Erik Skibsted

Perturbation of embedded eigenvalues in the generalized

We discuss the perturbation of continuum eigenvalues without analyticity assumptions. Among our results, we show that generally a small perturbation removes these eigenvalues in accordance with Fermis Golden Rule. Thus, generically (in a Baire category sense), the Schrödinger operator has no embedded non-threshold eigenvalues.

N

For a quantum mechanical two-bodys-wave resonance we prove that the evolution of square integrable approximations of the Gamow function is outgoing and exponentially damped. An error estimate is given in terms of resonance energy and explicity. We obtain the Breit-Wigner form. The results are used in an α-decay model to prove general validity of the exponential decay law for periods of several lifetimes.

-body problem

We prove propagation estimates (of strong type) for long-rangeN-body Hamiltonians. Emphasis is on phase-space analysis in the free channel region.

Truncated Gamow functions, α-decay and the exponential law

We prove smoothness of the 2-cluster-2-cluster and 2-cluster-N-cluster scattering amplitudes under a general short range condition on the potential and under a discreteness assumption on the channel energies.

Propagation estimates for N-body Schroedinger operators

We consider two-body Schrodinger operators with multiplicative, exponentially decreasing and form-compact potentials. It is proved that the evolution of sharp cutoff approximations of a resonance function is outgoing and exponentially damped. Except for the choices of cutoff radii, shown to be determined by outgoing asymptotics, an explicit error estimate is given in terms of time variable, resonance energy, and width.

SMOOTHNESS OF N-BODY SCATTERING AMPLITUDES

We prove that the spectrum for a large class ofN-body Stark Hamiltonians is purely absolutely continuous. We need slow decay at infinity and local singularities of at most Coulomb type. In particular our results include the usual models for atoms and molecules.

On the evolution of resonance states

We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling.

Spectral analysis of

Abstract Let V 1 : S n−1 → ℝ be a Morse function and define V 0(x) = V 1(x/|x|). We consider the scattering theory of the Hamiltonian in L 2(ℝ n ), n ≥ 2, where V is a short-range perturbation of V 0. We introduce two types of wave operators for channels corresponding to local minima of V 1 and prove completeness of these wave operators in the appropriate energy ranges.

N

We prove asymptotic completeness for short- and long-rangeN-body Stark Hamiltonians with local singularities of at most Coulomb type. Our results include the usual models for atoms and molecules.

-body Stark Hamiltonians

We study regularity of bound states pertaining to embedded eigenvalues of a self-adjoint operator H, with respect to an auxiliary operator A that is conjugate to H in the sense of Mourre. We work within the framework of singular Mourre theory which enables us to deal with confined massless Pauli–Fierz models, our primary example, and many-body AC-Stark Hamiltonians. In the simpler context of regular Mourre theory, our results boil down to an improvement of results obtained recently in [8, 9].