Rellich’s theorem and N-body Schrödinger operators

We show an optimal version of Rellich’s theorem for generalized [Formula: see text]-body Schrödinger operators. It applies to singular potentials, in particular, to a model for atoms and molecules with infinite mass and finite extent nuclei. Our proof relies on a Mourre estimate [10] and a functional calculus localization technique.

SPECTRAL AND SCATTERING THEORY FOR SOME ABSTRACT QFT HAMILTONIANS

We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. These Hamiltonians are of the form H = dΓ(ω) + V acting on the bosonic Fock space Γ(𝔥), where ω is a massive one-particle Hamiltonian acting on 𝔥 and V is a Wick polynomial Wick(w) for a kernel w satisfying some decay properties at infinity. We describe the essential spectrum of H, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the asymptotic completeness of the scattering theory, which means that the CCR representations given by the asymptotic fields are of Fock type, with the asymptotic vacua equal to the bound states of H. As a consequence, H is unitarily equivalent to a collection of second quantized Hamiltonians.

SCATTERING OF MASSLESS DIRAC FIELDS BY A KERR BLACK HOLE

For the massless Dirac equation outside a slow Kerr black hole, we prove asymptotic completeness. We introduce a new Newman–Penrose tetrad in which the expression of the equation contains no artificial long-range perturbations. The main technique used is then a Mourre estimate. The geometry near the horizon requires us to apply a unitary transformation before we find ourselves in a situation where the generator of dilations is a good conjugate operator. The results are eventually re-interpreted geometrically to provide the solution to a Goursat problem on the Penrose compactified exterior.

A MOURRE ESTIMATE FOR A SCHRÖDINGER OPERATOR ON A BINARY TREE

Let G be a binary tree with vertices V and H be a Schrödinger operator acting on ℓ2 (V). A decomposition of the space ℓ2 (V) into invariant subspaces is exhibited yielding a conjugate operator A for use in the Mourre estimate. We show that for potentials q satisfying a first order difference decay condition, a Mourre estimate for H holds.

STARK HAMILTONIAN WITH UNBOUNDED RANDOM POTENTIALS

Spectral properties of one-dimensional Schrödinger operators with unbounded potentials are studied. The main example is the Stark Hamiltonian with unbounded Anderson-type random perturbations. In this case, it is shown that if the perturbation is o(x) then the spectrum is the real line and absolutely continuous except for eigenvalues with no accumulation points. If the perturbation is larger than O(x), then the Hamiltonian has no absolutely continuous spectrum. The methods of proof involve the Mourre estimate and trace-class perturbation theory as recently used by Simon and Spencer.