On the spectral analysis of many-body systems

TL;DR

This paper analyzes the spectral properties of complex many-body quantum systems with particle interactions and creation-annihilation processes, extending spectral analysis techniques to more general quantum models.

Contribution

It computes the Hamiltonian algebra for such systems and establishes the Mourre estimate, generalizing N-body spectral analysis to systems with variable particle numbers.

Findings

Describes the essential spectrum of many-body systems.

Proves the Mourre estimate for systems with particle creation and annihilation.

Shows the Hamiltonian algebra is graded by a semilattice.

Abstract

We describe the essential spectrum and prove the Mourre estimate for quantum particle systems interacting through k-body forces and creation-annihilation processes which do not preserve the number of particles. For this we compute the ``Hamiltonian algebra'' of the system, i.e. the C*-algebra C generated by the Hamiltonians we want to study, and show that, as in the N-body case, it is graded by a semilattice. Hilbert C*-modules graded by semilattices are involved in the construction of C. For example, if we start with an N-body system whose Hamiltonian algebra is B and then we add field type couplings between subsystems, then the many-body Hamiltonian algebra C is the imprimitivity algebra of a graded Hilbert B-module.