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A Proof of the Gutzwiller Semiclassical Trace Formula using Coherent Sates Decomposition
 Commun. in Math. Phys
, 1999
"... The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral ..."
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Cited by 35 (5 self)
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The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck’s constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of H. Later, using the theory of Fourier integral operators, mathematicians gave rigorous proofs of the formula in various settings. Here we show how the use of coherent states allows us to give a simple and direct proof. 1
The Semiclassical Trace Formula and Propagation of Wave Packets
 J. Funct. Anal
, 1994
"... We study spectral and propagation properties of operators of the form S ¯ h = P N j=0 ¯h j P j where 8j P j is a differential operator of order j on a manifold M , asymptotically as ¯h ! 0. The estimates are in terms of the flow fOE t g of the classical Hamiltonian H(x; p) = P N j=0 oe P j (x; ..."
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Cited by 21 (4 self)
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We study spectral and propagation properties of operators of the form S ¯ h = P N j=0 ¯h j P j where 8j P j is a differential operator of order j on a manifold M , asymptotically as ¯h ! 0. The estimates are in terms of the flow fOE t g of the classical Hamiltonian H(x; p) = P N j=0 oe P j (x; p) on T M , where oe P j is the principal symbol of P j . We present two sets of results. (I) The "semiclassical trace formula", on the asymptotic behavior of eigenvalues and eigenfunctions of S ¯ h in terms of periodic trajectories of H . (II) Associated to certain isotropic submanifolds ae T M we define families of functions f/ ¯ h g and prove that 8t fexp(\Gammait¯hS h )(/ ¯ h )g is a family of the same kind associated to OE t (). Introduction and description of results. In this paper we present some results concerning spectral and propagation properties of a class of differential operators "with small parameter", ¯h (Planck's constant). We have in mind operators of the form a(x...
Strichartz estimates without loss on manifolds with hyperbolic trapped geodesics
 GAFA
"... Abstract. In [21], Doi proved that the L 2 t H 1 2 x local smoothing effect for Schrödinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L 1 → L ∞ dispersive estimates still hold without loss for e ..."
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Cited by 8 (0 self)
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Abstract. In [21], Doi proved that the L 2 t H 1 2 x local smoothing effect for Schrödinger equation on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L 1 → L ∞ dispersive estimates still hold without loss for e it ∆ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension. The influence of the geometry on the behaviour of solutions of linear or non linear partial differential equations has been widely studied recently, and especially in the context of wave or Schrödinger equations. In particular, the understanding of the smoothing effect for the Schrödinger flow and Strichartz type estimates has been related to the global behaviour of the geodesic flow on the manifold (see for example the works by Doi [21] and Burq [11]). Let us recall that for the Laplacian ∆ on a ddimensional noncompact Riemannian manifold (M, g), the local smoothing effect for bounded time t ∈ [0, T] and Schrödinger waves u = eit∆u0: M × R → C is the estimate χe it ∆ u0  L2 ((0,T);H1/2(M)) ≤ CT u0  L2 (M), ∀u0 ∈ L 2 (M) where CT> 0 is a constant depending a priori on T and χ is a compactly supported smooth function (the asumption on χ can of course be weakened in many cases, e.g for M = Rd) [19]. In other words, although the solution is only L2 in space uniformly in time, it is actually half a derivative better (locally) in an L2intime sense. For its description in geometric setting, the picture now is fairly complete: the so called “nontrapping condition ” stating roughly that every geodesic maximally extended goes to infinity, is known to be essentially necessary and sufficient (modulo reasonable conditions near infinity) [11]. Another tool for analyzing nonlinear Schrödinger equations is the family of socalled Strichartz estimates introduced by [39]: for Schrödinger waves on Euclidean space Rd with initial data u0, (0.1) e it ∆ u0  L p ((0,T);L q (R d)) ≤ CT u0  L 2 (R d) if p, q ≥ 2, 2 d p q
Wave invariants at elliptic closed geodesics
 Geom. Funct. Anal
, 1997
"... The purpose of this article is to provide an effective method for calculating the wave invariants associated to a nondegenerate elliptic closed geodesic γ of a compact Riemannian manifold (M, g): that is, the coefficients in the singularity expansion ..."
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Cited by 8 (1 self)
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The purpose of this article is to provide an effective method for calculating the wave invariants associated to a nondegenerate elliptic closed geodesic γ of a compact Riemannian manifold (M, g): that is, the coefficients in the singularity expansion
SemiClassical Asymptotics in Magnetic Bloch Bands
, 2002
"... This article gives a simple construction of wave packets localized near semiclassical trajectories for an electron subject to external electric and magnetic fields. We assume that the magnetic and electric potentials are slowly varying perturbations of the potential of a constant magnetic field and ..."
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Cited by 7 (1 self)
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This article gives a simple construction of wave packets localized near semiclassical trajectories for an electron subject to external electric and magnetic fields. We assume that the magnetic and electric potentials are slowly varying perturbations of the potential of a constant magnetic field and a periodic lattice potential, respectively.
BohrSommerfeld Quantization Rules in the Semiclassical Limit
"... We study onedimensional quantum mechanical systems in the semiclassical limit. We construct a lowest order quasimode (~) for the Hamiltonian H(~) when the energy E and Planck's constant ~ satisfy the appropriate BohrSommerfeld conditions. This means that (~) is an approximate solution of the Schrö ..."
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Cited by 6 (2 self)
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We study onedimensional quantum mechanical systems in the semiclassical limit. We construct a lowest order quasimode (~) for the Hamiltonian H(~) when the energy E and Planck's constant ~ satisfy the appropriate BohrSommerfeld conditions. This means that (~) is an approximate solution of the Schrödinger equation in the sense that k[H(~) \Gamma E] (~)k ^ C~3=2 k(~)k. It follows that H(~) has some spectrum within a distance C~3=2 of E. Although the result has a long history, our timedependent construction technique is novel and elementary.
On the Pointwise Behaviour of SemiClassical Measures
, 1994
"... In this paper we concern ourselves with the small ¯h asymptotics of the inner products of the eigenfunctions of a Schrodingertype operator with a coherent state. More precisely, let / ¯ h j and E ¯ h j denote the eigenfunctions and eigenvalues of a Schrodingertype operator H ¯ h with discrete s ..."
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Cited by 1 (0 self)
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In this paper we concern ourselves with the small ¯h asymptotics of the inner products of the eigenfunctions of a Schrodingertype operator with a coherent state. More precisely, let / ¯ h j and E ¯ h j denote the eigenfunctions and eigenvalues of a Schrodingertype operator H ¯ h with discrete spectrum. Let / (x;¸) be a coherent state centered at the point (x; ¸) in phase space. We estimate as ¯h ! 0 the averages of the squares of the inner products j (/ a (x;¸) ; / ¯ h j ) j 2 over an energy interval of size ¯h around a fixed energy, E. This follows from asymptotic expansions of the form X j ' ` E j (¯h) \Gamma E ¯h ' j (/ a (x;¸) ; / ¯ h j ) j 2 ¸ 1 X k=0 c k (a)¯h \Gamman+ 1 2 +k for certain test functions ' and Schwartz amplitudes a of the coherent state. We compute the leading coefficient in the expansion, which depends on whether the classical trajectory through (x; ¸) is periodic or not. In the periodic case the iterates of the trajectory contribute ...