Asymptotics of the d’Alembertian with potential on a pseudo-Riemannian manifold

Abstract

Let ◻ \square be the Laplace-d’Alembert operator on a pseudo-Riemann- ian manifold ( M , g ) (M,g) . We derive a series expansion for the fundamental solution G ( x , y ) G(x,y) of ◻ + H \square +H , H ∈ C ∞ ( M ) H\in C^{\infty }(M) , which behaves well under various symmetric space dualities. The qualitative properties of this expansion were used in our paper in Invent. Math. 129 (1997), 63–74, to show that the property of vanishing logarithmic term for G ( x , y ) G(x,y) is preserved under these dualities.