Inverse spectral problems of transmission eigenvalue problem for anisotropic media with spherical symmetry assumptions

Abstract

Abstract We investigate the inverse spectral problem of the interior transmission eigenvalue problem for an anisotropic medium supported in D := { x : r = | x | ≤ 1 } ${D:=\{{\rm x}:r=|{\rm x}|\leq 1\}}$ : { α ⁢ Δ ⁢ u + k 2 ⁢ n ⁢ u = 0 , Δ ⁢ v + k 2 ⁢ v = 0 , x ∈ D , $\left\{\begin{aligned} \displaystyle\alpha\Delta u+k^{2}nu&\displaystyle=0,\\ \displaystyle\Delta v+{k^{2}}v&\displaystyle=0,\quad\mathrm{x}\in D,\end{% aligned}\right.$ with the boundary conditions u = v ${u=v}$ , α ⁢ ν ⋅ ∇ ⁡ u = ν ⋅ ∇ ⁡ v ${\alpha\nu\cdot\nabla u=\nu\cdot\nabla v}$ for x ∈ ∂ ⁡ D ${{\rm x}\in\partial D}$ , where α and n are physical parameters. In the spherical symmetry case, we consider the case α ≠ 1 ${\alpha\neq 1}$ , whereas most previous work deals with α = 1 ${\alpha=1}$ only. In this paper we prove that all transmission eigenvalues (including multiplicity) uniquely determine n and α under the condition a := ∫ 0 1 n ⁢ ( r ) / α ⁢ 𝑑 r ≤ 1 ${a:=\int_{0}^{1}{\sqrt{n(r)/\alpha}\,dr}\leq 1}$ , and provide construction algorithms. In particular, when a = 1 ${a=1}$ one needs an additional condition for unique recovery and reconstruction.