Annales scientifiques de l'École normale supérieure
The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of ... more The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C ∞ Anosov flows. More general results have been recently proved by Giulietti-Liverani-Pollicott [GiLiPo] but our approach is different and is based on the study of the generator of the flow as a semiclassical differential operator.
The purpose of this note is to give a mathematical explanation of a formula for the scattering ma... more The purpose of this note is to give a mathematical explanation of a formula for the scattering matrix for a manifold with infinite cylindrical ends or a waveguide. This formula, which is well known in the physics literature, is sometimes referred to as the Mahaux-Weidenm\"uller formula. We show that a version of this formula given below gives the standard scattering
Let f (z,z) be a positive bi-homogeneous hermitian form on C n , of degree m. A theorem proved by... more Let f (z,z) be a positive bi-homogeneous hermitian form on C n , of degree m. A theorem proved by Quillen and rediscovered by Catlin and D'Angelo states that for N large enough, z,z N f (z,z) can be written as the sum of squares of homogeneous polynomials of degree m+N. We show this works for N ≥ C f ((n+m) log n) 3 where C f has a natural expression in terms of coefficients of f , inversely proportional to the minimum of f on the sphere. The proof uses a semiclassical point of view on which 1/N plays a role of the small parameter h.
Using tools from semiclassical analysis, we give weighted L ∞ estimates for eigenfunctions of str... more Using tools from semiclassical analysis, we give weighted L ∞ estimates for eigenfunctions of strictly convex surfaces of revolution. These estimates give rise to new sampling techniques and provide improved bounds on the number of samples necessary for recovering sparse eigenfunction expansions on surfaces of revolution. On the sphere, our estimates imply that any function having an s-sparse expansion in the first N spherical harmonics can be efficiently recovered from its values at m sN 1/6 log 4 (N) sampling points.
We present a conjecture relating the density of quantum resonances for an open chaotic system to ... more 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.
Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmoni... more Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmonic inversion the distribution of the imaginary parts of the resonances is determined. With increasing opening of the systems, a spectral gap is observed for thick as well as for thin repellers and for the latter case it is compared with the known topological pressure bounds. The maxima of the distributions are found to coincide for a large range of the distance to radius parameter with half of the classical escape rate. This confirms theoretical predictions based on rigorous mathematical analysis for the spectral gap and on numerical experiments for the maxima of the distributions.
We present microwave experiments on the symmetry reduced five-disk billiard studying the transiti... more We present microwave experiments on the symmetry reduced five-disk billiard studying the transition from a closed to an open system. The measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied. For the closed system this counting function shows the Weyl asymptotic with a leading exponent equal to 2. By opening the system successively this exponent decreases smoothly to a noninteger value. For the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed. The results can be interpreted as a first experimental indication for the fractal Weyl conjecture for resonances.
In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous me... more In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous media one observes "breathing" in the sense of the amplitude of the wave going up and down on a much faster scale than the motion of the wave-see Fig. 2 below. In this paper we investigate this phenomenon in the simplest case of stationary waves in which the evolution corresponds to relaxation to a nonlinear ground state. The particular model is the popular δ 0 impurity in the cubic nonlinear Schrödinger equation on the line. We give asymptotics of the amplitude on a finite but relevant time interval and show their remarkable agreement with numerical experiments, see Fig. 1. We stress the nonlinear origin of the "breathing patterns" caused by the selection of the ground state depending on the initial data, and by the non-normality of the linearized operator.
We show that the zeta function for the dynamics generated by the map z → z 2 +c, c < −2, can be e... more 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.
Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigensta... more Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigenstates are, in average, uniformly distributed on energy surfaces. We show that if N is a hypersurface in the position space satisfying a simple dynamical condition, the restrictions of eigenstates to N are also quantum ergodic.
A well known result of Jaffard states that an arbitrary region on a torus controls, in the L 2 se... more A well known result of Jaffard states that an arbitrary region on a torus controls, in the L 2 sense, solutions of the free stationary and dynamical Schrödinger equations. In this note we show that the same result is valid in the presence of a potential, that is for Schrödinger operators, −∆ + V , V ∈ C ∞ .
Journal of the European Mathematical Society, 2013
For the Schrödinger equation (i∂ t +)u = 0 on a torus, an arbitrary non-empty open set provides c... more For the Schrödinger equation (i∂ t +)u = 0 on a torus, an arbitrary non-empty open set provides control and observability of the solution: u| t=0 L 2 (T 2) ≤ K T u L 2 ((0,T)×). We show that the same result remains true for (i∂ t + − V)u = 0 where V ∈ L 2 (T 2), and T 2 is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for V ∈ C(T 2) and conjectured for V ∈ L ∞ (T 2). The higher dimensional generalization remains open for V ∈ L ∞ (T n).
Annales scientifiques de l'École normale supérieure
The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of ... more The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C ∞ Anosov flows. More general results have been recently proved by Giulietti-Liverani-Pollicott [GiLiPo] but our approach is different and is based on the study of the generator of the flow as a semiclassical differential operator.
The purpose of this note is to give a mathematical explanation of a formula for the scattering ma... more The purpose of this note is to give a mathematical explanation of a formula for the scattering matrix for a manifold with infinite cylindrical ends or a waveguide. This formula, which is well known in the physics literature, is sometimes referred to as the Mahaux-Weidenm\"uller formula. We show that a version of this formula given below gives the standard scattering
Let f (z,z) be a positive bi-homogeneous hermitian form on C n , of degree m. A theorem proved by... more Let f (z,z) be a positive bi-homogeneous hermitian form on C n , of degree m. A theorem proved by Quillen and rediscovered by Catlin and D'Angelo states that for N large enough, z,z N f (z,z) can be written as the sum of squares of homogeneous polynomials of degree m+N. We show this works for N ≥ C f ((n+m) log n) 3 where C f has a natural expression in terms of coefficients of f , inversely proportional to the minimum of f on the sphere. The proof uses a semiclassical point of view on which 1/N plays a role of the small parameter h.
Using tools from semiclassical analysis, we give weighted L ∞ estimates for eigenfunctions of str... more Using tools from semiclassical analysis, we give weighted L ∞ estimates for eigenfunctions of strictly convex surfaces of revolution. These estimates give rise to new sampling techniques and provide improved bounds on the number of samples necessary for recovering sparse eigenfunction expansions on surfaces of revolution. On the sphere, our estimates imply that any function having an s-sparse expansion in the first N spherical harmonics can be efficiently recovered from its values at m sN 1/6 log 4 (N) sampling points.
We present a conjecture relating the density of quantum resonances for an open chaotic system to ... more 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.
Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmoni... more Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmonic inversion the distribution of the imaginary parts of the resonances is determined. With increasing opening of the systems, a spectral gap is observed for thick as well as for thin repellers and for the latter case it is compared with the known topological pressure bounds. The maxima of the distributions are found to coincide for a large range of the distance to radius parameter with half of the classical escape rate. This confirms theoretical predictions based on rigorous mathematical analysis for the spectral gap and on numerical experiments for the maxima of the distributions.
We present microwave experiments on the symmetry reduced five-disk billiard studying the transiti... more We present microwave experiments on the symmetry reduced five-disk billiard studying the transition from a closed to an open system. The measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied. For the closed system this counting function shows the Weyl asymptotic with a leading exponent equal to 2. By opening the system successively this exponent decreases smoothly to a noninteger value. For the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed. The results can be interpreted as a first experimental indication for the fractal Weyl conjecture for resonances.
In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous me... more In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous media one observes "breathing" in the sense of the amplitude of the wave going up and down on a much faster scale than the motion of the wave-see Fig. 2 below. In this paper we investigate this phenomenon in the simplest case of stationary waves in which the evolution corresponds to relaxation to a nonlinear ground state. The particular model is the popular δ 0 impurity in the cubic nonlinear Schrödinger equation on the line. We give asymptotics of the amplitude on a finite but relevant time interval and show their remarkable agreement with numerical experiments, see Fig. 1. We stress the nonlinear origin of the "breathing patterns" caused by the selection of the ground state depending on the initial data, and by the non-normality of the linearized operator.
We show that the zeta function for the dynamics generated by the map z → z 2 +c, c < −2, can be e... more 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.
Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigensta... more Quantum ergodicity theorem states that for quantum systems with ergodic classical flows, eigenstates are, in average, uniformly distributed on energy surfaces. We show that if N is a hypersurface in the position space satisfying a simple dynamical condition, the restrictions of eigenstates to N are also quantum ergodic.
A well known result of Jaffard states that an arbitrary region on a torus controls, in the L 2 se... more A well known result of Jaffard states that an arbitrary region on a torus controls, in the L 2 sense, solutions of the free stationary and dynamical Schrödinger equations. In this note we show that the same result is valid in the presence of a potential, that is for Schrödinger operators, −∆ + V , V ∈ C ∞ .
Journal of the European Mathematical Society, 2013
For the Schrödinger equation (i∂ t +)u = 0 on a torus, an arbitrary non-empty open set provides c... more For the Schrödinger equation (i∂ t +)u = 0 on a torus, an arbitrary non-empty open set provides control and observability of the solution: u| t=0 L 2 (T 2) ≤ K T u L 2 ((0,T)×). We show that the same result remains true for (i∂ t + − V)u = 0 where V ∈ L 2 (T 2), and T 2 is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for V ∈ C(T 2) and conjectured for V ∈ L ∞ (T 2). The higher dimensional generalization remains open for V ∈ L ∞ (T n).
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Papers by M. Zworski