Lᵖ spectral theory and heat dynamics of locally symmetric spaces

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In this paper we first derive several results concerning the Lᵖ spectrum of arithmetic locally symmetric spaces whose -rank equals one. In particular, we show that there is an open subset of consisting of eigenvalues of the Lᵖ Laplacian if p <2 and that corresponding eigenfunctions are given by certain Eisenstein series. On the other hand, if p>2 there is at most a discrete set of real eigenvalues of the Lᵖ Laplacian. These results are used in the second part of this paper in order to show that the dynamics of the Lᵖ heat semigroups for p<2 is very different from the dynamics of the Lᵖ heat semigroups if p≥ 2.