No abstract

Let S 1 be a unit circle with the standard measure, P k be the orthogonal projection in L 2 (S 1 ) on the subspace spanned by e ijx , j=0, \1, ..., \k, and B be the operator of multiplication by a smooth function b in L 2 (S 1 ). The classical Szego limit theorem states that under some assumptions on bThe operators P k coincide with the spectral projections P * of the selfadjoint operator (&d 2 Âdx 2 ) 1Â2 in L 2 (S 1 ) corresponding to the intervals [0, *) with k<* k+1. Following V. Guillemin [G], we obtain a generalization of this theorem for P * being the spectral projections of an elliptic selfadjoint (pseudo)differential operator A on a manifold without boundary. We also study the case where A is the operator of an elliptic boundary value problem.Moreover, we consider an arbitrary sufficiently smooth function instead of the logarithm. In other words, we obtain asymptotics of the functional

We study asymptotic distribution of eigenvalues of the Laplacian on a bounded domain in R n . Our main results include an explicit remainder estimate in the Weyl formula for the Dirichlet Laplacian on an arbitrary bounded domain, sufficient conditions for the validity of the Weyl formula for the Neumann Laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula. In particular, we show that the Weyl formula holds true for the Neumann Laplacian on a Lip α -domain whenever (d − 1)/α < d , prove that the remainder in this formula is O(λ (d−1)/α ) and give an example where the remainder estimate O(λ (d−1)/α ) is order sharp. We use a new version of variational technique which does not require the extension theorem.

In this paper we develop a new approach to the theory of Fourier integral operators. It allows us to represent the Schwartz kernel of a Fourier integral operator by one oscillatory integral with a complex phase function. We consider Fourier integral operators associated with canonical transformations, having in mind applications to hyperbolic equations. As a by-product we obtain yet another formula for the Maslov index.