\(C^*\)-cross products and a generalized quantum mechanical \(N\)-body problem. (English) Zbl 0968.47035

Damak, Mondher; Georgescu, Vladimir

Electron. J. Differ. Equ. 2000, Conf. 04, 51-69 (2000).

Summary: For each finite-dimensional real vector space \(X\) we construct a \(C^*\)-algebra \(C^X_0\) graded by the lattice of all subspaces of \(X\). Then we compute its quotient with respect to the algebra of compact operators. This allows us to describe the essential spectrum and to prove the Mourre estimate for the self-adjoint operators associated with \(C^X_0\).

MSC:

47L65	Crossed product algebras (analytic crossed products)
46L55	Noncommutative dynamical systems
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis
81R15	Operator algebra methods applied to problems in quantum theory
81V70	Many-body theory; quantum Hall effect
46L60	Applications of selfadjoint operator algebras to physics
46N50	Applications of functional analysis in quantum physics
47L90	Applications of operator algebras to the sciences

Keywords:

\(C^*\)-cross products; quantum mechanical \(N\)-body problem; algebra of compact operators; essential spectrum; Mourre estimate