Spectral theory is a general term in mathematics, which refers to theories that extend the concepts of eigenfunction and eigenvalue from square matrices to wider classes of linear operators in very different spaces. Such theories naturally arise in the study of systems of linear equations and their generalizations. Such theories are closely related to analytic functions, since the spectral properties of the operator are associated with the analytic functions of the spectral parameter.
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The term “spectral theory” itself was introduced by David Hilbert in the original formulation of the theory of Hilbert spaces , which was formulated using a quadratic form of an infinite number of variables. Therefore, the original version of the spectral theorem was formulated as an extension of the theorem on the reduction of a quadratic form to the principal axes . Later studies in quantum mechanics allowed us to explain the features of the spectrum of the atom , which was quite unexpected.
There are three basic formulations of the spectral theory, each of which has reason to be considered useful. After the initial formulation of Hilbert, more recent studies of the spectral theory of a normal operator in a Hilbert space were carried out under the needs of physics, in particular, studies carried out by von Neumann [1] . Further development of the theory was also able to include Banach algebras . These studies led to the Gelfand representation, which completely covers the commutative case, and later to noncommutative harmonic analysis.
The difference can be understood by drawing a parallel with Fourier analysis. On the one hand, the Fourier transform on the real axis is the spectral theory of differentiation as a differential operator. However, in practice it turns out that one has to work with the generalization of eigenfunctions (for example, by using the equipment of the Hilbert space). On the other hand, it is sufficient to simply construct a group algebra that satisfies the basic properties of the Fourier transform, and this can be done using the Pontryagin duality .
The spectral properties of operators on Banach spaces can also be investigated, for example, compact operators on a Banach space have spectral properties quite similar to those of matrices.
Physical notes
The oscillations were explained precisely by the methods of spectral theory,
Spectral theory is closely related to the study of localized vibrations of various objects, from atoms and molecules in chemistry to acoustic waveguides. These vibrations have frequencies (natural frequencies of vibrations). The applied question is how to calculate these frequencies. This is a rather difficult task, since each body has not only the basic tone (corresponding to the lowest frequency), but also a multitude of overtones, the sequence of which is very nontrivial.
Mathematical theory at the technical level is not tied to this kind of physical considerations, although there are many examples of mutual influence. For the first time, the term spectrum in this sense was apparently taken by Gilbert in 1897 from Wilhelm Wirtinger’s article on Hill’s differential equation , and then the term was picked up by his students, including Erhard Schmidt and Hermann Weyl .
Only twenty years later, after the Schrödinger formulation of quantum mechanics, was the connection established between the mathematical spectrum of the operator and the spectrum of the atom. Although, as noted by Henri Poincaré , the connection with the mathematical model of oscillations was suspected much earlier, however, it was rejected by rather simple quantitative arguments, for example, the inability to explain Balmer’s frequency series . Thus, the name of the spectral theory was not logically connected with its ability to explain the spectrum of the atom, it was just a coincidence.
See also
- Can you hear the shape of the drum?
Notes
- ↑ John von Neumann. The mathematical foundations of quantum mechanics; Volume 2 in Princeton Landmarks in Mathematics series . - Reprint of translation of original 1932. - Princeton University Press, 1996. - ISBN 0-691-02893-1 .