Embedded Eigenvalues for operators and Fredholm properties of pencils

Altaweel, Nifeen and Elton, Daniel (2021) Embedded Eigenvalues for operators and Fredholm properties of pencils. PhD thesis, Lancaster University.

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Abstract

The perturbation problem for operators is considered one of the differential equations with operator coefficients; a possible example of this problem is embedded eigenvalues, which serves as a prototype of this problem. My research is concerned with two main tasks; first, highlighting the idea of the existence of embedded eigenvalues (trapped modes) of different operators. These include the stability of the embedded eigenvalues within the spectrum for the operator on a cylindrical domain. Common threads will be taken from these problems to subsequently develop a more generalised understanding of the existence of embedded eigenvalues. The second task is to study the Fredholm properties of an operator pencil. In particular, we detect and approximate the spectra of the Fredholm operator pencils via a Green’s kernel (contour integral) by considering exponential solutions of differential equations with operator coefficients. The arguments for this task act on a class of weighted function spaces which can be modelled on Sobolev spaces. One of the main motivation behind this research is to gain a deeper understanding the development of aspects of the theory of ordinary differential equations with operator coefficients by concentrating on some specific examples of trapped modes. The results of our first task showed that, in different cases, for sufficiently small potential functions our operator has an eigenvalue which is contained in the essential spectrum, and hence is an embedded eigenvalue. According to the result of the second task, it was directly established that Fredholm operator pencil and the index could be calculated without the need to consider the adjoint operator. Also, we leveraged certain concepts to go from the semi-Fredholm property to the Fredholm property using some of the results of the current thesis.