Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

Spectral theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic calculations present in many mathematical algorithms. In our work we use the CAS Mathematica to implement for the first time on a computer analytical algorithms developed by us and others within the Operator Theory. The main goal of this paper is to show how the symbolic computation capabilities of Mathematica allow us to explore the spectra of several classes of singular integral operators. For the one-dimensional case, nontrivial rational examples, computed with the automated process called [ASpecPaired-Scalar], are presented. For the matrix case, nontrivial essentially bounded and rational examples, computed with the analytical algorithms [AFact], [SInt], and [ASpecPaired-Matrix], are presented. In both cases, it is possible to check, for each considered paired singular integral operator, if a complex number (chosen arbitrarily) belongs to its spectrum.