Sentence examples for completeness of wave from inspiring English sources

As an application to the scattering problems, the decay of local energy and the completeness of wave operators are obtained.

We prove the asymptotic completeness of wave operators associated with the scattering of a quantum particle in a field of classical particles in the dispersive case when the free hamiltonian has the formH0=p(D) withpelliptic of degreem⩾1 and satisfying some convexity hypotheses.

Smoothness also implies existence and completeness of wave operators.

Step 4 This suffices to get existence and completeness of wave operators.

The first is the basic one with four important results: the equivalence of many conditions giving the definition, the connection to spectral analysis, the implications for existence and completeness of wave operators and, finally, a perturbation result.

While Agmon did not discuss scattering in his announcement, Lavine [417] noted that his estimates and Lavine's theory of local smoothness implied existence and completeness of wave operators (and later, both Agmon and Hörmander presented other approaches to get completeness).

In this work, we establish precise microlocalized propagation estimates in three-body problems and give a proof for the asymptotic completeness of waves operators in three-body long range scattering for a class of long range potentials of the form Va xa) = V(1)a xa) + V(2)a xa) with V(1)a ≥ 0 decaying like O(|xa| −ϵ′) for some ϵ′ > 1/2 and V(2)a decaying like O(|xa| −γ) for some γ > 2(1 − ϵ′)/ϵ′.

These conditions are shown to be optimal for existence and completeness of a wave operator.

In the framework of time-dependent geometric scattering theory, we study the existence and completeness of the wave operators for perturbations of the Riemannian metric for the Laplacian on a complete manifold of dimension n.

From an analytical point of view, our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper.

One of the main results are conditions on the rate of decay, depending on geometric properties of the underlying manifold, that guarantee the existence and completeness of the wave operators.