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Exploring the spectra of singular integral operators with rational coefficients

José C. Pereira
Ana C. Conceição
Journal / Anthology

Proceedings of the 1st International Conference on Algebraic and Symbolic Computation – SYMCOMP 2013, ECCOMAS, Portugal, Lisbon, September 9-10, 2013
Year: 2013
Page range: 175-194

Spectral Theory has many applications in the main scientific research areas (Structural Mechanics, Aeronautics, Quantum Mechanics, Ecology, Probability Theory, Electrical Engineering, among others) and the importance of their study is globally acknowledge. We present some results on the spectra of some classes of singular integral operators, with rational coefficients, defined on the unit circle. It is shown how the symbolic computation capabilities of the computer algebra system Mathematica can be used to check, for each considered class of singular integral operators, if a complex number (chosen arbitrarily) belongs to its spectrum. The implementation of the spectral algorithm with Mathematica makes the results of lengthy and complex calculations available in a simple way. Some nontrivial examples, computed with the [ASpec-Tab] algorithm are presented.


Singular Integral Operators, Spectral Algorithm, Rational Function Factorization, Factorization Index, Symbolic Computation, Wolfram Mathematica