is to use the semiclassical theory based on dynamical zeta functions consisting of periodic orbits.This may allow for a derivation of the semiclassical measure.More generally, the periodic-orbit approach might be able to give support to the factorization conjecture.

Figure 1 .
Figure 1.(a) Visualization of the three-disk scattering system with radius a of the disks and distance R of their centers.Its fundamental domain is shown by red lines.The dimensionless Birkhoff coordinate s along the disk's boundary starts at the symmetry line and is chosen in clockwise direction.(b) Exemplary resonance state in position space at ka = 58.86− 0.35i.Note that for clarity this figure uses R/a = 2.5, while the value R/a = 2.1, with disks closer to each other, is used throughout the paper.

Figure 2 .
Figure 2. Invariant sets for R/a = 2.1 on the boundary phase space (s, p), (a) backward-trapped set Γ b , (b) forward-trapped set Γ f , and (c) chaotic saddle Γ s .(d) Natural measure µ nat on the boundary phase space supported by the backwardtrapped set Γ b , where the mean value of non-zero pixels is set to one.(e) Temporal distance t ϵ to an ϵ-surrounding of the chaotic saddle Γ s for ϵ = 10 −4 .(f) Stable manifold (blue) and unstable manifold (red, restricted to Γ b ) of the periodic orbit, which has to be crossed to exit the system (inset).

Figure 3 .
Figure 3. Spectrum for R/a = 2.1 for various intervals in Re ka and Im ka ∈ [−1, 0] showing about 10 5 resonance poles.The value of Im ka = − γnatτ 2 corresponding via Eq.(2) to the natural decay rate γ nat is shown as a line (dashed).The poles corresponding to resonance states used for averaging in Sec.4.3 are marked in red.

Figure 4 .
Figure 4. Fractal Weyl law for density of states, Eq. (4), approximated with bin size ∆κa = 50 and cutoff Ca = 1 for the poles shown in Fig.3.A power-law fit in the interval κa ∈ [10 2 , 10 5 ] yields the power-law exponent δ = 0.85.The fit function is shown in the main panel (shifted, dashed line) and in the inset for κa ∈ [10000, 11000] (not shifted, solid line).The circles highlight the density of states in the smaller intervals at larger values of κa.
(b) including two representative

8 Figure 5 .
Figure 5. Resonance states for R/a = 2.1 at four representative values of |Im ka|, increasing from left to right.(a) Intensity |ψ(r)| 2 in position space (fundamental domain confined to a region of size 0.5a × 0.7a) for Re ka ≈ 2 • 10 4 .The inset (left) is a magnification by a factor 150 showing fluctuations on the scale of the wavelength.(b) Husimi representation H(x) on the boundary phase space x = (s, p) of a disk for Re ka ≈ 2 • 10 4 (left, right) and even larger wavenumber Re ka ≈ 10 5 (second and third from left).The inset (second from left) shows fluctuations on the scale of the Planck cell.In all figures the average value (in (a) on fundamental domain, in (b) on backward-trapped set) is scaled to one and intensities greater than the maximal value of the colorbar are shown with darkest color.

Figure 6 .
Figure 6.Ray-segment scars in a magnification of the resonance state from Fig. 5(a, second from right) at ka = 20258.74− 0.50i shown for the region 0.25a × 0.2a (corresponding to about 800 × 650 wavelengths).

Figure 7 .
Figure 7. Factorization of an exemplary resonance state in Husimi representation H(x) on the boundary phase space (left) into an average ⟨H(x)⟩ γ of resonance states with similar decay rate (middle) and fluctuations η(x) (right).The resonance state is taken from Fig. 5(b, second from right) and the average is also shown in Fig. 8(b, second from right).

Figure 9 .
Figure 9. Distribution of (a) the intensities I = |ψ(r)| 2 (blue) and I = H(x), x ∈ Γ b (red) and (b) the fluctuations η(r) (blue) and η(x) (red) for the resonance states of Fig. 5, colored by increasing |Im ka| from light to dark.For comparison an exponential distribution with mean one is shown (black).
(a) the distribution of the intensities |ψ(r)| 2 and the Husimi representation H(x) for x ∈ Γ b are shown for the resonance states from Fig. 5.One finds different distributions for different Im ka and for the two representations, in particular no exponential distribution.