New methods in spectral theory of N-body Schrödinger operators

We develop a new scheme of proofs for spectral theory of the [Formula: see text]-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem which is unified with exponential decay estimates studied previously only for [Formula: see text]-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with ‘zeroth order’ operators. In particular, they do not rely on Mourre’s differential inequality technique.

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.

-Δ is Positive Definite on a "Spiny Urchin"

In a recent note [3] in this department C. Clark has shown that Rellich's theorem on the compactness of the imbedding is valid if G is the "spiny urchin" domain obtained by removing from the plane the union of the sets Sk (k = 1, 2,…) defined in polar coordinates by

On Rellich's Theorem Concerning Infinitely Narrow Tubes

Let G be a region in Eućlidean n-space En and consider the eigenvalue problem Δ2u = λu on G, with boundary conditions u = 0 on Γ, the boundary of G. (To be precise, we are considering the eigenvalue problem for the self-adjoint 2 realization L associated with the Laplacian -Δ2and zero boundary condition, acting in L2(G), cf Browder [2]). If G is bounded, the spectrum of this problem is discrete, but Rellich showed in 1952 [6] that the spectrum could also be discrete for certain unbounded regions which he introduced and called "infinitely narrow tubes".