MROZ, K. and STROHMAIER, A., 2016. Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature. Journal of Spectral Theory, in press.
Let (M, g) be a compact, d-dimensional Riemannian manifold without boundary.
Suppose further that (M, g) is either two dimensional and has no conjugate points
or (M, g) has non-positive sectional curvature. The goal of this note is to show that
the long time parametrix obtained for such manifolds by B´erard can be used to prove
a logarithmic improvement for the remainder term of the Riesz means of the counting
function of the Laplace operator.
This paper was accepted for publication in the Journal of Spectral Theory and the accepted version is also available in arXiv http://arxiv.org/pdf/1409.1869v2.pdf