Abstract
Herein, we considered the Schrödinger operator with a potential q on a disk and the map that associates to q the corresponding Dirichlet-to-Neumann (DtN) map. We provide some numerical and analytical results on the range of this map and its stability for the particular class of one-step radial potentials.
Highlights
Let Ω ⊂ R2 be a bounded domain with smooth boundary ∂Ω
We focus on the range of the Dirichlet-to-Neumann map (DtN) in terms of the relevant coefficients (c0, c1 ), i.e., the range of the map Λh in (27): R(Λh )
We considered the relationship between the potential in the Schrödinger equation and the associated DtN map in one of the simplest situations, i.e., for a subset of radial one-step potentials in two-dimension
Summary
Let Ω ⊂ R2 be a bounded domain with smooth boundary ∂Ω. For each q ∈ L∞ (Ω), consider the so called Dirichlet-to-Neumann map (DtN) given by: Λq : H 1/2 (∂Ω). We were interested in the following map: This has an important role in inverse problems, where the aim is to recover the potential q from boundary measurements. We still restrict ourselves to the bounded set F to simplify Even in this simple case, a complete analytic answer to the previous questions (range of the DtN map and sharp stability conditions) is unknown. We obtain a stability constant depending on γ−1 b−3 , which is uniformly bounded for b > b0 and fixed γ (see Theorem 3 below) Note, that this constant blows up as γ → 0. The Dirichlet-to-Neumann map, or voltage-to-current map, is given by: In this case, the relationship between piecewise constant radial conductivities and the eigenvalues of the DtN map is known [8] through a suitable recurrence formula.
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