A new look at Mourre’s commutator theory. (English) Zbl 1167.47010
A new approach to Mourre’s commutator theory is developed. Let \(H\) be a selfadjoint operator in a Hilbert space \({\mathfrak H}\). Let \(\tau\in C_c^\infty (\mathbb{R})\) and set \(H_\tau=H\tau(H)\). The limiting absorption principle (LAP) for \(H_\tau\) implies that for \(H\). That means, the strict Mourre estimate for \(H_\tau\) implies the LAP for \(H\). The authors also show the LAP for the reduced resolvent and study the regularity of eigenvectors of \(H\). An example is constructed where the reduced LAP holds but the original Mourre estimate does not.
Reviewer: Michael Demuth (Goslar)
MSC:
47A40 | Scattering theory of linear operators |
47B25 | Linear symmetric and selfadjoint operators (unbounded) |
47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |
47N50 | Applications of operator theory in the physical sciences |
81U99 | Quantum scattering theory |