A perturbation problem for transmission eigenvalues
In this paper, the authors study the following scattering problem. Consider the Helmholtz equation for the incident wave \(\nu\) of monochromatic radiation with frequency \(\omega\) \[ \triangle\nu+k^{2}\nu=0\text{ \ in \ }\mathbb{R}^{d}, \] with \(d=2,3\), in a bounded region \(D\) with the refractive index \(n\) being a bounded real-valued function such that \(\sup\left( n-1\right) =\bar{D}\), and \(k\) is the wave number proportional to \(\omega\). The total field \(u\) is decomposed as \(u=u^{s}+\nu\), where the scattered field \(u^{s}\in H_{loc} ^{2}\left( \mathbb{R}^{d}\right) \) satisfies \[ \triangle u^{s}+k^{2}nu^{s}=-k^{2}\left( n-1\right) \nu,\text{ \ in \ }\mathbb{R}^{d}, \] with the outgoing Sommerfeld radiation condition \[ \lim_{r\rightarrow\infty}r^{\frac{d-1}{2}}\left( \frac{\partial u^{s} }{\partial r}-iku^{s}\right) =0, \] uniformly with respect to \(\hat{x}=x/\left\vert x\right\vert \), \(r=\left\vert x\right\vert \). \(k\) is a non-scattering wave number if \(u^{s}\) is zero outside \(D\). Non-scattering wave numbers are a subset of real transmission eigenvalues. From the previous results in the literature, if the media \(D\) is Lipschitz and \(n\) a bounded function, an infinite discrete set of real transmission eigenvalues is shown to exist under the assumption that either \(n-1\geq\alpha>0\), or else, \(1-n\geq\alpha>0\) a.e. in \(D\). Moreover, when \(D\) is a ball of radius \(a\) centered at the origin and \(n(r)\) is a radial function, the existence of real transmission eigenvalues is known for \(n\in C^{2}[0,a]\) under the additional assumption \(\frac{1}{a}\int_{0}^{a} \sqrt{n\left( \rho\right) }d\rho\neq1\). The aim of the present paper is to extend the above results and to provide examples of existence of real transmission eigenvalues for classes of refractive index that do not satisfy the above conditions. The authors use a perturbation method based on the implicit function theorem, meaning that the ''irregular'' refractive index is a perturbation of a refractive index for which a real transmission eigenvalues is known to exist. Several examples of spherical perturbations of spherically symmetric media are presented and partial results for general media are obtained.
scattering theory for inhomogeneous media
scientific article; zbMATH DE number 7463878
scientific article