Important Issues on Spectral Properties of a Transmission Eigenvalue Problem

Nowadays, inverse scattering is an important field of interest for many mathematicians who deal with partial differential equations theory, and the research in inverse scattering is in continuous progress. There are many problems related to scattering by an inhomogeneous media. Here, we study the transmission eigenvalue problem corresponding to a new scattering problem, where boundary conditions differ from any other interior problem studied previously. more specifically, instead of prescribing the difference Cauchy data on the boundary which is the classical form of the problem, we consider the case when the difference of the trace of the fields is proportional to the normal derivative of the field. Typical concerns related to TEP (transmission eigenvalue problem) are Fredholm property and solvability, the discreteness of the transmission eigenvalues, and their existence. In this article, we provide answers for all these concerns in a given interior transmission problem for an inhomogeneous media. We use the variational method and a very important theorem on the existence of transmission eigenvalues to arrive at the conclusion of the existence of the transmission eigenvalues.

A note on generalized invertibility and invariants of infinite matrices

The classes of band-dominated operators and the subclass of operators in the Wiener algebra $${\mathcal {W}}$$ W are known to be inverse closed. This paper studies and extends partially known results of that type for one-sided and generalized invertibility. Furthermore, for the operators in the Wiener algebra $${\mathcal {W}}$$ W invertibility, the Fredholm property and the Fredholm index are known to be independent of the underlying space $$l^p$$ l p , $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ . Here this is completed by the observation that even the kernel and a suitable direct complement of the range as well as generalized inverses of operators in $${\mathcal {W}}$$ W are invariant w.r.t. p.

EQUIVALENCE OF ELLIPTICITY AND THE FREDHOLM PROPERTY IN THE WEYL-HÖRMANDER CALCULUS

The main result is that the ellipticity and the Fredholm property of a $\Psi $ DO acting on Sobolev spaces in the Weyl-Hörmander calculus are equivalent when the Hörmander metric is geodesically temperate and its associated Planck function vanishes at infinity. The proof is essentially related to the following result that we prove for geodesically temperate Hörmander metrics: If $\lambda \mapsto a_{\lambda }\in S(1,g)$ is a $\mathcal {C}^N$ , $0\leq N\leq \infty $ , map such that each $a_{\lambda }^w$ is invertible on $L^2$ , then the mapping $\lambda \mapsto b_{\lambda }\in S(1,g)$ , where $b_{\lambda }^w$ is the inverse of $a_{\lambda }^w$ , is again of class $\mathcal {C}^N$ . Additionally, assuming also the strong uncertainty principle for the metric, we obtain a Fedosov-Hörmander formula for the index of an elliptic operator. At the very end, we give an example to illustrate our main result.