ESSENTIAL SPECTRUM, QUASI-ORBITS AND COMPACTIFICATIONS: APPLICATION TO THE HEISENBERG GROUP

Let H be the Heisenberg group and (H) over bar= H boolean OR Sigma(H) be its radial compactification. Our main result is a decomposition of the essential spectrum of the Schrodinger-type operator T = -Delta + V, where V is a continuous real function on (H) over bar. Our result extends classical results of HVZ-type from Euclidean spaces to the Heisenberg group. While many features are preserved in the Heisenberg group case, there are also some notable differences. First, the action of H on itself extends to an action of H on (H) over bar and we compute the quasi-orbits (the closure of the orbits) of the action, whose structure is more complicated in the Heisenberg Following [16], we show that the essential spectrum of any operator T contained in (or affiliated to) C((H) over bar) (sic) H is the union of the spectra of a family (T alpha)(alpha is an element of F) of simpler operators indexed by a family of quasi-orbits that cover (H) over bar \ H, that is, sigma(ess)(T) = boolean OR(alpha is an element of F) sigma(T-alpha). We obtain similar results also for an other H-equivariant compactification of H.