Anders Melin

Parametrix constructions for right invariant differential operators on nilpotent groups

Algebras of right invariant pseudo-differential operators are constructed on any graded nilpotent group G . We obtain parametrices in such algebras for right invariant differential operators P such that P and its adjoint satisfy the hypoellipticity condition of Rockland .

Local Smoothing for the Backscattering Transform

An analysis of the backscattering data for the Schrödinger operator in odd dimensions n ≥ 3 motivates the introduction of the backscattering transform . This is an entire analytic mapping and we write where B N v is the Nth order term in the power series expansion at v = 0. In this paper we study estimates for B N v in H (s) spaces, and prove that Bv is entire analytic in v ∈ H (s) ∩ ℰ′ when s ≥ (n − 3)/2.

On the general inversion problem

We consider the problem of direct and inverse scattering for the Schrodinger operator Hv = −Δ + v(x) in odd space dimensions with a short range potential. It will be shown that the wave operators are built up from a family of operators A⊙, ⊙ ɛ S n −1, which satisfy the equation HνAϕ = A⊙Ho. The corresponding operator kernels are supported in the set where (y - x,⊙) ≥ O, and they can be described in detail. By introducing polar coordinates for y - x one finds also that these kernels have several properties in common with their one-dimensional analogues. The potential can be easily computed from a special trace of A ⊙ * A⊙, and this operator in turn is given from a factorization of the scattering matrix into upper and lower triangular parts with respect to the direction ⊙. Finally we give some remarks on the so called miracle, which was introduced by R. G. Newton.

Multilinear singular integral operators in backscattering

An analysis of the backscattering data for the Schrodinger operator Hν = −Δ + ν in Rd, where d ⩾ 3 is odd and ν ∈ C0∞(Rd; R), motivates the introduction of the backscattering transform B : C0∞(Rd; C) → C∞(Rd; C). This is an entire analytic mapping that can be extended to much larger spaces. The N:th term, BNν, in the Taylor expansion of Bν at ν = 0 is a multilinear singular integral operator. The complexity of BN increases with N, and for fixed N the singularity of its Schwartz kernel increases strongly with dimension. In spite of this it can be proved that BNν gains smoothness when N increases and no regularity assumptions are imposed on ν. Considerations in inverse scattering motivate the study of the continuity properties of BN in the Sobolev spaces Wp,k(Rn), or rather in subspaces defined by imposing further decay conditions at infinity. The scaling properties of ν ⟼ BNν and the nonlinearity of this operator put restrictions on p, k for BN to be continuous on Wp,k. We present here some such continuity...

Intertwining Operators in Inverse Scattering

In these notes we are going to present some technique which is a multidimensional analogue of some methods which are nowadays standard in scattering theory on the real line for the Schrodinger operator. These methods are based on the construction of operators intertwining the Schrodinger operator with the ‘free operator’ obtained when the potential term is removed. We refer to the monograph [5] by V. A. Marchenko and to the paper [6] for a detailed presentation of this technique.