Quantum Fidelity Decay of Quasi-Integrable Systems

We show, via numerical simulations, that the fidelity decay behavior of quasi-integrable systems is strongly dependent on the location of the initial coherent state with respect to the underlying classical phase space. In parallel to classical fidelity, the quantum fidelity generally exhibits Gaussian decay when the perturbation affects the frequency of periodic phase space orbits and power-law decay when the perturbation changes the shape of the orbits. For both behaviors the decay rate also depends on initial state location. The spectrum of the initial states in the eigenbasis of the system reflects the different fidelity decay behaviors. In addition, states with initial Gaussian decay exhibit a stage of exponential decay for strong perturbations. This elicits a surprising phenomenon: a strong perturbation can induce a higher fidelity than a weak perturbation of the same type.Comment: 11 pages, 11 figures, to be published Phys. Rev.

Rigidity around Poisson Submanifolds

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's linearization theorem, also proving that Conn's theorem is, indeed, just a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem; another interesting case corresponds to spheres inside duals of compact semisimple Lie algebras, our result can be used to fully compute the resulting Poisson moduli space.Comment: 43 pages, v3: published versio

Symplectic Microgeometry II: Generating functions

We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an application, we describe hamiltonian flows as special symplectic micromorphisms whose local generating functions are the solutions of Hamilton-Jacobi equations. We obtain a purely categorical formulation of the temporal evolution in classical mechanics.Comment: 27 pages, 1 figur

Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field.Comment: To appear in Inventiones Mathematica

Time Dependent Resonance Theory

An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photon-radiation field, and Auger states of the helium atom, as well as in spectral geometry and number theory. We present a dynamic (time-dependent) theory of such quantum resonances. The key hypotheses are (i) a resonance condition which holds generically (non-vanishing of the {\it Fermi golden rule}) and (ii) local decay estimates for the unperturbed dynamics with initial data consisting of continuum modes associated with an interval containing the embedded eigenvalue of the unperturbed Hamiltonian. No assumption of dilation analyticity of the potential is made. Our method explicitly demonstrates the flow of energy from the resonant discrete mode to continuum modes due to their coupling. The approach is also applicable to nonautonomous linear problems and to nonlinear problems. We derive the time behavior of the resonant states for intermediate and long times. Examples and applications are presented. Among them is a proof of the instability of an embedded eigenvalue at a threshold energy under suitable hypotheses.Comment: to appear in Geometrical and Functional Analysi

Nonautonomous Hamiltonians

We present a theory of resonances for a class of non-autonomous Hamiltonians to treat the structural instability of spatially localized and time-periodic solutions associated with an unperturbed autonomous Hamiltonian. The mechanism of instability is radiative decay, due to resonant coupling of the discrete modes to the continuum modes by the time-dependent perturbation. This results in a slow transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. The ideas are closely related to the authors' work on (i) a time dependent approach to the instability of eigenvalues embedded in the continuous spectra, and (ii) resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. The theory is applied to a general class of Schr\"odinger equations. The phenomenon of ionization may be viewed as a resonance problem of the type we consider and we apply our theory to find the rate of ionization, spectral line shift and local decay estimates for such Hamiltonians.Comment: To appear in Journal of Statistical Physic

Equilibrium Configuration of Black Holes and the Inverse Scattering Method

The inverse scattering method is applied to the investigation of the equilibrium configuration of black holes. A study of the boundary problem corresponding to this configuration shows that any axially symmetric, stationary solution of the Einstein equations with disconnected event horizon must belong to the class of Belinskii-Zakharov solutions. Relationships between the angular momenta and angular velocities of black holes are derived.Comment: LaTeX, 14 pages, no figure