Second-order right-invariant partial differential equations on a Lie group (Q915997)

The author gives a simple, relatively explicit eigenfunction expansion for some Laplacian operators and shows that some more explicit results may be obtained for the sub-Laplacians than is the case for the corresponding elliptic Laplace operators defined from a basis for the Lie algebra. It follows, in particular, that some Laplacians have no singular continuous spectrum when the Lie group G is simply connected nilpotent because the spectrum is (0,\(\infty)\), absolutely continuous and with uniform multiplicity (Theorem 4.1. p. 342). The paper embodies not only all the previous papers by the author on the same subject, but also many results which are listed in the other references. In section 5 the author points out that a spectral representation for the Nelson-Stinespring Laplace operator \(\Delta\) can be constructed by applying the Mackey-Blattner imprimitivity theorem. Finally, this long paper which is nearly a monograph, it contains a large number of remarks, lemmas, assumptions, theorems, examples and proofs which are too complicated to be reproduced here.