arXiv subjects

Naotaka Shoji

Publications and source records attributed to Naotaka Shoji.

3 recordsLinked to original sources

arXiv2020

Non-scattering energies for acoustic-type equations on manifolds with a single flat end

In this paper, we consider the scattering theory for acoustic-type equations on non-compact manifolds with a single flat end. Our main purpose is to show an existence result of non-scattering energies. Precisely, we show a Weyl-type lower bound for the number of non-scattering energies. Usually a scattered wave occurs for every incident wave by the inhomogeneity of the media. However, there may exist suitable wavenumbers and patterns of incident waves such that the corresponding scattered wave vanishes. We call (the square of) this wavenumber a non-scattering energy in this paper. The problem of non-scattering energies can be reduced to a well-known interior transmission eigenvalues problem.

math-ph↗

arXiv2018

Interior transmission eigenvalue problems on compact manifolds with boundary conductivity parameters

In this paper, we consider an interior transmission eigenvalue (ITE) problem on some compact $C^{\infty }$-Riemannian manifolds with a common smooth boundary. In particular, these manifolds may have different topologies, but we impose some conditions of Riemannian metrics, indices of refraction and boundary conductivity parameters on the boundary. Then we prove the discreteness of the set of ITEs, the existence of infinitely many ITEs, and its Weyl type lower bound. For our settings, we can adopt the argument by Lakshtanov and Vainberg, considering the Dirichlet-to-Neumann map. As an application, we derive the existence of non-scattering energies for time-harmonic acoustic equations. For the sake of simplicity, we consider the scattering theory on the Euclidean space. However, the argument is applicable for certain kinds of non-compact manifolds with ends on which we can define the scattering matrix.

math.SP↗

arXiv2017

On $T$-coercive interior transmission eigenvalue problems on compact manifolds with smooth boundary

In this paper, we consider an interior transmission eigenvalue problem on two compact Riemannian manifolds with common smooth boundary. We suppose that a couple of these manifolds is equipped with locally anisotropic type Riemannian metric tensors, i.e., these two tensors are not equivalent in a neighborhood of common boundary. Here we note that we do not assume that these manifolds are diffeomorphic. In addition, we impose some conditions of the refractive indices in a neighborhood of common boundary. Then we prove that the set of ITEs form infinite discrete set and the existence of ITE-free region. In order to prove our results, we employ so-called the $T$-coercivity method.

math.SP↗