Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds

In this paper, we adapt the well-known \emph{local} uniqueness results of Borg-Marchenko type in the inverse problems for one dimensional Schr{\"o}dinger equation to prove \emph{local} uniqueness results in the setting of inverse \emph{metric} problems. More specifically, we consider a class of spherically symmetric manifolds having two asymptotically hyperbolic ends and study the scattering properties of massless Dirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationary scattering is encoded by a countable family of one-dimensional Dirac equations. This allows us to define the corresponding transmission coefficients $T(\lambda,n)$ and reflection coefficients $L(\lambda,n)$ and $R(\lambda,n)$ of a Dirac wave having a fixed energy $\lambda$ and angular momentum $n$. For instance, the reflection coefficients $L(\lambda,n)$ correspond to the scattering experiment in which a wave is sent from the \emph{left} end in the remote past and measured in the same left end in the future. The main result of this paper is an inverse uniqueness result local in nature. Namely, we prove that for a fixed $\lambda \not=0$, the knowledge of the reflection coefficients $L(\lambda,n)$ (resp. $R(\lambda,n)$) - up to a precise error term of the form $O(e^{-2nB})$ with B\textgreater{}0 - determines the manifold in a neighbourhood of the left (resp. right) end, the size of this neighbourhood depending on the magnitude $B$ of the error term. The crucial ingredients in the proof of this result are the Complex Angular Momentum method as well as some useful uniqueness results for Laplace transforms.Comment: 24 page

Understanding earwig phenology in top fruit orchards

Earwigs, Forficula auricularia, are key generalist predators to a variety of orchard pests. However, numbers of earwigs have declined in both organic and IPM orchards in recent years. Both Integrated and Organic fruit growers have tried to re-establish earwig populations, thus far with little success. To understand earwig population dynamics and to find measures to increase natural orchard populations, we conducted a detailed phenological survey of earwigs in orchards. Earwigs were sampled while sheltering during daytime in artificial refuges. They move into the trees from the third nymph stage onwards. In most orchards, a small second brood is produced in summer, and this has a positive impact on population size in fall. We see only minor differences in phenology between apple and pear orchards, mainly caused by differences in alternative hiding places. Earwigs show an inexplicable reduction in numbers at the timing of moulting into adults. When earwig phenology is correlated with pest phenology in apple and pear, its use for pest control of major pests is clear