We consider nonlinear Schrödinger equations in R³. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although self-adjoint perturbation turns embedded eigenvalues into resonances, this class of non-self adjoint perturbations turn an embedded eigenvalue into two eigenvalues with the distance to the continuous spectrum given to the leading order by the Fermi golden rule.

We consider the nonlinear Schrödinger equation (1.1) i∂tψ + ∂ 2 xψ = −|ψ | 2σ ψ on the line R with σ> 2. This is exactly the L 2-supercritical case and these equations are locally well-posed in H 1 (R) = W 1,2 (R). Let φ = φ(·, α) be the

We prove L p estimates for charge transfer Hamiltonians, including matrix and inhomogeneous generalizations; such equations appear naturally in the study of multi-soliton systems. 1

We consider the critical focusing NLS in 1-d of the form (1.1) i∂tψ + ∂ 2 xψ = −|ψ | 4 ψ, i = √ −1, ψ = ψ(t, x), and ψ complex valued. It is well-known that this equation permits standing wave solutions of the form φ(t, x) = e iαt φ0(x, α), α> 0

n ∈ Z, x, ω ∈ [0, 1] with real-analytic potential function V (x). If L(E, ω0)> 0 for all E ∈ (E′, E ′′) and some Diophantine ω0, then the integrated density of states is absolutely continuous for almost every ω close to ω0, see [GolSch2]. In this work we apply the methods and results of [GolSch2] to establish the formation of a dense set of gaps in ⋃ sp H(x, ω) ∩ (E x ′, E ′′). Our approach is based on multi-scale arguments, and is therefore both constructive as well as quantitative. We show how resonances between eigenfunctions of one scale lead to ”pre-gaps ” at a larger scale. Then we show how these pre-gaps cannot be filled more than a finite (and uniformly bounded) number of times. To accomplish this, we relate a pre-gap to pairs of complex zeros of the Dirichlet determinants off the unite circle using the techniques of [GolSch2]. Of basic importance to our entire construction are the finite-volume description of Anderson localization as well as the separation of Dirichlet eigenvalues in a finite volume which were obtained in [GolSch2]. Another essential ingredient is the elimination of triple resonances from Chan [Cha], a special case of which is reproduced here.

The often ignored quantum probability flux is fundamental for a genuine understanding of scattering theory as, in particular, expressed in the flux-across-surfaces theorem. This work splits into two parts. First we show how the flux enters into scattering theory and we give an elementary proof of the free flux-across-surfaces theorem. At least heuristically, the free theorem together with completeness of the wave operators implies the full fluxacross -surfaces theorem. Therefore, in the second part, we discuss the proof of asymptotic completeness in potential scattering---the main focus of mathematical scattering theory so far. Of course this is well known, however, we found that the presentations of the proof (we looked at) showed no awareness of the crucial physical ingredient, namely the current positivity condition, a condition on the quantum flux. We wish to present here our understanding of the issues involved and we wish to emphasize that the arguments are all straightforward and natural: The proof uses Riemann-Lebesgue, compactness of operators and the current positivity condition.

Perturbation problems for operators with embedded eigenvalues are generally challenging since the embedded eigenvalues cannot be separated from the rest of the spectrum. In this paper we study a perturbation problem for embedded eigenvalues for the bilaplacian with an added potential, when the underlying domain is a cylinder. We show that the set of nearby potentials, for which a simple embedded eigenvalue persists, forms a smooth manifold of finite codimension.

We consider linear nonautonomous second order parabolic equations on R^N. Under an instability condition, we prove the existence of two complementary Floquet bundles, one spanned by a positive entire solution- the principal Floquet bundle, the other one consisting of sign-changing solutions. We establish an exponential separation between the two bundles, showing in particular that a class of signchanging solutions are exponentially dominated by positive solutions.

Introduction The physical picture underlying nonrelativistic quantum electrodynamics is that of charged particles which interact through the exchange of photons and dissipate energy through emission of photons. In situations where the velocities of the particles are small compared to the propagation speed of the photons the interaction is given through effective, instantaneous pair potentials. If, in addition, also accelerations are small, then dissipation through radiation can be neglected in good approximation. Instead of full nonrelativistic QED we consider the massless Nelson model. This model describes N spinless particles coupled to a scalar Bose eld of zero mass. The content of this work is a mathematical derivation of the time-dependent Schrodinger equation for N particles with Coulombic pair potentials from the massless Nelson model with ultraviolet cutos. The key mechanism in our derivation is adiabatic decoupling without a spectral gap. Before we turn to a more caref

Abstract: We consider a quantum particle coupled (with strength λ) to a spatial array of independent non-interacting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at large times for small, but finite λ. Our proof relies on an expansion around the kinetic scaling limit (λ ց 0, while time and space scale as λ−2) in which the particle satisfies a Boltzmann equation. We also show an equipartition theorem: the distribution of the kinetic energy of the particle tends to a Maxwell-Boltzmann distribution, up to a correction of O(λ2). KEY WORDS: diffusion, kinetic limit, quantum brownian motion