Spektralgeometrie

Spektralgeometrie is a field of mathematics that studies the relationship between the geometric properties of a space and the spectral properties of differential operators defined on that space. The spectrum of an operator, such as the Laplace-Beltrami operator, refers to the set of eigenvalues and corresponding eigenvectors. These eigenvalues can be thought of as vibrating frequencies of the space.

The fundamental idea in spektralgeometrie is that the eigenvalues of these operators encode information about the geometry of the underlying manifold. For example, certain geometric quantities like the volume or the curvature of a Riemannian manifold can, in principle, be recovered from its spectrum. This is often referred to as the "inverse spectral problem," which asks whether a given spectrum uniquely determines a geometric object.

A significant result in this area is the Weyl law, which provides an asymptotic formula for the distribution of eigenvalues of the Laplace-Beltrami operator on a compact Riemannian manifold. This law shows that the density of eigenvalues is related to the volume of the manifold.

Spektralgeometrie has connections to various areas of mathematics, including differential geometry, spectral theory, harmonic analysis, and mathematical physics. It has found applications in areas such as quantum mechanics, signal processing, and computer graphics. The study involves analyzing how changes in the geometry of a space affect the solutions to certain differential equations and the behavior of related operators.