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Distribution of Resonances for Open Quantum Maps
Abstract
We analyze simple models of classical chaotic open systems and of their quantizations (open quantum maps on the torus). Our models are similar to models recently studied in atomic and mesoscopic physics. They provide a numerical confirmation of the fractal Weyl law for the density of quantum resonances of such systems. The exponent in that law is related to the dimension of the classical repeller (or trapped set) of the system. In a simplified model, a rigorous argument gives the full resonance spectrum, which satisfies the fractal Weyl law. For this model, we can also compute a quantity characterizing the fluctuations of conductance through the system, namely the shot noise power: the value we obtain is close to the prediction of random matrix theory.
... On the other hand very little is known about spectral gaps for systems with P (1/2) > 0 and Theorem 1 appears to be the first general result in this case, albeit for a special class of systems. Examples of systems with P (1/2) > 0 and a spectral gap were previously given in [NoZw07], discussed below, and [DyZa]. ...
... The closest to the present paper is the work of Nonnenmacher-Zworski [NoZw05,NoZw07] who studied open quantum baker's maps, in particular the Walsh quantization for the cases M = 3, 4, A = {0, 2}, χ ≡ 1 in the notation of our paper (as well as obtaining numerical results for other maps and quantizations). The Walsh quantization is obtained by replacing F N , F N/M in (1.2) by the Walsh Fourier transform, which is the Fourier transform on the group (Z M ) k . ...
... The Walsh quantization is obtained by replacing F N , F N/M in (1.2) by the Walsh Fourier transform, which is the Fourier transform on the group (Z M ) k . Eigenvalues of Walsh quantizations are computed explicitly in [NoZw07,§5], which proves fractal Weyl law and shows concentration of resonances around decay rate P (1)/2. Moreover [NoZw07] shows that there is a spectral gap for M = 3, A = {0, 2} but not for M = 4, A = {0, 2}. ...
We study eigenvalues of quantum open baker's maps with trapped sets given by linear arithmetic Cantor sets of dimensions . We show that the size of the spectral gap is strictly greater than the standard bound for all values of , which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus.
... [37,34,39]. In Section 2.6 and Section 8, we will also look at an alternative (non-Weyl) quantization method for the torus and baker map that was studied in [39,45,32,3]. However, most of this article will focus on the Balazs-Voros quantization (1.2), though similar techniques also apply to other related Weyl quantizations such as the symmetrized Saraceno quantization [37]. ...
... Finally, we will look at an alternate Walsh (non-Weyl) quantization of the baker map studied in [39,45,32,3] which has highly degenerate eigenspaces. In this case, the random band-limited wave statistics can be replaced by statistics of actual eigenbases of the quantized operator. ...
... Eigenstates for the Walsh quantized baker map. In this section we briefly define the Walsh quantization of the baker map, which has been studied in [39,45,32,3]. We will provide further details in Section 8. Fix D ≥ 2 and consider the classical D-baker map on T 2 , B(q, p) = (Dq mod 1, p + ⌊Dq⌋ D ). ...
We extend the approach from [arXiv:2110.15301] to prove windowed spectral projection estimates and a generalized Weyl law for the (Weyl) quantized baker's map on the torus. The spectral window is allowed to shrink. As a consequence, we obtain a strengthening of the quantum ergodic theorem from [arXiv:math-ph/0412058] to hold in shrinking spectral windows, a Weyl law on uniform spreading of eigenvalues, and statistics of random band-limited waves. Using similar techniques, we also investigate generic random eigenbases of a different (non-Weyl) quantization, the Walsh-quantized D-baker map, which has high degeneracies in its spectrum. For generic random eigenbases, we prove gaussian eigenstate statistics and QUE with high probability in the semiclassical limit.
... In Sect. 3 we recall some fundamental notions and results from standard semiclassical calculus needed in this paper. We use this to describe the procedure to quantize complex-valued function on dilated tori T 2d α , generalizing the quantization approach presented in [4,21]. ...
... In this section we begin by reviewing some basic notions and properties of semiclassical calculus in R d , as can be found for instance in [8,19,30]. Afterwards, we will review the Toeplitz quantizaton of functions on the torus, as presented in [4,21], which roughly speaking consist in restricting the semiclassical calculus to periodic symbols and to function spaces given by tempered distributions which are both periodic in space and in semiclassical frequency. ...
... We essentially follow the approach of [4,21] who considered the case α = 1 of the subsequent. For α > 0, we define the torus ...
We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean 0, variance 1 and bounded fourth moment. We prove that the eigenvalues of the perturbed operator satisfy a Weyl law with probability close to one, which proves in particular a conjecture by Christiansen and Zworski (Commun Math Phys 299:305–334, 2010).
... In Section 3 we recall some fundamental notions and results from standard semiclassical calculus needed in this paper. We use this to describe the procedure to quantize complex-valued function on dilated tori T 2d α , generalizing the quantization approach presented in [4,21]. ...
... In this section we begin by reviewing some basic notions and properties of semiclassical calculus in R d , as can be found for instance in [8,19,30]. Afterwards, we will review the Toeplitz quantizaton of functions on the torus, as presented in [4,21], which roughly speaking consist in restricting the semiclassical calculus to periodic symbols and to function spaces given by tempered distributions which are both periodic in space and in semiclassical frequency. ...
... Quantization of the torus. We essentially follow the approach of [4,21] who considered the case α = 1 of the subsequent. For α > 0, we define the torus ...
We study the eigenvalues of the Toeplitz quantization of complex-valued functions on the torus subject to small random perturbations given by a complex-valued random matrix whose entries are independent copies of a random variable with mean 0, variance 1 and bounded fourth moment. We prove that the eigenvalues of the perturbed operator satisfy a Weyl law with probability close to one, which proves in particular a conjecture by T.~Christiansen and M.~Zworski.
... [15,16] for reviews, and Refs. [14][15][16][17][18][19][21][22][23][24][25][26][27][28][29][30][31][32][33][34] for some examples. ...
... See Refs. [14][15][16][17][18][19][21][22][23][24][25][26][27][28][29][30][31][32][33][34] for examples. While these heuristic models have stimulated many findings in the recent past, they do not represent a step by step reduction of a scattering system in the sense of Ref. [20] and it remains unclear to which degree they faithfully represent resonances. ...
... Ultimately, we hope that the uniform hyperbolicity of the proposed model system can be proven. This would result in a new sibling among the uniformly hyperbolic model systems, which, in contrast to the three-disc scatterer [1- 3,63] and the open baker map [15,[23][24][25]27], is free from distracting diffraction effects. This should be useful for future investigations of fractal Weyl conjectures and further topics in chaotic scattering. ...
We consider quantum maps induced by periodically-kicked scattering systems and discuss the computation of their resonance spectra in terms of complex scaling and sufficiently weak absorbing potentials. We also show that strong absorptive and projective openings, as commonly used for open quantum maps, fail to produce the resonance spectra of kicked scattering systems, even if the opening does not affect the classical trapped set. The results are illustrated for a concrete model system whose dynamics resembles key features of ionization and exhibits a trapped set which is organized by a topological horseshoe at large kick strength. Our findings should be useful for future tests of fractal Weyl conjectures and investigations of dynamical tunneling.
... On the other hand, very little is known about spectral gaps for systems with P(1/2) > 0 and Theorem 1 appears to be the first general result in this case, albeit for a special class of systems. Examples of systems with P(1/2) > 0 and a spectral gap were previously given in [NoZw07], discussed below, and [DyZa]. ...
... The closest to the present paper is the work of Nonnenmacher-Zworski [NoZw05,NoZw07] who studied open quantum baker's maps, in particular the Walsh quantization for the cases M = 3, 4, A = {0, 2}, χ ≡ 1 in the notation of our paper (as well as obtaining numerical results for other maps and quantizations). The Walsh quantization is obtained by replacing F N , F N /M in (1.2) by the Walsh Fourier transform, which is the Fourier transform on the group (Z M ) k . ...
... The Walsh quantization is obtained by replacing F N , F N /M in (1.2) by the Walsh Fourier transform, which is the Fourier transform on the group (Z M ) k . Eigenvalues of Walsh quantizations are computed explicitly in [NoZw07,§5], which proves fractal Weyl law and shows concentration of resonances around decay rate P(1)/2. Moreover, [NoZw07] shows that there is a spectral gap for M = 3, A = {0, 2} but not for M = 4, A = {0, 2}. ...
We study eigenvalues of quantum open baker's maps with trapped sets given by linear arithmetic Cantor sets of dimensions . We show that the size of the spectral gap is strictly greater than the standard bound for all values of , which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus.
... However, only the upper bounds for the number of resonances could be rigorously proven [41, 49, 15, 42]. To avoid the complexity of " realistic " scattering systems, one can study simpler models, namely quantized open maps on a compact phase space, for instance the quantized open baker's map studied in [30, 31, 20] (see §2.1). Such a model is meant to mimick the propagator of the nonselfadjoint Hamiltonian H W, , in the case where the classical flow at energy E is chaotic in the interacting region. ...
... In §4.1 we numerically check this fractal law for an asymmetric version of the open baker's map; apart from extending the results of [30], this model allows to specify more precisely the dimension d appearing in the scaling law. To our knowledge, so far the only system for which the asymptotics corresponding to (1.1) could be rigorously proven is the " Walsh quantization " of the symmetric open baker's map [31], which will be described in §6. After counting resonances, the next step consists in studying the long-living resonant eigenstates ϕ j or˜ϕor˜ or˜ϕ j . ...
... In the case of the " standard " quantized open baker, the classification remains open. In §6 we consider a solvable model, the Walsh quantization of the open baker's map, introduced in [30, 31]. For that model, one can explicitly construct some semiclassical measures and partially answer the above questions. ...
We study the spectrum of quantized open maps as a model for the resonance spectrum of quantum scattering systems. We are particularly interested in open maps admitting a fractal repeller. Using the 'open baker's map' as an example, we numerically investigate the exponent appearing in the fractal Weyl law for the density of resonances; we show that this exponent is not related with the 'information dimension', but rather the Hausdorff dimension of the repeller. We then consider the semiclassical measures associated with the eigenstates: we prove that these measures are conditionally invariant with respect to the classical dynamics. We then address the problem of classifying semiclassical measures among conditionally invariant ones. For a solvable model, the 'Walsh-quantized' open baker's map, we manage to exhibit a family of semiclassical measures with simple self-similar properties.
... Further studies lead to the verification of the fractal Weyl law [69] [53] [54] (see §6.1). Such discrete time models have several advantages. ...
... Such discrete time models have several advantages. Firstly, the long time dynamics of the classical map κ is sometimes very easy to analyze; this is the case for instance for the open baker's map studied in [53] [54], which we will study explicitly in §6.1.1. Secondly, the corresponding open quantum maps are often very explicit matrices, which can be numerically diagonalized, much more easily so than Schrödinger operators like P θ (). ...
... . The open baker's map. The spectra of several types of quantum open baker's maps were analyzed in [53] [54] [50] ...
We present an overview of mathematical results and methods relevant for the spectral study of semiclassical Schrödinger (or wave) operators of scattering systems, in cases where the corresponding classical dynamics is chaotic; more precisely, we assume that in some energy range, the classical Hamiltonian flow admits a fractal set of trapped trajectories, which hosts chaotic (hyperbolic) dynamics. The aim is then to connect the information on this trapped set with the distribution of quantum resonances in the semiclassical limit.
Our study encompasses several models sharing these dynamical characteristics: free motion outside a union of convex hard obstacles, scattering by certain families of compactly supported potentials, geometric scattering on manifolds with (constant or variable) negative curvature. We also consider the toy model of open quantum maps, and sketch the construction of quantum monodromy operators associated with a Poincaré section for a scattering flow.
The semiclassical density of long-living resonances exhibits a fractal Weyl law, related to the fact that the corresponding metastable states are 'supported' by the fractal trapped set (and its outgoing tail). We also describe a classical condition for the presence of a gap in the resonance spectrum, equivalently a uniform lower bound on the quantum decay rates, and present a proof of this gap in a rather general situation, using quantum monodromy operators.
... Our main new result is Theorem 15 page 26 which provides an upper bound o α n−1/2 (with n = dim X), for the number of resonances in the spectral domain ℜ (λ) ∈ [α, α + √ α], ℑ (λ) > −β (all β) in the semiclassical limit |α| → ∞. The use of escape functions on phase space for resonances has been introduced by B. Helffer and J. Sjöstrand [25] and used in many situations [46,42,45,44,54,24,35]. In particular in [24], the authors consider the geodesic flow associated to Schottky groups and provide an upper bound for the density of Ruelle resonances (see also [8]). ...
... • The case E = 0 is excluded in Theorem 15 because the vicinity of the origin ξ = 0 is excluded in (35). If one were able to construct an escape function such that in addition X (G m ) (x, ξ) ≤ C, ∀ (x, ξ), with some C independent of m then E = 0 would not be excluded. ...
... Since the vector field X is linear in the fibers of the bundle T * X we get the same estimates (33) and (35). We can now proceed as in Section 4.1: G m is a h-semiclassical symbol , G m ∈ S +0 and quantization gives ...
Using a semiclassical approach we show that the spectrum of a smooth Anosov
vector field V on a compact manifold is discrete (in suitable anisotropic
Sobolev spaces) and then we provide an upper bound for the density of
eigenvalues of the operator (-i)V, called Ruelle resonances, close to the real
axis and for large real parts.
... It is interesting to note that this direction of research has been further driven by mathematicians. In particular, the fractal Weyl law has been conjectured under rigorous mathematical settings [12][13][14][15][16][17][18][19]. In accordance with these works, numerical [20][21][22][23][24][25][26], and even experimental works [30][31][32], which have attempted to verify the conjecture, followed. ...
... (2. 15) In the following, we rewrite ( q, p) as (q, p), and U as U, respectively. ...
In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a numerically sound method for computing quantum resonance states, did not exist. To address this challenge, we recently introduced a new class of quantum maps (Mertig and Shudo 2018 Phys. Rev. E 97 042216). For these quantum maps, we showed that, quantum resonance states can numerically be computed using theoretically grounded methods such as complex scaling or weak absorbing potentials (Mertig and Shudo 2018 Phys. Rev. E 97 042216). However, proving uniform hyperbolicty for this class of quantum maps was not straight forward. Going beyond that work this article generalises the class of scattering maps and provides mathematical proofs for their uniform hyperbolicity. In particular, we show that the suggested class of two-dimensional symplectic scattering maps satisfies the topological horseshoe condition and uniform hyperbolicity. In order to prove these properties, we follow the procedure developed in the paper by Devaney and Nitecki (1979 Commun. Math. Phys. 67 137–46). Specifically, uniform hyperbolicity is shown by identifying a proper region in which the non-wandering set satisfies a sufficient condition to have the so-called sector bundle or cone field. Since no quantum map is known where both a proof of uniform hyperbolicity and a methodologically sound method for numerically computing quantum resonance states exist simultaneously, the present result should be valuable to further test fractal Weyl laws and related topics such as chaotic eigenfunction hypothesis.
... Despite some numerical evidence, for models based on partial differential equations there are no rigorous results showing that these bounds are optimal. However, recent joint work with Stéphane Nonnenmacher [14] shows that the bounds are optimal for some discrete models of chaotic scattering based on open quantum maps. Here, some of the ideas of [18] are explained in detail by proving a simpler result about the number of (complex) eigenvalues of a chaotic potential with a complex absorbing barrier . ...
... Numerical investigations in different settings of semiclassical three bump potentials [8],[9], Schottky quotients [6], three disc scattering [10], and Cantor-like Julia sets for z → z 2 + c, c < −2 [20], suggest that for µ ≃ Ch and δ ≃ 1 the estimate (1.8) is optimal. A different model was recently considered in [14]: quantum resonances were defined using an open quantum map with a classical " trapped set " corresponding to K intersected with a hypersurface transversal to the flow. The numerical results and a simple linear algebraic toy model suggest that the fine estimate (1.5) is optimal. ...
We present results of recent work with Johannes Sjostrand (18) on upper bounds of the number of semiclassical resonances for systems with chaotic classical dynamics. These upper bounds are interpreted as "fractal Weyl laws for resonances" since the exponent is now related to the dimension of the trapped set of the classical system. Despite some numerical evidence, for models based on partial differential equations there are no rigorous results showing that these bounds are optimal. However, recent joint work with Stephane Nonnenmacher (14) shows that the bounds are optimal for some discrete models of chaotic scattering based on open quantum maps. Here, some of the ideas of (18) are explained in detail by proving a simpler result about the number of (complex) eigenvalues of a chaotic potential with a complex absorbing bar- rier. That corresponds to a model popular in computational chemistry - see the work of Stefanov (19) for a recent mathematical treatment and references. The energy interval we consider has a fixed length, rather than the length Ch, which leads to further, more serious, simplifications.
... Schottky quotients [12], three disc scattering [19], and Cantor-like Julia sets for z → z 2 + c, c < −2 [35], suggest that for µ ≃ Ch and δ ≃ 1 the estimate (1.4) is optimal. A different model was recently considered in [25]: quantum resonances were defined using an open quantum map with a classical " trapped set " corresponding to K intersected with a hypersurface transversal to the flow – seeFig.5. The numerical results and a simple linear algebraic toy model suggest that the fine estimate (1.3) is optimal – seeFig.3. ...
... A particularly nice feature of the theory developed in [13] is allowing arbitrary homogeneous polynomials as potentials (see [ EvenFigure 3. An example of numerical results of [25]. The eigenvalues of a 3N × 3N matrix A 3N model the resonances in a disc of size h ∼ 1/N. ...
We consider bounds on the number of semiclassical resonances in neighbourhoods of the size of the semiclassical parameter, h , around energy levels at which the flow is hyperbolic. We show that the number of resonances is bounded by , where is essentially the dimension of the trapped set on the energy surface. We note that in a confined setting, this dimension is equal to , where n is the dimension of the physical space and the bound, , corresponds to the optimal bound on the number of eigenvalues. Although no lower bounds of this type are rigorously known in the setting of semiclassical differential operators, the corresponding bound is optimal for certain models based on open quantum maps (see [26])
... Localisation results which relay on numerical range are given in [1]. A lot of results, often related to some dynamical assumptions, are obtained by microlocal arguments (see for instance [4], [22], [17] and [18]). For the magnetic Schrödinger operator, we also mention [5] where localization of resonances is obtained by perturbation methods. ...
For general non-symmetric operators A, we prove that the moment of order of negative real-parts of its eigenvalues is bounded by the moment of order of negative eigenvalues of its symmetric part As an application, we obtain Lieb-Thirring estimates for non self-adjoint Schr\"odinger operators. In particular, we recover recent results by Frank, Laptev, Lieb and Seiringer \cite{FLLS}. We also discuss moment of resonances of Schr\"odinger self-adjoint operators.
... Open quantum maps have been widely used to model the scattering behavior of open quantum systems, with relations between the two settings established by Nonnenmacher-Sjöstrand-Zworski [NSZ11]. In this paper, we consider the toy Here N ∈ N is taken to be a multiple of M ∈ N, F N is the unitary discrete Fourier transform on C N defined in (2.1) below, and χ N/M ∈ C N/M is the discretization of a cutoff function χ ∈ C ∞ c ((0, 1); [0, 1]) given by Open quantum baker's maps have previously been studied by Keating-Novaes-Prado-Sieber [KNPS06], Keating-Nonnenmacher-Novaes-Sieber [KNNS08], Nonnenmacher-Zworski [NZ05,NZ07], and Dyatlov-Jin [DJ17,DJ18]; see the introduction to [DJ17] for a more detailed review of the literature. ...
We prove two results on Fractal Uncertainty Principle (FUP) for discrete Cantor sets with large alphabets. First, we give an example of an alphabet with dimension where the FUP exponent is exponentially small as the size of the alphabet grows. Secondly, for we show that a similar alphabet has a large FUP exponent, arbitrarily close to the optimal upper bound of , if we dilate the Fourier transform by a factor satisfying a generic Diophantine condition. We give an application of the latter result to spectral gaps for open quantum baker's maps.
... It is interesting to note that this direction of research has been further driven by mathematicians. In particular, the fractal Weyl law has been conjectured under rigorous mathematical settings [12,13,14,15,16,17,18,19]. In accordance with these works, numerical [20,21,22,23,24,25,26], and even experimental works [30,31,32], which have attempted to verify the conjecture, followed. ...
In recent years fractal Weyl laws and related quantum eigenfunction hypothesis have been studied in a plethora of numerical model systems, called quantum maps. In some models studied there one can easily prove uniform hyperbolicity. Yet, a numerically sound method for computing quantum resonance states, did not exist. To address this challenge, we recently introduced a new class of quantum maps. For these quantum maps, we showed that, quantum resonance states can numerically be computed using theoretically grounded methods such as complex scaling or weak absorbing potentials. However, proving uniform hyperbolicty for this class of quantum maps was not straight forward. Going beyond that work this article generalizes the class of scattering maps and provides mathematical proofs for their uniform hyperbolicity. In particular, we show that the suggested class of two-dimensional symplectic scattering maps satisfies the topological horseshoe condition and uniform hyperbolicity. In order to prove these properties, we follow the procedure developed in the paper by Devaney and Nitecki. Specifically, uniform hyperbolicity is shown by identifying a proper region in which the non-wandering set satisfies a sufficient condition to have the so-called sector bundle or cone field. Since no quantum map is known where both a proof of uniform hyperbolicity and a methodologically sound method for numerically computing quantum resonance states exist simultaneously, the present result should be valuable to further test fractal Weyl laws and related topics such as chaotic eigenfunction hypothesis.
... / 1 . For a rigorous analysis of the analogy, see papers of Degli Esposti, Nonnenmacher, and Winn [5] and Nonnenmacher and Zworski [16]. Heuristics can be found in earlier works of Balázs and Voros [1] and Saraceno and Voros [19]. ...
We find Weyl upper bounds for the quantum open baker’s map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound \mathcal O(N^\delta) , where \delta is the dimension of the trapped set of the baker’s map and (2 \pi N)^{-1} is the semiclassical parameter, which improves upon the previous result of \mathcal O(N^{\delta + \epsilon}) . Furthermore, we derive a Weyl upper bound with explicit dependence on the inner radius of the annulus for quantum open baker’s maps with Gevrey cutoffs.
... It is appropriate also to mention an alternative line of research going back to [26] and relying upon semiclassical methods to address resonances in quantum chaotic systems [27][28][29][30]. Note also a growing interest in studying the resonance statistics in such and related systems by rigorous mathematical methods, both in semiclassical [31,32] and random matrix framework [33][34][35]. ...
We develop a general non-perturbative characterisation of universal features of the density of S-matrix poles (resonances) describing waves incident and reflected from a disordered medium via a single M-channel waveguide/lead. Explicit expressions for are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D and 3D media. In the case of perfectly coupled lead with a few channels () the most salient features are tails for narrow resonances reflecting exponential localization and for broad resonances reflecting states located in the vicinity of the attached wire. For multimode wires with , intermediate asymptotics is shown to emerge reflecting diffusive nature of decay into wide enough contacts.
... The quantum open baker's map can then be seen as the discrete analogue of this Fourier integral operator with the corresponding semiclassical parameter (2πN ) −1 . For a rigorous analysis of the analogy, see papers of Degli Esposti-Nonnenmacher-Winn [ENW06] and Nonnenmacher-Zworski [NZ06]. Heuristics can be found in earlier works of Balázs-Voros [BV89] and Saraceno-Voros [SV94]. ...
We find Weyl upper bounds for the quantum open baker's map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound where is the dimension of the trapped set of the baker's map and is the semiclassical parameter, which improves upon the previous result of . Furthermore, we derive a Weyl upper bound with explicit dependence on the inner radius of the annulus for quantum open baker's maps with Gevrey cutoffs.
... This question is related with the gap question discussed in the next section. This conjectural fractal Weyl's law has been tested numerically on various chaotic systems, with variable success: Schrödinger operator with a smooth potential [21], hyperbolic surfaces by [17] and [2], discrete time analogues of scattering systems (quantized open maps) in [27], and even experimentally in the case of the scattering by N disks, see [32]. ...
The study of wave propagation outside bounded obstacles uncovers the existence of resonances for the Laplace operator, which are complex-valued generalized eigenvalues, relevant to estimate the long time asymptotics of the wave. In order to understand distribution of these resonances at high frequency, we employ semiclassical tools, which leads to considering the classical scattering problem, and in particular the set of trapped trajectories. We focus on "chaotic" situations, where this set is a hyperbolic repeller, generally with a fractal geometry. In this context, we derive fractal Weyl upper bounds for the resonance counting; we also obtain dynamical criteria ensuring the presence of a resonance gap. We also address situations where the trapped set is a normally hyperbolic submanifold, a case which can help analyzing the long time properties of (classical) Anosov contact flows through semiclassical methods.
... (Related experiments by Barkhofen et al[B * 13] confirmed the pressure gaps of Theorem 9, seeFigures 10 and 26). Fractal Weyl laws have been considered (and numerically checked) for various open chaotic quantum maps -see[NZ07],[DJ16] and references given there. The have also been proposed in other types of chaotic systems, ranging from dielectric cavities to communication and social networks -see recent review articles by Cao-Wiersig[CW15] ...
We provide an introduction to mathematical theory of scattering resonances and survey some recent results.
... Aside from some very special exceptions (e.g. the Walsh quantized Baker map [24]) the position of the resonances of such complex systems cannot be determined exactly. It has, however, been a very successful approach to gain coarser information on general properties of the resonance distribution in terms of the underlying classical dynamics and geometry, respectively. ...
In many non-integrable open systems in physics and mathematics resonances
have been found to be surprisingly ordered along curved lines in the complex
plane. In this article we provide a unifying approach to these resonance chains
by generalizing dynamical zeta functions. By means of a detailed numerical
study we show that these generalized zeta functions explain the mechanism that
creates the chains of quantum resonance and classical Ruelle resonances for
3-disk systems as well as geometric resonances on Schottky surfaces. We also
present a direct system-intrinsic definition of the continuous lines on which
the resonances are strung together as a projection of an analytic variety.
Additionally, this approach shows that the existence of resonance chains is
directly related to a clustering of the classical length spectrum on multiples
of a base length. Finally, this link is used to construct new examples where
several different structures of resonance chains coexist.
... The only system for an asymptotic of the form of (2) could be proved is a simplistic "open quantum baker's map" [24]. Even in that model, the resonance spectrum may, for some choices of parameters, degenerate to a much smaller density (equivalent with the case of a single periodic geodesic). ...
The subject area referred to as 'wave chaos', 'quantum chaos' or 'quantum chaology' has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability, etc. After giving a rough account on 'what is quantum chaos?', I intend to list some pending questions, some of them having been raised a long time ago, some others more recent. The choice of problems (and of references) is of course partial and personal.
... The work of Sjöstrand [52] on semiclassical bounds for resonance counting has led to a general expectation for chaotic scattering systems that the number of resonances near the continuous spectrum should satisfy a power law with exponent equal to half of the dimension of the classical trapped set. Recently a large number of theoretical [56,15,37,53,36,38,35,9], numerical [48,24,25,27,49] and experimental [26,28,46,1,23] studies have appeared in support of this conjectural 'fractal Weyl law'. ...
We study the distribution of resonances for geometrically finite hyperbolic
surfaces of infinite area by countting resonances numerically. The resonances
are computed as zeros of the Selberg zeta function, using an algorithm for
computation of the zeta function for Schottky groups. Our particular focus is
on three aspects of the resonance distribution that have attracted attention
recently: the fractal Weyl law, the spectral gap, and the concentration of
decay rates.
... Also let us remark that with the condition of "minimal captivity", the dynamics of the canonical map F in the vicinity of the trapped set K is univalued and can be identified with the classical dynamics of a "open quantum map". Within this identification, the results of S. Nonnenmacher et M. Zworski [24] about asymptotic spectral radius and Weyl law of these open quantum maps are very similar to the results presented in this paper. ...
We consider a simple model of an open partially expanding map. Its trapped
set K in phase space is a fractal set. We first show that there is a well
defined discrete spectrum of Ruelle resonances which describes the asymptotics
of correlation functions for large time and which is parametrized by the
Fourier component \nu on the neutral direction of the dynamics. We introduce a
specific hypothesis on the dynamics that we call "minimal captivity". This
hypothesis is stable under perturbations and means that the dynamics is
univalued on a neighborhood of K. Under this hypothesis we show the existence
of an asymptotic spectral gap and a Fractal Weyl law for the upper bound of
density of Ruelle resonances in the semiclassical limit \nu -> infinity. Some
numerical computations with the truncated Gauss map illustrate these results.
... LEMMA 2.9 (Theorem 9 from Hiai and Petz (1998)). Suppose λ 1 , · · · , λ n have the joint density function f (λ 1 , · · · , λ n ) as in (31). Then {P n ; n ≥ 1} satisfies the large deviation principle with speed {n 2 ; n ≥ 1} and a good rate function I(µ), which takes the unique minimum at µ 0 and I(µ 0 ) = 0. ...
Let V be the m x m upper-left corner of an n x n Haar-invariant unitary matrix. Let lambda (1),..., lambda(m) be the eigenvalues of V. We prove that the empirical distribution of a normalization of lambda(1) ,..., lambda(m) goes to the circular law, that is, the uniform distribution on {z is an element of C; |z| <= 1} as m -> infinity with m/n -> 0. We also prove that the empirical distribution of lambda(1) ,..., lambda(m) goes to the arc law, that is, the uniform distribution on {z is an element of C; |z| <= 1} as m/n -> 1. These explain two observations by. Zyczkowski and Sommers (2000). (C) 2012 American Institute of Physics. [doi: 10.1063/1.3672885]
... The natural question is: "Are these bounds sharp?" The fractal Weyl bound (7) is conjectured to be sharp for α > 0 large enough, a fact which has been tested numerically on a number of examples, but could be proved only for a very specific toy model [8]. On the opposite, the bounds (9) for eigenmodes of the damped wave equation are not expected to be sharp for all values of α. ...
Two different "wave chaotic" systems, involving complex eigenvalues or
resonances, can be analyzed using common semiclassical methods. In particular,
one obtains fractal Weyl upper bounds for the density of resonances/eigenvalues
near the real axis, and a classical dynamical criterion for a spectral gap.
... This has been shown numerically in various cases, including hyperbolic scattering [5] as well as quantum maps [9]. This asymptotics has been proved only for a very specific quantum maps [7], and represents an interesting challenge for more realistic systems. ...
Transfer operators have been used widely to study the long time properties of chaotic maps or flows. We describe quantum analogues of these operators, which have been used as toy models by the quantum chaos community, but are also relevant to study "physical" continuous time systems.
... The Toeplitz quantization of tori, or of more general classes of compact symplectic manifolds , has a long tradition and we refer to [6] for references in the case of tori, and to [4] for the case of compact symplectic Kähler manifolds. We take a lowbrow approach and our presentation which follows [15] is self-contained but assumes the knowledge of standard semiclassical calculus in R n . It is reviewed in §2.1 with detailed references to [9] and [10] provided. ...
For the Toeplitz quantization of complex-valued functions on a
2n-dimensional torus we prove that the expected number of eigenvalues of
small random perturbations of a quantized observable satisfies a natural Weyl
law. In numerical experiments the same Weyl law also holds for ``false''
eigenvalues created by pseudospectral effects.
We construct δ-regular sets with δ≥12 for which the analog of the Bourgain–Dyatlov fractal uncertainty principle fails for the Walsh–Fourier transform.
This article provides a broad review of recent developments on the fractal uncertainty principle and their applications to quantum chaos.
We derive open quantum maps from periodically kicked scattering systems and discuss the computation of their resonance spectra in terms of theoretically grounded methods, such as complex scaling and sufficiently weak absorbing potentials. In contrast, we also show that current implementations of open quantum maps, based on strong absorptive or even projective openings, fail to produce the resonance spectra of kicked scattering systems. This comparison pinpoints flaws in current implementations of open quantum maps, namely, the inability to separate resonance eigenvalues from the continuum as well as the presence of diffraction effects due to strong absorption. The reported deviations from the true resonance spectra appear, even if the openings do not affect the classical trapped set, and become appreciable for shorter-lived resonances, e.g., those associated with chaotic orbits. This makes the open quantum maps, which we derive in this paper, a valuable alternative for future explorations of quantum-chaotic scattering systems, for example, in the context of the fractal Weyl law. The results are illustrated for a quantum map model whose classical dynamics exhibits key features of ionization and a trapped set which is organized by a topological horseshoe.
Because resonances are defined by meromorphic continuation, direct computation via the resolvent is very difficult. The product definition (2.23) of the Selberg zeta function has the same difficulty; the formula does not apply in the region of interest. However, for hyperbolic surfaces without cusps, the dynamical zeta function introduced in §15.3 provides a suitable alternative. The transfer operator is trace-class for any value of , so analytic continuation is not required.
In Chapters 4 and 5 we worked out the resolvent kernels for the elementary surfaces. This provides a set of model resolvents for funnels and cusps in particular, which are the only possible end types in a non-elementary geometrically finite hyperbolic surface by Theorem 2. 23.
This is a review on theoretical and experimental studies on dielectric microcavities, which play a significant role in fundamental and applied research. The basic concepts and theories are introduced. Experimental techniques for fabrication of microcavities and optical characterization are described. Starting from undeformed cavities, the review moves on to weak deformation, intermediate deformation with mixed phase space, and then strong deformation with full ray chaos. Non-Hermitian physics such as avoided resonance crossings and exceptional points are covered along with various dynamical tunneling phenomena. Some specific topics such as unidirectional output, beam shifts, wavelength-scale microcavities, and rotating microcavities are discussed. The open microdisk and microsphere cavities are ideal model systems for the studies on wave chaos and non-Hermitian physics.
In past ten years, modern societies developed enormous communication and
social networks. Their classification and information retrieval processing
become a formidable task for the society. Due to the rapid growth of World Wide
Web, social and communication networks, new mathematical methods have been
invented to characterize the properties of these networks on a more detailed
and precise level. Various search engines are essentially using such methods.
It is highly important to develop new tools to classify and rank enormous
amount of network information in a way adapted to internal network structures
and characteristics. This review describes the Google matrix analysis of
directed complex networks demonstrating its efficiency on various examples
including World Wide Web, Wikipedia, software architecture, world trade, social
and citation networks, brain neural networks, DNA sequences and Ulam networks.
The analytical and numerical matrix methods used in this analysis originate
from the fields of Markov chains, quantum chaos and Random Matrix theory.
We study the statistical properties of spectrum and eigenstates of the Google
matrix of the citation network of Physical Review for the period 1893 - 2009.
The main fraction of complex eigenvalues with largest modulus is determined
numerically by different methods based on high precision computations with up
to p=16384 binary digits that allows to resolve hard numerical problems for
small eigenvalues. The nearly nilpotent matrix structure allows to obtain a
semi-analytical computation of eigenvalues. We find that the spectrum is
characterized by the fractal Weyl law with a fractal dimension .
It is found that the majority of eigenvectors are located in a localized phase.
The statistical distribution of articles in the PageRank-CheiRank plane is
established providing a better understanding of information flows on the
network. The concept of ImpactRank is proposed to determine an influence domain
of a given article. We also discuss the properties of random matrix models of
Perron-Frobenius operators.
We review recent studies about the resonance spectrum of quantum scattering
systems, in the semiclassical limit and assuming chaotic classical dynamics.
Stationary quantum properties are related to fractal structures in the
classical phase space. We focus attention on a particular class of problems
that are chaotic maps in the torus with holes. Among the topics considered are
the fractal Weyl law, the formation of a spectral gap and the morphology of
eigenstates. We also discuss the situation when the holes are only partially
transparent and the use of random matrices for a statistical description.
In this review article we develop a basic part of the mathematical theory involved in the description of a particle (classical
and quantal) placed in the Euclidean space ℝ
N
under the influence of a magnetic field B, emphasising the structure of the family of observables.
We study the baker’s map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization
properties of eigenstates, in the semiclassical régime. Simple counterexamples show that quantum unique ergodicity fails for
this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl
entropies of eigenstates. The central tool of the proofs is an “entropic uncertainty principle”.
For asymptotically hyperbolic manifolds with hyperbolic trapped sets we prove
a fractal upper bound on the number of resonances near the essential spectrum,
with power determined by the dimension of the trapped set. This covers the case
of general convex cocompact quotients (including the case of connected trapped
sets) where our result implies a bound on the number of zeros of the Selberg
zeta function in disks of arbitrary size along the imaginary axis. Although no
sharp fractal lower bounds are known, the case of quasifuchsian groups,
included here, is most likely to provide them.
Wave scattering in chaotic systems can be characterized by its spectrum of resonances, z(n)=E(n)-iΓ(n)/2, where E(n) is related to the energy and Γ(n) is the decay rate or width of the resonance. If the corresponding ray dynamics is chaotic, a gap is believed to develop in the large-energy limit: almost all Γ(n) become larger than some γ. However, rare cases with Γ<γ may be present and actually dominate scattering events. We consider the statistical properties of these supersharp resonances. We find that their number does not follow the fractal Weyl law conjectured for the bulk of the spectrum. We also test, for a simple model, the universal predictions of random matrix theory for density of states inside the gap and the hereby derived probability distribution of gap size.
We study the semiclassical quantization of Poincar\'e maps arising in
scattering problems with fractal hyperbolic trapped sets. The main application
is the proof of a fractal Weyl upper bound for the number of
resonances/scattering poles in small domains near the real axis. This result
encompasses the case of several convex (hard) obstacles satisfying a no-eclipse
condition.
Statistical properties of nonsymmetric real random matrices of size M, obtained as truncations of random orthogonal N×N matrices, are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong nonorthogonality, M/N=const, the behavior typical to real Ginibre ensemble is found. In the case M=N-L with fixed L, a universal distribution of resonance widths is recovered.
We study the properties of spectrum and eigenstates of the Google matrix of a
directed network formed by the procedure calls in the Linux Kernel. Our results
obtained for various versions of the Linux Kernel show that the spectrum is
characterized by the fractal Weyl law established recently for systems of
quantum chaotic scattering and the Perron-Frobenius operators of dynamical
maps. The fractal Weyl exponent is found to be that
corresponds to the fractal dimension of the network . The
eigenmodes of the Google matrix of Linux Kernel are localized on certain
principal nodes. We argue that the fractal Weyl law should be generic for
directed networks with the fractal dimension .
For a class of quantized open chaotic systems satisfying a natural dynamical
assumption, we show that the study of the resolvent, and hence of scattering
and resonances, can be reduced to the study of a family of open quantum maps,
that is of finite dimensional operators obtained by quantizing the Poincar\'e
map associated with the flow near the set of trapped trajectories.
These notes describe a new method to investigate the spectral properties if quantum scattering Hamiltonians, developed in collaboration with J. Sj\"ostrand and M.Zworski. This method consists in constructing a family of "quantized transfer operators" associated with a classical Poincar\'e section near some fixed classical energy E. These operators are finite dimensional, and have the structure of "open quantum maps". In the semiclassical limit, the family encode the quantum dynamics near the energy E. In particular, the quantum resonances of the form E+z, for z=O(h), are obtained as the roots of . Comment: 18 pages, 3 figures
We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent , where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we find the Weyl exponent where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation where is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law. Comment: 4 pages, 5 figures. Research done at Quantware http://www.quantware.ups-tlse.fr/
We study the properties of the Google matrix generated by a coarse-grained Perron-Frobenius operator of the Chirikov typical map with dissipation. The finite-size matrix approximant of this operator is constructed by the Ulam method. This method applied to the simple dynamical model generates directed Ulam networks with approximate scale-free scaling and characteristics being in certain features similar to those of the world wide web with approximate scale-free degree distributions as well as two characteristics similar to the web: a power-law decay in PageRank that mirrors the decay of PageRank on the world wide web and a sensitivity to the value alpha in PageRank. The simple dynamical attractors play here the role of popular websites with a strong concentration of PageRank. A variation in the Google parameter alpha or other parameters of the dynamical map can drive the PageRank of the Google matrix to a delocalized phase with a strange attractor where the Google search becomes inefficient.
In this survey we consider some recent results of investigations of stochastic motion in classical and quantum dynamical systems. We discuss in detail the phenomenon of transient, or temporary stochasticity in quantum mechanics. Results of numerical simulation of this phenomenon are given. Estimates are made of quantum effects in the quasiclassical region. A simple classical model of quantum stochasticity is discussed.
We compute the divisor of Selberg's zeta function for convex cocompact, torsion-free discrete groups T acting on a real hyperbolic space of dimension n+l. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X=Γ\ \sp n+1 together with the Euler characteristic of X compactified to a manifold with boundary. If n is even, the singularities of the zeta function associated to the Euler characteristic of X are identified using work of U. Bunke and M. Olbrich.
We study analytically the time evolution in decaying chaotic systems and discuss in detail the hierarchy of characteristic time scales that appeared in the quasiclassical region. There exist two quantum time scales: the Heisenberg time t_H and the time t_q=t_H/\sqrt{\kappa T} (with \kappa >> 1 and T being the degree of resonance overlapping and the transmission coefficient respectively) associated with the decay. If t_q < t_H the quantum deviation from the classical decay law starts at the time t_q and are due to the openness of the system. Under the opposite condition quantum effects in intrinsic evolution begin to influence the decay at the time t_H. In this case we establish the connection between quantities which describe the time evolution in an open system and their closed counterparts. Comment: 3 pages, REVTeX, no figures, replaced with the published version (misprints corrected, references updated)
The problem of spectral asymptotics, in particular the problem of the asymptotic distribution of eigenvalues, is one of the central problems of the spectral theory of partial differential operators, and is very important for the general theory of partial differential operators. Apart from applications in quantum mechanics, radio physics, continuum media mechanics (elasticity, hydrodynamics, theory of shells), etc., there are also applications to mathematics itself and deep though non-obvious links with differential geometry, dynamical systems theory and ergodic theory; even the term “spectral geometry” has arisen. All these circumstances make this topic very attractive for a mathematician.
A new method for exact quantization of general bound Hamiltonian systems is presented. It is the quantum analogue of the classical Poincare surface-of-section (SOS) reduction of classical dynamics. The quantum Poincare mapping is shown to be the product of the two generalized (non-unitary but compact) on-shell scattering operators of the two scattering Hamiltonians which are obtained from the original bound one by cutting the f-dimensional configuration space (CS) the along the (f-1)-dimensional configurational SOS and attaching the flat quasi-one-dimensional waveguides instead. The quantum Poincare mapping has fixed points at the eigenenergies of the original bound Hamiltonian. The energy-dependent quantum propagator (E-H)-1 can be decomposed in terms of the four energy-dependent propagators which propagate from and/or to CS to and/or from configurational SOS (which may generally be composed of many disconnected parts). I show that in the semiclassical limit (h(cross) to 0) the quantum Poincare mapping converges to the Bogomolny propagator and explain how the higher-order semiclassical corrections can be obtained systematically.
We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on the hyperbolic space n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp (C|s|) where is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp (C|s|n+1) , and it gives new bounds on the number of resonances (scattering poles) of \n+1 . The proof of this result is based on the application of holomorphic L2-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider \n+1 as the simplest model of quantum chaotic scattering.
A symbolic representation of the states of N qubit that leads naturally to a class of quantum baker's maps, defined as shift maps with respect to the symbolic representation, is presented. For each of the maps in this class, there is a product basis such that the action of the map on an arbitrary basis state is equivalent to a shift of the string of qubits to the left, followed by controlled phase changes on the rightmost qubit. This result is a potential starting point for a generalization of the method of classical symbolic dynamics to the chaotic quantum maps.
The statistical distribution of lifetimes in a simple quantum model with absorption is numerically investigated. It displays a number of characteristic features that can be related to the classical diffusive mechanism of absorption, described by a Fokker-Planck equation.
We investigate the transport properties of open quantum chaotic systems in the semiclassical limit. We show how the transmission spectrum, the conductance fluctuations, and their correlations are influenced by the underlying chaotic classical dynamics, and result from the separation of the quantum phase space into a stochastic and a deterministic phase. Consequently, sample-to-sample conductance fluctuations lose their universality, while the persistence of a finite stochastic phase protects the universality of conductance fluctuations under variation of a quantum parameter.
We analyze certain eigenstates of the quantum baker's map and demonstrate, using the Walsh-Hadamard transform, the emergence of the ubiquitous Thue-Morse sequence, a simple sequence that is at the border between quasiperiodicity and chaos, and hence is a good paradigm for quantum chaotic states. We show a family of states that are also simply related to the Thue-Morse sequence and are strongly scarred by short periodic orbits and their homoclinic excursions. We give approximate expressions for these states and provide evidence that these and other generic states are multifractal.
0. Introduction.- I. Semiclassical Microlocal Analysis.- 1. Introduction to Semiclassical Microlocal Analysis.- 2. Propagation of Singularities in the Interior of a Domain.- 3. Propagation of Singularities near the Boundary.- II. Local and Microlocal Semiclassical Asymptotics.- 4. LSSA in the Interior of a Domain.- 5. Standard LSSA near the Boundary.- 6. Schrodinger Operators with Strong Magnetic Field.- 7. Dirac Operators with Strong Magnetic Field.- III. Estimates of the Spectrum.- 8. Estimates of the Negative Spectrum.- 9. Estimates of the Spectrum in an Interval.- IV. Asymptotics of Spectra.- 10. Weylian Asymptotics of Spectra.- 11. Schrodinger, Dirac Operators with Strong Magnetic Field.- 12. Miscellaneous Asymptotics.- References.
Introduction 1. Local symplectic geometry 2. The WKB-method 3. The WKB-method for a potential minimum 4. Self-adjoint operators 5. The method of stationary phase 6. Tunnel effect and interaction matrix 7. h-pseudodifferential operators 8. Functional calculus for pseudodifferential operators 9. Trace class operators and applications of the functional calculus 10. More precise spectral asymptotics for non-critical Hamiltonians 11. Improvement when the periodic trajectories form a set of measure 0 12. A more general study of the trace 13. Spectral theory for perturbed periodic problems 14. Normal forms for some scalar pseudodifferential operators 15. Spectrum of operators with periodic bicharacteristics References Index Index of notation.
Trace formulae provide one of the most elegant descriptions of the classical-quantum correspondence. One side of a formula is given by a trace of a quantum object, typically derived from a quantum Hamiltonian, and the other side is described in terms of closed orbits of the corresponding classical Hamiltonian. In algebraic situations, such as the original Selberg trace formula, the identities are exact, while in general they hold only in semi-classical or high-energy limits. We refer to a recent survey \cite{Ur} for an introduction and references. In this paper we present an intermediate trace formula in which the original trace is expressed in terms of traces of quantum monodromy operators directly related to the classical dynamics. The usual trace formulae follow and in addition this approach allows handling effective Hamiltonians.
We demonstrate the utility of the periodic-orbit description of chaotic motion by computing from a few periodic orbits highly accurate estimates of a large number of quantum resonances for the classically chaotic 3-disk scattering problem. The symmetry decompositions of the eigenspectra are the same for the classical and the quantum problem, and good agreement between the periodic-orbit estimates and the exact quantum poles is observed.
A scattering process can be described by suitably closing the system and
considering the first return map from the entrance onto itself. This scattering
map may be singular and discontinuous, but it will be measure preserving as a
consequence of the recurrence theorem applied to any region of a simpler map.
In the case of a billiard this is the Birkhoff map. The semiclassical
quantization of the Birkhoff map can be subdivided into an entrance and a
repeller. The construction of a scattering operator then follows in exact
analogy to the classical process. Generically, the approximate unitarity of the
semiclassical Birkhoff map is inherited by the S-matrix, even for highly
resonant scattering where direct quantization of the scattering map breaks
down.
The authors derive a semiclassical secular equation which applies to quantized (compact) billiards of any shape. Their approach is based on the fact that the billiard boundary defines two dual problems: the 'inside problem' of the bounded dynamics, and the 'outside problem' which can be looked upon as a scattering from the boundary as an obstacle. This duality exists both on the classical and quantum mechanical levels, and is therefore very useful in deriving a semiclassical quantization rule. They obtain a semiclassical secular equation which is based on classical input from a finite number of classical periodic orbits. They compare their result to secular equations which were derived by other means, and provide some numerical data which illustrate their method when applied to the quantization of the Sinai billiard.
The solution of a stationary k-dimensional Schrodinger equation H Psi =E Psi in the semiclassical limit h(cross) to 0 is reduced to a discrete (k-1)-dimensional quantum map psi '=T psi where the integral kernel (the matrix) T is built through classical trajectories corresponding to the classical Poincare map of the given problem. High-excited energy eigenvalues obey the quantization condition zeta s(E)=0 where the function zeta s(E)=det(1-T) coincides with the Selberg zeta function defined as the product over primitive periodic orbits. Different properties of the constructed Poincare map are discussed, in particular the Riemann-Siegel relation for the dynamical zeta function.
The quantal Baker's map is described by a unitary operator acting in a N-dimensional Hilbert space. Its properties are similar to the classical transformation and it becomes that in the classical limit (N → ∞, → 0). The eigenangles are effectively all irrational leading to no recurrences. The quantal and classical time evolutions cease to have any relation to each other for times larger than log2N.
IntroductionQuantum Time Evolution, Path Integrals, and Semiclassical LimitClassical DynamicsTime Domain: Trace of the PropagatorEnergy Domain: Equilibrium-Point QuantizationEnergy Domain: Periodic-Orbit QuantizationSemiclassical Averages of Quantum ObservablesQuantum BilliardsMatrix HamiltoniansAtomic and Molecular HamiltoniansMolecular Vibrograms: Spectroscopy in the Time DomainThe Molecular Transition State and Its ResonancesSemiclassical Electronic Regimes in Atomic and Solid-State SystemsDiscussion and Conclusions
We show that the zeta function for the dynamics generated by the map z --> z(2)+c, c < -2, can be estimated in terms of the dimension of the corresponding Julia set. That implies a geometric upper bound on the number of its zeros, which are interpreted as resonances for this dynamical systems. The method of proof of the upper bound is used to construct a code for counting the number of zeros of the zeta function. The numerical results support the conjecture that the upper bound in terms of the dimension of the Julia set is optimal. Abstract We show that the zeta function for the dynamics generated by the map z → z 2 +c, c < −2, can be estimated in terms of the dimension of the corresponding Julia set. That implies a geometric upper bound on the number of its zeros, which are interpreted as resonances for this dynamical systems. The method of proof of the upper bound is used to construct a code for counting the number of zeros of the zeta function. The numerical results support the conjecture that the upper bound in terms of the dimension of the Julia set is optimal.
We study Anosov diieomorphisms on manifolds in which somèholes' are cut. The points that are mapped into those holes disappear and never return. The holes studied here are rectangles of a Markov partition. Such maps generalize Smale's horseshoes and certain open billiards. The set of nonwandering points of a map of this kind is a Cantor-like set called repeller. We construct invariant and conditionally invariant measures on the sets of nonwandering points. Then we establish ergodic, statistical, and fractal properties of those measures.
This note deals with the classical and quantum dynamics on convex co-compact surfaces. We review the recent developements of the theory and compare the asymptotic behaviour of both classical and quantum observables. We show rigourously that classical decay rate is larger than the quantum decay rate. This is well known in the physics literature on chaotic scattering but has never been verified mathematically.
Random contractions (subunitary random matrices) appear naturally when considering quantized chaotic maps within a general
theory of open linear stationary systems with discrete time. We analyze statistical properties of complex eigenvalues of generic
N × N random matrices  of such a type, corresponding to systems with broken time reversal invariance. Deviations from unitarity are characterized
by rank M≤N and a set of eigenvalues 0<T
i≤1, i=1,..., M of the matrix
[^(T)] = [^1] - [^(A)]f [^(A)]\hat T = \hat 1 - \hat A^\dag \hat A
. We solve the problem completely by deriving the joint probability density of N complex eigenvalues and calculating all n-point correlation functions. In the limit N≫M, n, the correlation functions acquire the universal form found earlier for weakly non-Hermitian random matrices.
We develop semiclassical (ℏ→0) treatments of the quantum baker's map paying special attention to (a) the discrete and finite nature of its Hilbert space, and (b) the asymptotic analysis of the true quantum objects themselves. New techniques are described at the theoretical level (exact symbolic decomposition of the propagator) and at the numerical level (color graphics display of operators in the Kirkwood representation), both of which are of general interest for other quantized chaotic maps. Periodic orbit theory is tested numerically, and we find that log (ℏ) corrections need to be included before an accurate calculation of the spectrum from periodic orbits can be achieved. However, the asymptotic analysis of the traces of the propagator reveals unexpected departures from standard semiclassical theory; those anomalies, which we compute explicitly in a few cases, are traced ultimately to the discontinuity of the map and to the compactness of the phase space. The analysis is far from complete but the results point towards the necessity of careful assessment of semiclassical “folklore” when applied to chaotic maps.
This paper presents numerical evidence that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like ħ−(D(KE)+1)/2 as ħ→0. Here, KE denotes the subset of the energy surface {H=E} which stays bounded for all time under the flow generated by the classical Hamiltonian H and D(KE) denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like ħ−n, this suggests that the quantity (D(KE)+1)/2 represents the effective number of degrees of freedom in chaotic scattering problems. The calculations were performed using a recursive refinement technique for estimating the dimension of fractal repellors in classical Hamiltonian scattering, in conjunction with tools from modern quantum chemistry and numerical linear algebra.
This Letter summarizes numerical results from [J. Comp. Phys. (to appear)] which show that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like . Here, KE denotes the subset of the classical energy surface which stays bounded for all time under the flow of H and denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like ℏ−n, this suggests that the quantity represents the effective number of degrees of freedom in chaotic scattering problems.
Quantization on a phase space q, p in the form of a torus (or periodized plane) with dimensions Δq, Δp requires the Planck's constant take one of the values h = ΔqΔp/N, where N is an integer. Corresponding to a linear classical map T of points q, p is a unitary operator U mapping quantum states that are periodic in q and p; the construction of U involves techniques from number theory. U has eigenvalues exp(iα). The ‘eigenangles’ α must be multiples of , where n(N) is the lowest common multiple of the lengths of the classical ‘cycles’ mapped under T by those rational points in q, p which are multiples of and (i.e. n(N) is the ‘period of T mod N′), at least for odd N. If T is hyperbolic, n is a very erratic function of N, and the classical limit N → ∞ is very different from the ‘Bohr-Sommerfeld’ behaviour for parabolic maps. The degeneracy structure of the eigenangle spectrum is related to the distribution of cycle lengths. Computation of the quantal Wigner function shows that eigenstates of U do not correspond to individual cycles.
For a separable or nonseparable system an approximate solution of the Schrödinger equation is constructed of the form . From the single-valuedness of the solution, assuming that A is single-valued, a condition on S is obtained from which follows A. Einstein's generalized form of the Bohr-Sommerfeld-Wilson quantum conditions. This derivation, essentially due to L. Brillouin, yields only integer quantum numbers. We extend the considerations to multiple valued functions A and to approximate solutions of the form In this way we deduce the corrected form of the quantum conditions with the appropriate integer, half-integer or other quantum number (generally a quarter integer). Our result yields a classical mechanical principle for determining the type of quantum number to be used in any particular instance. This fills a gap in the formulation of the “quantum theory”, since the only other method for deciding upon the type of quantum number—that of Kramers—applies only to separable systems, whereas the present result also applies to nonseparable systems.In addition to yielding this result, the approximate solution of the Schrödinger equation—which can be constructed by classical mechanics—may itself prove to be useful.
A quantum analogue of the Baker's transformation is constructed using a specially developed quantization procedure. We obtain a unitary operator acting on an N-dimensional Hilbert space, with N finite (and even), that has similar properties to the classical baker's map, and reduces to it in the classical limit, which corresponds here to N → ∞. The operator can be described as a very simple, fully explicit N × N matrix. Generalized Baker's maps are also quantized and studied. Numerical investigations confirm that this model has nontrivial features which ought to represent quantal manifestations of classical chaoticity. The quasi-energy spectrum is given by irrational eigenangles, leading to no recurrences. Most eigenfunctions look irregular, but some exhibit puzzling regular features, such as peaks at coordinate values belonging to periodic orbits of the classical baker's map. We compare the quantal and classical time-evolutions, as applied to initially coherent quasi-classical states: the evolving states stay in close agreement for short times but seem to lose all relationship to each other beyond a critical time of the order of .
Assuming the validity of random matrices for describing the statistics of a closed chaotic quantum system, we study analytically some statistical properties of the S-matrix characterizing scattering in its open counterpart. In the first part of the paper we attempt to expose systematically ideas underlying the so-called stochastic (Heidelberg) approach to chaotic quantum scattering. Then we concentrate on systems with broken time-reversal invariance coupled to continua via Mopen channels. A physical realization of this case corresponds to the chaotic scattering in ballistic microstructures pierced by a strong enough magnetic flux. By using the supersymmetry method we derive an explicit expression for the density of S-matrix poles (resonances) in the complex energy plane. When all scattering channels are considered to be equivalent our expression describes a crossover from the chi2 distribution of resonance widths (regime of isolated resonances) to a broad power-like distribution typical for the regime of overlapping resonances. The first moment is found to reproduce exactly the Moldauer–Simonius relation between the mean resonance width and the transmission coefficient. Under the same assumptions we derive an explicit expression for the parametric correlation function of densities of eigenphases thetaa of the S-matrix (taken modulo 2pi). We use it to find the distribution of derivatives taua=[partial-derivative]thetaa/[partial-derivative]E of these eigenphases with respect to the energy (partial delay times) as well as with respect to an arbitrary external parameter. We also find the parametric correlations of the Wigner–Smith time delay tauw(E)=(1/M)[summation]a [partial-derivative]thetaa/[partial-derivative]E at two different energies E–Omega/2 and E+Omega/2 as well as at two different values of the external parameter. The relation between our results and those following from the semiclassical approach as well as the relevance to experiments are briefly discussed.
We give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area-preserving maps on the torus. Examples are the irrational translations, the skew translations, the hyperbolic automorphisms and some of their perturbations.
Low order classical perturbation theory is used to obtain semiclassical eigenvalues for a system of three anharmonically coupled oscillators. The results in the low energy region studied here agree well with the "exact" quantum values. The latter had been calculated by matrix diagonalization using a large basis set.
We show that the class of quantum baker's maps defined by Schack and Caves have the proper classical limit provided the number of momentum bits approaches infinity. This is done by deriving a semi-classical approximation to the coherent-state propagator. Yes Yes
Thermal fluctuations at equilibrium and excess fluctuations in the presence of transport in open multiprobe conductors are related to the scattering matrix of the conductor. The fluctuation-dissipation theorem for multiprobe conductors is discussed. A general expression for the excess noise in the presence of transport is derived. These results are applied to conductors which exhibit the quantized Hall effect. If backscattering is suppressed, excess noise is also suppressed.
We construct a new example of a quantum map, the quantized version of the D-transformation, which is the natural extension to two dimensions of the tent map. The classical, quantum and semiclassical behavior is studied. We also exhibit some relationships between the quantum versions of the D-map and the parity projected baker's map. The method of construction allows a generalization to dissipative maps which includes the quantization of a horseshoe. (c) 1996 American Institute of Physics.
We present a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. Mathematical arguments justifying this conjecture are discussed. Numerical evidence based on computation of resonances of systems of n disks on a plane are presented supporting this conjecture. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.
In the semiclassical limit of open ballistic quantum systems, we demonstrate the emergence of instantaneous decay modes guided by classical escape faster than the Ehrenfest time. The decay time of the associated quasibound states is smaller than the classical time of flight. The remaining long-lived quasibound states obey random-matrix statistics, renormalized in compliance with the recently proposed fractal Weyl law for open systems [W.T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)]. We validate our theory numerically for a model system, the open kicked rotator.
this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex co-compact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous spectrum is related to the dimension of the limit set of the corresponding Kleinian group. Figure 1. Tesselation by the Schottky group generated by inversions in three symmetrically placed circles each cutting the unit circle in an 110
We give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area-preserving maps on the torus. Examples are the irrational translations, the skew translations, the hyperbolic automorphisms and some of their perturbations. 1 Introduction Perhaps the simplest trace of the ergodicity of a Hamiltonian dynamical system one expects to find in the corresponding quantum system is the equipartition of its eigenfunctions in the classical limit. Such a phenomenon has been proved to occur in several cases. For the geodesic flow on compact Riemannian manifolds it is proved in [Sc] [Z1] [CdV]; for Hamiltonian flows on 2n in [HMR] and for smooth convex two-dimensional ergodic billiards in [GL]. In this paper we study the quantization and the classical limit of certain area-preserving ergodic maps on the two-torus T (2) , viewed as phase space, with canonical coordinates (q; p) 2 [0; a[Theta[0; b[. We will present a rather CEREMADE, Universit'e Paris-Daup...
We represent both the states and the evolution of a quantum computer in phase space using the discrete Wigner function. We study properties of the phase space representation of quantum algorithms: apart from analyzing important examples, such as the Fourier Transform and Grover's search, we examine the conditions for the existence of a direct correspondence between quantum and classical evolutions in phase space. Finally, we describe how to directly measure the Wigner function in a given phase space point by means of a tomographic method that, itself, can be interpreted as a simple quantum algorithm. Comment: 16 pages, 7 figures, to appear in Phys Rev A