ABSTRACT
A fundamental ingredient in the construction of the heat kernel of Lτ for τ ∈ R\ {0} is the spectral analysis of Lτ , which we now present. The contents in this chapter and the constructions of the heat kernel and Green function for Lτ , τ ∈ R \ {0}, in the following two chapters are the τ -versions of those in the paper [42].
