On the remainder in the Weyl formula for the Euclidean disk

We prove a 2-terms Weyl formula for the counting function N(mu) of the spectrum of the Laplace operator in the Euclidean disk with a sharp remainder estimate O(mu^2/3)

Semi-classical trace formulas and heat expansions

in the recent paper [Journal of Physics A, 43474-0288 (2011)], B. Helffer and R. Purice compute the second term of a semi-classical trace formula for a Schr\"odinger operator with magnetic field. We show how to recover their formula by using the methods developped by the geometers in the seventies for the heat expansions.Comment: To appear in "Analysis of Partial Differential Equations

The action of pseudo-differential operators on functions harmonic outside a smooth hyper-surface

We consider a smooth hyper-surface Z of a closed Riemannian manifold X. Let P be the Poisson operator associating to a smooth function on Z its harmonic extension on X\Z. If A is a pseudo-differential operator on X of degree <3, we prove that B=P^* A P is a pseudo-differential operator on Z and calculate the principal symbol of B