FredholmKriterium

The Fredholm criterion is a mathematical theorem that provides conditions for the existence and uniqueness of solutions to Fredholm integral equations of the second kind. These equations are of the form:

$ \phi(x) = f(x) + \lambda \int_a^b K(x, y) \phi(y) dy $

where $ \phi(x) $ is the unknown function, $ f(x) $ is a known function, $ K(x, y) $ is the kernel, $ \lambda $ is a scalar parameter, and the integral is taken over a finite interval $ [a, b] $.

The Fredholm criterion states that the homogeneous equation:

$ \phi(x) = \lambda \int_a^b K(x, y) \phi(y) dy $

has a non-trivial solution if and only if $ \lambda $ is an eigenvalue of the integral operator.

Furthermore, the original non-homogeneous equation has a unique solution for any $ f(x) $ if and only if $ \lambda $ is not an eigenvalue. If $ \lambda $ is an eigenvalue, then a solution exists if and only if $ f(x) $ is orthogonal to all solutions of the corresponding homogeneous adjoint equation. If a solution exists in this case, it is not unique.

This criterion is fundamental in the study of integral equations and has applications in various fields of physics and engineering, including electromagnetism, quantum mechanics, and heat transfer. It connects the theory of integral equations to eigenvalue problems in linear algebra and functional analysis.