Eigenintegrals

Eigenintegrals are a concept in functional analysis and operator theory, which generalizes the idea of eigenvalues and eigenvectors to integral equations. An eigenintegral is a function that, when integrated against a given kernel, results in a scalar multiple of the same function. This concept is analogous to how an eigenvector of a matrix is a vector that, when multiplied by the matrix, results in a scalar multiple of the same vector.

The study of eigenintegrals involves solving integral equations of the form:

f(x) = λ ∫ K(x, y) f(y) dy

where f(x) is the eigenintegral, λ is the eigenvalue, and K(x, y) is the kernel of the integral equation. The kernel K(x, y) is a function that defines the integral equation, and it plays a crucial role in determining the existence and properties of eigenintegrals.

Eigenintegrals have applications in various fields, including physics, engineering, and economics. In physics, for example, eigenintegrals are used to study quantum systems and wave functions. In engineering, they are used in the analysis of systems governed by integral equations, such as those describing heat transfer or fluid flow. In economics, eigenintegrals are used in input-output analysis to study the interdependencies between different sectors of an economy.

The theory of eigenintegrals is closely related to the theory of compact operators and spectral theory. Compact operators are operators that map a Banach space into a space with a compact closure, and they have a rich spectral theory that includes the study of eigenvalues and eigenvectors. The theory of eigenintegrals extends this spectral theory to integral equations, providing a powerful tool for analyzing a wide range of mathematical and physical problems.