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22
Almost global existence for Hamiltonian semilinear KleinGordon equations with small Cauchy data on Zoll manifolds, preprint (2005). [BG06] D. Bambusi and B. Grébert, Birkhoff normal form for PDEs with tame modulus
 Lecture Notes in Mathematics
"... B. Grébert, J. Szeftel This paper is devoted to the proof of almost global existence results for KleinGordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods ..."
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Cited by 19 (10 self)
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B. Grébert, J. Szeftel This paper is devoted to the proof of almost global existence results for KleinGordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds. 1
Eigenfrequencies for damped wave equations on Zoll manifolds
, 2002
"... The eigenfrequencies associated to a damped wave equation are known to belong to a band parallel to the real axis. Under the assumption of periodicity of the geodesic flow we study the asymptotic distribution of the eigenfrequencies in the band. We show that the set of eigenfrequencies exhibits a c ..."
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Cited by 12 (6 self)
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The eigenfrequencies associated to a damped wave equation are known to belong to a band parallel to the real axis. Under the assumption of periodicity of the geodesic flow we study the asymptotic distribution of the eigenfrequencies in the band. We show that the set of eigenfrequencies exhibits a cluster structure determined by the Morse index of the closed geodesics and the damping coefficient averaged along the geodesic flow. The asymptotics for the multiplicities of the clusters are also obtained.
Singular Bohr–Sommerfeld Rules for 2d Integrable Systems
, 2003
"... This article gives Bohr–Sommerfeld rules for semiclassical completely integrable systems with two degrees of freedom with nondegenerate singularities (Morse–Bott singularities) under the assumption that the energy level of the first Hamiltonian is nonsingular. The more singular case of focus– foc ..."
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Cited by 4 (0 self)
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This article gives Bohr–Sommerfeld rules for semiclassical completely integrable systems with two degrees of freedom with nondegenerate singularities (Morse–Bott singularities) under the assumption that the energy level of the first Hamiltonian is nonsingular. The more singular case of focus– focus singularities was treated in previous works by San Vũ Ngoc. The case of one degree of freedom has been studied by Colin de Verdière and Parisse. The results are applied to some famous examples: the geodesics of the ellipsoid, the 1:2resonance, and Schrödinger operators on the sphere S 2. A numerical test shows that the semiclassical Bohr–Sommerfeld rules match very accurately the “purely quantum” computations.
Perturbations of selfadjoint operators with periodic classical flow
, 2003
"... We consider nonselfadjoint perturbations of a selfadjoint hpseudodifferential operator in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength ǫ of the perturbation satisfies h δ0 < ǫ ≤ ǫ0 for some δ0 ∈]0,1/2 [ and a suffi ..."
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Cited by 4 (2 self)
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We consider nonselfadjoint perturbations of a selfadjoint hpseudodifferential operator in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength ǫ of the perturbation satisfies h δ0 < ǫ ≤ ǫ0 for some δ0 ∈]0,1/2 [ and a sufficiently small ǫ0> 0. We get a complete asymptotic description of all eigenvalues in certain rectangles [−1/C,1/C] + iǫ[F0 − 1/C,F0 + 1/C]. In particular we are able to treat the case when ǫ> 0 is small but independent of h.
The Trace Formula and the Distribution of Eigenvalues of Schrödinger Operators on Manifolds all of whose Geodesics are closed
, 1995
"... We investigate the behaviour of the remainder term R(E) in the Weyl formula #fnjE n Eg = Vol(M) (4ß) d=2 \Gamma(d=2 + 1) E d=2 +R(E) for the eigenvalues E n of a Schrödinger operator on a ddimensional compact Riemannian manifold all of whose geodesics are closed. We show that R(E) is of the f ..."
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Cited by 3 (0 self)
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We investigate the behaviour of the remainder term R(E) in the Weyl formula #fnjE n Eg = Vol(M) (4ß) d=2 \Gamma(d=2 + 1) E d=2 +R(E) for the eigenvalues E n of a Schrödinger operator on a ddimensional compact Riemannian manifold all of whose geodesics are closed. We show that R(E) is of the form E (d\Gamma1)=2 \Theta( p E), where \Theta(x) is an almost periodic function of Besicovitch class B 2 which has a limit distribution whose density is a boxshaped function. This is in agreement with a recent conjecture of Steiner [19, 1]. Furthermore we derive a trace formula and study higher order terms in the asymptotics of the coefficients related to the periodic orbits. The periodicity of the geodesic flow leads to a very simple structure of the trace formula which is the reason why the limit distribution can be computed explicitly.
Asymptotics of Rydberg States for the Hydrogen Atom
, 1997
"... The asymptotics of Rydberg states, i.e., highly excited bound states of the hydrogen atom Hamiltonian, and various expectations involving these states are investigated. We show that suitable linear combinations of these states, appropriately rescaled and regarded as functions either in momentum spac ..."
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Cited by 3 (1 self)
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The asymptotics of Rydberg states, i.e., highly excited bound states of the hydrogen atom Hamiltonian, and various expectations involving these states are investigated. We show that suitable linear combinations of these states, appropriately rescaled and regarded as functions either in momentum space or configuration space, are highly concentrated on classical momentum space or configuration space Kepler orbits respectively, for large quantum numbers. Expectations of momentum space or configuration space functions with respect to these states are related to timeaverages of these functions over Kepler orbits. 1 Section I. Introduction Let H be the hydrogen atom Hamiltonian H = \Gamma 1 2 \Delta \Gamma jxj \Gamma1 acting in L 2 (R 3 ), with \Delta the 3dimensional Laplacian. The purpose of this article is to investigate the asymptotics of Rydberg states of the Hamiltonian H, i.e., states with large principal quantum number k, and to investigate the asymptotics of various ex...