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A boundary-singular two-dimensional partial data inverse problem. (arXiv:1809.06395v1 [math.AP])
来源于:arXiv
We consider uniqueness in an inverse Schr\"odinger problem in a bounded
domain in $\mathbb{R}^2$ given the Dirichlet-to-Neumann map on part of the
boundary. On the remaining boundary we impose a new type of singular boundary
condition with unknown parameter. Owing to recent results on this class of
boundary conditions, we discuss the necessity of an extra point condition to
well-define the data for the inverse problem. Our results are two-fold. At a
single frequency the inverse problem displays non-uniqueness, since an unknown
boundary condition can spoil `seeing' the Schr\"odinger potential via the
Dirichlet-to-Neumann map. On the other hand, taking as input data the
Dirichlet-to-Neumann map at every frequency $\lambda\in\mathbb{R}$ for which it
is well-defined yields full uniqueness of the potential and all the boundary
conditions. We adapt recent methods in related two-dimensional inverse problems
and develop new techniques to cope with the singularity in the boundary
condition. 查看全文>>