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.

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Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg's `quantum-theoretical Umdeutung (reinterpretation) of classical observables', which lies at the basis of quantization theory. Similarly, Bohr's correspondence principle (in somewhat revised form) and Schroedinger's wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit h -> 0 of small Planck's constant (in a finite system), in the limit N goes to infinity of a large system with $N$ degrees of freedom (at fixed h), and through decoherence and consistent histories. The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of C*-algebras and the concepts of superselection sectors and macroscopic observables. In these limits, the classical world does not emerge as a sharply defined objective reality, but rather as an approximate appearance relative to certain ``classical" states and observables. Decoherence subsequently clarifies the role of such states, in that they are ``einselected", i.e. robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories. This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the elimination of certain states and observables from quantum theory. Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it.

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Between classical and quantum

This selective review is written as an introduction to the mathematical theory of the Schrodinger equation for N particles. Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results. The central theme is the interplay between the spectral theory of N-body Hamiltonians and the space–time and phase-space analysis of bound states and scattering states.

The quantum N-body problem

We consider a linear Schrodinger equation with a nonlinear perturbation in R 3 . Assume that the linear Hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition. We prove that if the initial data is sufficiently small and is near a nonlinear ground state, then the solution approaches to certain nonlinear ground state as the time tends to infinity. Furthermore, the difference between the wave function solving the nonlinear Schrodinger equation and its asymptotic profile can have two different types of decay: The resonancedominated solutions decay as t 1=2 or the dispersion-dominated solutions decay at least like t 3=2 . c 2002 John Wiley & Sons, Inc.

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Asymptotic Dynamics of Nonlinear Schrödinger Equations: Resonance-Dominated and Dispersion-Dominated Solutions

This paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics.

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Schrödinger operators in the twentieth century

We give a new derivation of the minimal velocity estimates (SiSo1) for unitary evolutions. LetH andA be selfadjoint operators on a Hilbert space H. The starting point is Mourre's inequality i(H,A) ≥ � > 0, which is supposed to hold in form sense on the spectral subspace Hof H for some interval � ⊂ R. The second assumption is that the multiple commutators ad (k) A (H) are well- behaved fork = 1...n (n ≥ 2) . Then we show that, for a dense set of 's in Hand allm < n−1, t = exp(−iHt) is contained in the spectral subspace A ≥ �t as t → ∞, up to an error of order t −m in norm. We apply this general result to the case where H is a Schrodinger operator on R n and A the dilation generator, proving that t(x) is asymptotically supported in the set |x| ≥ t √ � up to an error of order t −m in norm.

Minimal escape velocities

We give a new derivation of the minimal velocity estimates for unitary evolutions with some optimal bounds.

W. Hunziker

Papers

The quantum N-body problem

Minimal escape velocities

Minimal Escape velocities