The continuous spectrum of the Dirac operator on noncompact Riemannian symmetric spaces of rank one (Q2758988)

Let \(X=G/K\) be a Riemannian symmetric space of noncompact type. Let \(D\) be the Dirac operator acting on spinors on \(X\). Recently \textit{S. Goette} and \textit{U. Semmelmann} [Proc. Am. Math. Soc. 130, 915-923 (2002)] calculated the \(L^2\) point spectrum of \(D\): let \(X\) be irreducible, then this spectrum is nonempty if and only if \(X=SU(p,q)/S(U(p)\times U(q))\) with \(p+q\) odd, and in this case it consists of the one point 0. In the paper under review, the authors study the \(L^2\) continuous spectrum of \(D\) for spaces \(X\) of rank one. The result is: for these spaces, the continuous spectrum of \(D\) is \(\mathbb{R}\), except complex hyperbolic spaces \(H^n(\mathbb{C})\) with \(n\) even, in which case it is \((-\infty, -1/2 ]\cup[1/2, +\infty)\) (so we see that this spectral gap takes place precisely as soon as there is a point spectrum). For real hyperbolic spaces the result was already known.