Spectral analysis: theory and numerical results

This paper explains how spectral theory characterises an operator, acting on a Banach or Hilbert space, and so helps to solve an equation of that operator, or characterise its solution. Sobolev spaces are discussed, and then Spectral theory is applied to a Laplace operator with Dirichlet boundary conditions, and the eigenvalues characterised. An adapted version of the Rayleigh–Ritz Approximation technique is then used to estimate the eigenvalues.