## SPECTRAL THEORY OF DAMPED QUANTUM CHAOTIC SYSTEMS

Citations: | 1 - 0 self |

### BibTeX

@MISC{Nonnenmacher_spectraltheory,

author = {Stéphane Nonnenmacher},

title = {SPECTRAL THEORY OF DAMPED QUANTUM CHAOTIC SYSTEMS},

year = {}

}

### OpenURL

### Abstract

Abstract. We investigate the spectral distribution of the damped wave equation on a compact Riemannian manifold, especially in the case of a metric of negative curvature, for which the geodesic flow is Anosov (very chaotic). The final objective is to obtain conditions (in terms of the geodesic flow on X, the structure of the damping function) for which the energy of the waves decays exponentially fast, at least for smooth enough initial data. The spectrum of the equation amounts to a nonselfadjoint spectral problem. Using semiclassical methods, we derive estimates and upper bounds for the high frequency spectral distribution, in terms of dynamically defined quantities, like the value distribution of the time-averaged damping. We also consider the toy model of damped quantized chaotic maps, for which we derive similar estimates, as well as a new upper bound for the spectral radius depending on the set of minimally damped trajectories. Contents

### Citations

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Citation Context ...rty due to the instability of the classical flow (see Thms 15 and 18 below). In the next section we recall the definition and properties of Anosov manifolds. 3.1. A short reminder on Anosov manifolds =-=[KatHas95]-=-. At the “antipode” of the completely integrable case, one finds the “strongly chaotic” flows, namely the Anosov (or uniformly hyperbolic) flows. Uniform hyperbolicity means that at each point ρ ∈ S ∗... |

35 | Uniform semiclassical estimates for the propagation of quantum observables - Bouzouina, Robert |

35 | Fractal upper bounds on the density of semiclassical resonances
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Citation Context ... this upper bound improves the bound o(� −d+1 ) of Thm. 11 by a fractional power of � (the name “fractal Weyl upper bound” takes its origin in the counting of resonances of chaotic scattering systems =-=[SjoZwo07]-=-). For any α ∈ [¯q,q+] one has ˜ H(α) ≥ −sup µµ(ϕ + ) ≥ −λmax(d−1), so the exponent of � in (3.12) is negative, allowing the presence of (many) quantum decay rates arbitrary close to q+. These fractal... |

30 |
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Citation Context ... rather see it as a measure of the “thickness” of K. This bound reminds us of the lower bound obtained by Anantharamanwhen describing the localizationof eigenstates of theLaplacian onAnosov manifolds =-=[Anan08]-=-: in constant curvature, she shows the semiclassical measures associated with any sequence of eigenstates cannot be supported on a set of entropy smaller than (d − 1)Λ/2, thus forbidding the eigenstat... |

30 | Corrigendum to “Semiclassical non-concentration near hyperbolic orbits - Christianson |

28 |
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Citation Context ...∀s ∈ R, ˜ H(s) def = sup { hKS(µ)−µ(ϕ + ), µ ∈ M, µ(q) = s } , where we recall that ϕ + is the infinitesimal unstable Jacobian (3.3). We are now ready to state our large deviation result. Theorem 13. =-=[Kif90]-=-Assume the geodesic flow on S ∗ X is Anosov. Then, for any closed interval I ⊂ R and for any q ∈ C ∞ (S ∗ X), the time averages 〈q〉t satisfy (3.8) limsup t→∞ 1 t logµL{ρ ∈ S ∗ X, 〈q〉t(ρ) ∈ I} ≤ sup s∈... |

22 |
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(Show Context)
Citation Context ...e main question we want to address is the distribution of eigenvalues τn, and in particular of their imaginary parts, in the high-frequency limit Re τn ≫ 1. A semiclassical reformulation was used in (=-=[Sjo00]-=-): take � ≪ 1 and consider eigenvalues τn ≈ �−1 , by writing √ 2z (1.8) τ = with z ∈ D(1/2, C�). � The equation (1.5) becomes (1.9) (P (�, z) − z)u = 0, P (�, z) = − �2∆ 2 − i�√2z a = − �2∆ 2 − i�a + ... |

18 | Nonnenmacher Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold - Anantharaman, S - 2007 |

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Citation Context ...xp(tHp0) on T ∗X, with unit speed on the energy shell p −1 0 (1/2) = S∗X. The function iq(x, ξ) is the subprincipal symbol of P (�). 1.3. From resolvent estimates to energy decay. (see [Leb93] [Hit03]=-=[BuHi07]-=-[Chris09]). 1.3.1. From the resolvent to the semigroup. We want to expand the semigroup e itA using the resolvent (τ − A) −1 . This resolvent always satisfies the bound ∥ −1 (τ − A) ∥ ≤ H→H 1 , Im τ >... |

11 |
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Citation Context ...ansforming the DWE into an inhomogeneous equation, and by invoking a Parseval identity w.r.to the time variable. We only sketch the argument, which first appeared in the context of obstacle scattering=-=[Mora75]-=-. Call u(t, x) the solution of (1.1) with the data u(0) = (I − Π0)v(0). Apply a smooth cutoff in time χ ∈ C∞ (R), χ(t) = 0 for t ≤ 1, χ(t) = 1 for t ≥ 2. The function w(t, x) def = χ(t)u(t, x) satisfi... |

11 |
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Citation Context ...quency limit, a natural connection can be made with the classical ray dynamics on X, equivalently the geodesicflow Φt ontheunit cotangent bundle S∗X (see§1.4). Using this connection, Rauch and Taylor =-=[RauTay75]-=- showed that the uniform exponential decay (1.6) is equivalent with the Geometric Control Condition (GCC), which states that every geodesic meets the damping region {x ∈ X, a(x) > 0} (due to the compa... |

10 | Quantum Monodromy and Non-concentration Near a Closed Semi-hyperbolic Orbit
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(Show Context)
Citation Context ...jectories [BuHi07]. Christianson studied thecase where theset K consists inasinglehyperbolic closed geodesic: the energy decay is bounded by a stretched exponential O(e−γst1/2 ‖v‖Hs) [Chris07, Thm 5’]=-=[Chris11]-=-. The hyperbolicity of the geodesic induces a strong dispersion of the waves, responsible for this fast decay. 1 under the condition that, for any T > 0, the set of T-periodic geodesics has measure ze... |

9 |
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Citation Context ...cription of the spectrum “generated” by the torus, which lives in some region of D(1/2, C�). 18E − E + ρ t Φ( ρ) J ( ρ) t + Figure 3.1. Structure of the Anosov flow near an orbit Φ t (ρ) Asch-Lebeau =-=[AschLeb]-=- addressed the question 2 for the case of the 2D standard sphere, where they show that, if the damping function q is real analytic, then (under some generic condition) there exists ϵ1 > 0 such that, f... |

9 | Eigenfrequencies and expansions for damped wave equations
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(Show Context)
Citation Context ... Φt = exp(tHp0) on T ∗X, with unit speed on the energy shell p −1 0 (1/2) = S∗X. The function iq(x, ξ) is the subprincipal symbol of P (�). 1.3. From resolvent estimates to energy decay. (see [Leb93] =-=[Hit03]-=-[BuHi07][Chris09]). 1.3.1. From the resolvent to the semigroup. We want to expand the semigroup e itA using the resolvent (τ − A) −1 . This resolvent always satisfies the bound ∥ −1 (τ − A) ∥ ≤ H→H 1 ... |

9 | Energy decay for the damped wave equation under a pressure condition
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(Show Context)
Citation Context ... will see that such a phenomenon may occur also in the case of Anosov geodesic flows. 3.0.4. Anosov manifolds. The opposite case of a fully chaotic flow was considered by Anantharaman [?] and Schenck =-=[Schenck-pressure]-=-. Such a flow is obtained if the manifold (X, g) has a negative sectional curvature everywhere: this negative curvature is responsible for the uniform hyperbolicity of the flow, and conjugated with th... |

8 | Comparison theorems for spectra of linear operators and spectral - Markus, Matsaev - 1982 |

8 |
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(Show Context)
Citation Context ... this Fredholm spectrum is given by the annulus {z ∈ C, e −ta+ ≤ |z| ≤ e −ta− }. Noticethat itispossible tohave apositivegap G > 0andat thesame timea− = 0 (failure of the GCC), see e.g. an example in =-=[Ren94]-=-. This reflects the fact that the spectrum of the semigroup e −itA is not controlled by the spectrum of generator A, a frequent problem for nonnormal generators like A. 1.3. Beyond Geometric control: ... |

7 | Recent developments in mathematical quantum chaos
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- 2009
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Citation Context ... Anosov manifold belongs to the field of “quantum chaos”. The methods we will use below occur in various problems of this field, e.g. the study of eigenstates of the Laplacian P0(�) on such manifolds =-=[Zel09]-=-. 3.2. Fractal Weyl upper bounds for the quantum decay rates. Inthissection we address Question 2, that is the asymptotic distribution of the quantum decay rates, for the case of Anosov manifolds. 3.2... |

6 |
Exponential stabilization without geometric control
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- 2011
(Show Context)
Citation Context ... /2) < q+. For a uniform damping q ≡ ¯q, P(¯q − φ + /2) = ¯q + P(−φ + /2) is always larger than ¯q. So, the condition (3.16) can be satisfied only for sufficiently inhomogeneous damping functions. In =-=[Schenck10]-=- Schenck explains how to construct damping functions q for which the above condition is satisfied. In view of the original damping function a(x), the idea is to consider a damping function q ≤ 0, whic... |

5 | Spectral deviations for the damped wave equation
- Anantharaman
(Show Context)
Citation Context ...eyl upper bounds for the distribution of quantum decay rates 20 3.2. Finer spectral gaps for Anosov manifolds 23 3.3. A pressure estimate on the propagator 23 3.4. An arithmetic example in dimension 2=-=[Anan10]-=- 27 4. Spectral study of damped quantum maps 28 4.1. Damped quantum maps 28 4.2. Spectral bounds for damped quantum (Anosov) maps 29 4.3. A topological pressure condition for a gap 31 4.4. A topologic... |

5 | Entropy of semiclassical measures of the Walsh-quantized baker’s map - Anantharaman, Nonnenmacher - 2005 |

5 | Diophantine tori and spectral asymptotics for non-self adjoint operators
- Hitrik, Vu-Ngoc
(Show Context)
Citation Context ... Hitrik-Sjöstrand in a sequence of papers (see e.g. [HitSjo08]and reference therein). The case of nearly-integrable dynamics including KAM invariant tori has been studied by Hitrik-Sjöstrand-Vũ Ngo.c =-=[HSVN07]-=-. In these cases, one can transformthe Hamiltonian flow into a normal form near each invariant torus, which leads to a precise description of the spectrum “generated” by this torus. A Weyl law for the... |

3 | Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2. Ann. Scient. de l’école normale supérieure. arXiv:math/0703394v1 [math.SP
- Hitrik, Sjöstrand
- 2008
(Show Context)
Citation Context ... one needs to make specific assumptions on the geodesic flow on X. For instance, the case of a completely integrable dynamics has been considered by Hitrik-Sjöstrand in a sequence of papers (see e.g. =-=[HitSjo08]-=-and reference therein). The case of nearly-integrable dynamics including KAM invariant tori has been studied by Hitrik-Sjöstrand-Vũ Ngo.c [HSVN07]. In these cases, one can transformthe Hamiltonian flo... |

3 | Entropy of semiclassical measures in dimension 2
- Rivière
(Show Context)
Citation Context ...re, it is weaker than (3.27). A Proof of (3.30) should make use of local expansion rates, instead of the globally defined rates λmax and νmin, like in Rivière’s work on 2-dimensional Anosov manifolds =-=[Riv10]-=-. In the next subsection we sketch the proof of Thm 18. hal-00619146, version 1 - 5 Sep 2011 3.4.1. Sketch of proof for Thm 18. AswasthecaseforThm15,theproofproceedsby bounding the norm of the propaga... |

2 |
Entropy of quantum limits for 2-dimensional cat maps
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- 2010
(Show Context)
Citation Context ...e of dispersion estimate was used by S. Brooks, when studying the delocalization of the eigenstates of quantized hyperbolic automorphisms of the 2-dimensional torus (the so-called “quantum cat maps”) =-=[Bro10]-=-. To estimate this norm, it will be useful to replace this projector by a smoothed microlocal projector obtained by quantizing a symbol χ+ = χ+,α ∈ S −∞ 1/2−ǫ (T∗X), such that Op�(χ+) “dominates” Π+: ... |

2 |
On the spectrum of hyperbolic semigroups
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- 1995
(Show Context)
Citation Context ...), where G = inf {Im τn, τn ̸= 0} is the spectral gap, while def a− = lim T →∞ min S∗X ⟨a⟩T is the minimal asymptotic damping. We will sketch a proof of this decay estimate in §1.3.3. Koch and Tataru =-=[KoTa94]-=- studied the same question in a more general context (case of a manifold with boundaries, and of a damping taking place both in the 3“bulk” and on the boundary). They first showed that the minimum of... |

1 |
Equation des ondes amorties, Algebraie and geometric methods in mathematical physics, (Kaciveli
- Lebeau
- 1993
(Show Context)
Citation Context ... c > 0 such that, for T large enough, the function ⟨a⟩T ≥ c > 0 everywhere. We will show below (Thm 8) that, as a consequence, the quantum decay rates satisfy Im τn ≤ −c + o(1) when Re τn → ∞. Lebeau =-=[Leb93]-=- showed that GCC is equivalent with the uniform exponential decay of the energy for initial data in H, namely, there exists C > 0, γ > 0 such that, for any data v(0) ∈ H, (1.6) E(v(t)) ≤ C e −2γt E(v(... |

1 |
Resonance distribution in open quantum chaotic systems
- Nonnenmacher, Schenck
(Show Context)
Citation Context ...tum chaos, both on numerical and analytical aspects [DEG03]. In this section, we will consider a damped version of these quantized Anosov maps, aimed at modelling the damped wave system studied above =-=[NonSche08]-=-. 4.1. Damped quantum maps. In classical dynamics, one may study the flow on S ∗ X by setting up a Poincaré section Σ ⊂ S ∗ X, that is a finite set of hypersurfaces in S ∗ X transverse to the flow, an... |

1 | Quantum decay rates
- Nonnenmacher, Zworski
- 2009
(Show Context)
Citation Context ...ed in the case of the undamped operator P0(�) on an Anosov manifold [Anan08, AN1]. The adaptation to the case of the damped wave equation was written in [Sche10]. A similar estimate was also shown in =-=[NZ2]-=- in the case of chaotic scattering. 3.4. A “thickness” condition for a spectral gap. In this section we present a spectral gap condition expressed only in terms of the set of undamped trajectories (1.... |

1 |
Weyl Laws for Partially Open Quantum
- Schenck
- 2009
(Show Context)
Citation Context ... b⟩n, |λj(�)| ∈ [b− − ϵ, b+ + ϵ], log b− = lim n→∞ min M ⟨log b⟩n. 4.2.2. Fractal Weyl upper bounds. Here we state the analogue of Thm 13 for Anosov damped maps. The method is the one used by Schenck =-=[Schenck-map]-=-. The expression (4.2) can be pushed forward until the time n ≈ TEhr/2, keeping b (n) ∈ S1/2−ϵ. A similar decomposition holds on the opposite side, (M�) n = Op �(b (−n) )U n + O(� ∞ ), where b (−n) ∈ ... |

1 |
H.Christianson,Applications of Cutoff Resolvent Estimates to the Wave Equation
- Vol
(Show Context)
Citation Context ...data decays exponentially: (1.10) E(v(t)) 1/2 ≤ Cse −γst ‖v‖ H s , ∀v ∈ H s , ∀t ≥ 0. Theproofproceedsbycontrollingtheresolvent(τ −A) −1 inastrip{Reτ > C, |Imτ| ≤ c}, and then uses standard arguments =-=[Chris09]-=-. Here as well, the resolvent bounds are based on certain hyperbolic dispersion estimates. See the Corollaries 16 and 19 for the precise dynamical conditions, which involve the interplay between the f... |

1 |
tori and Weyl laws for non-selfadjoint operators in dimension two
- Sjöstrand
(Show Context)
Citation Context ...nian flow into a normal form near each invariant torus, which leads to a precise description of the spectrum “generated” by this torus. A Weyl law for the quantum decay rates was recently obtained in =-=[HitSjo11]-=- (for skew-adjoint perturbationsiθ(�)Op �(q), withθ(�) ≪ �). Ontheotherhand, Asch-Lebeau[AschLeb] addressed Question 1 for the case of the 2-dimensional standard sphere. They show that, if the damping... |

1 | S.NonnenmacherandM.Zworski,Semiclassical Resolvent Estimates in Chaotic Scattering - eXpr |

1 |
Eigenfunctions of the Laplacian and subsets of small topological pressure, preprint 2011
- Rivière
(Show Context)
Citation Context ... sharp (even in 2 dimensions), except maybe on manifolds of constant curvature. Rivière has recently improved Anantharaman’s lower bound on the support of semiclassical measures in variable curvature =-=[Riv11]-=-: he shows that any such support S must satisfy P(−ϕ + /2,Φ t ↾S) ≥ 0. If we follow the above analogy (and also following the results of §3.3.1), it seems natural to expect the following Conjecture 21... |