We consider scattering by general compactly supported semi-classical perturbations of the Euclidean Laplace-Beltrami operator. We show that if the suitably cut-off resolvent of the Hamiltonian quantizes a Lagrangian relation on the product cotangent bundle, the scattering amplitude quantizes the natural scattering relation. In the case when the resolvent is tempered, which is true under some no...

Let k be a positive integer. Suppose that f is a modular form of weight k + 1/2 on Γ0(4). The Shimura correspondence defined in [12] maps f to a modular form F of integral weight 2k. In addition, if f is an eigenform of the Hecke operator Tk+1/2(p), then F is also an eigenform of the Hecke operators T2k(p) with the same eigenvalue. For more on half-integral weight modular forms, see [12]. Let t...

Generalized Radon transforms such as the hyperbolic Radon transform cannot be implemented as efficiently in the frequency domain as convolutions, thus limiting their use in seismic data processing. We introduce a fast butterfly algorithm for the hyperbolic Radon transform. The basic idea is to reformulate the transform as an oscillatory integral operator and to construct a blockwise low-rank ap...

A simplified model in superconductivity theory studied by P. Krotkov and A. Chubukov [KC1, KC2] led to an integral operator K — see (1), (2). They guessed that the equation E0(a, T ) = 1 where E0 is the largest eigenvalue of the operator K has a solution (∗) T (a) = 1− τ(a) with τ(a) ∼ a when a goes to 0. τ(a) imitates the shift of critical (instability) temperature. We give a rigorous analysis...

We analyze in classical L(R)-spaces, n = 2 or n = 3, 1 < q < ∞, a singular integral operator arising from the linearization of a hydrodynamical problem with a rotating obstacle. The corresponding system of partial differential equations of second order involves an angular derivative which is not subordinate to the Laplacian. The main tools are Littlewood– Paley theory and a decomposition of the...

Let T be a Fourier integral operator on R n of order −(n − 1)/2. In [4] it was shown (among other things) that T maps the Hardy space H 1 to L 1. In this note we show that T is also of weak-type (1, 1). The main ideas are a decomposition of T into non-degenerate and degenerate components, and a factorization of the non-degenerate portion.

Let T be a Fourier integral operator on K" of order —(n — l)/2. Seeger, Sogge, and Stein showed (among other things) that T maps the Hardy space H to L. In this note we show that T is also of weak-type (1, 1). The main ideas are a decomposition of T into non-degenerate and degenerate components, and a factorization of the non-degenerate portion. 2000 Mathematics subject classification: primary ...

A system of PDOs(=Partial Differential Operators) has two non-commutativities, (i) one from [∂q, q] = 1 (Heisenberg relation), (ii) the other from [A, B] 6= 0 for A, B being matrices in general. Non-commutativity from Heisenberg relation is nicely controlled by using Fourier transformations (i.e. the theory of ΨDOs=pseudo-differential operators). Here, we give a new method of treating non-commu...