11/18/2023 0 Comments Spectra definition physics![]() This development leads to the Gelfand representation, which covers the commutative case, and further into non-commutative harmonic analysis. The further theory built on this to address Banach algebras in general. After Hilbert's initial formulation, the later development of abstract Hilbert spaces and the spectral theory of single normal operators on them were well suited to the requirements of physics, exemplified by the work of von Neumann. There have been three main ways to formulate spectral theory, each of which find use in different domains. Hilbert himself was surprised by the unexpected application of this theory, noting that "I developed my theory of infinitely many variables from purely mathematical interests, and even called it 'spectral analysis' without any presentiment that it would later find application to the actual spectrum of physics." The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |