Fredholm theory

 ( Mathematical Physics ) 

In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and on Hilbert space. It therefore forms a branch of operator theory and functional analysis. The theory is named in honour of Erik Ivar Fredholm.

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Fredholm equation of the first kind Much of Fredholm theory concerns itself with the following integral equation for f when g and K are given:

$g(x)=\backslash int\_a^b\; K(x,y)\; f(y)\backslash ,dy.$

This equation arises naturally in many problems in physics and mathematics, as the inverse of a differential equation. That is, one is asked to solve the differential equation

$Lg(x)=f(x)$

where the function is given and is unknown. Here, stands for a linear differential operator.

For example, one might take to be an elliptic operator, such as

$L=\backslash frac\{d^2\}\{dx^2\}\backslash ,$

in which case the equation to be solved becomes the Poisson equation.

A general method of solving such equations is by means of Green's functions, namely, rather than a direct attack, one first finds the function $K=K(x,y)$ such that for a given pair ,

$LK(x,y)\; =\; \backslash delta(x-y),$

where is the Dirac delta function.

The desired solution to the above differential equation is then written as an integral in the form of a Fredholm integral equation,

$g(x)=\backslash int\; K(x,y)\; f(y)\backslash ,dy.$

The function is variously known as a Green's function, or the kernel of an integral. It is sometimes called the nucleus of the integral, whence the term nuclear operator arises.

In the general theory, and may be points on any manifold; the real number line or -dimensional Euclidean space in the simplest cases. The general theory also often requires that the functions belong to some given function space: often, the space of square-integrable functions is studied, and appear often.

The actual function space used is often determined by the solutions of the eigenvalue problem of the differential operator; that is, by the solutions to

$L\backslash psi\_n(x)=\backslash omega\_n\; \backslash psi\_n(x)$

where the are the eigenvalues, and the are the eigenvectors. The set of eigenvectors span a Banach space, and, when there is a natural inner product, then the eigenvectors span a Hilbert space, at which point the Riesz representation theorem is applied. Examples of such spaces are the orthogonal polynomials that occur as the solutions to a class of second-order ordinary differential equations.

Given a Hilbert space as above, the kernel may be written in the form

$K(x,y)=\backslash sum\_n\; \backslash frac\{\backslash psi\_n(x)\; \backslash psi\_n(y)\}\; \{\backslash omega\_n\}.$

In this form, the object is often called the Fredholm operator or the Fredholm kernel. That this is the same kernel as before follows from the complete space of the basis of the Hilbert space, namely, that one has

$\backslash delta(x-y)=\backslash sum\_n\; \backslash psi\_n(x)\; \backslash psi\_n(y).$

Since the are generally increasing, the resulting eigenvalues of the operator are thus seen to be decreasing towards zero.

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Inhomogeneous equations The inhomogeneous Fredholm integral equation

$f(x)=-\; \backslash omega\; \backslash varphi(x)\; +\; \backslash int\; K(x,y)\; \backslash varphi(y)\backslash ,dy$

may be written formally as

$f\; =\; (K-\backslash omega)\; \backslash varphi$

which has the formal solution

$\backslash varphi=\backslash frac\{1\}\{K-\backslash omega\}\; f.$

A solution of this form is referred to as the resolvent formalism, where the resolvent is defined as the operator

$R(\backslash omega)=\; \backslash frac\{1\}\{K-\backslash omega\; I\}.$

Given the collection of eigenvectors and eigenvalues of K, the resolvent may be given a concrete form as

$R(\backslash omega;\; x,y)\; =\; \backslash sum\_n\; \backslash frac\{\backslash psi\_n(y)\backslash psi\_n(x)\}\{\backslash omega\_n\; -\; \backslash omega\}$

with the solution being

$\backslash varphi(x)=\backslash int\; R(\backslash omega;\; x,y)\; f(y)\backslash ,dy.$

A necessary and sufficient condition for such a solution to exist is one of Fredholm's theorems. The resolvent is commonly expanded in powers of $\backslash lambda=1/\backslash omega$, in which case it is known as the Liouville-Neumann series. In this case, the integral equation is written as

$g(x)=\; \backslash varphi(x)\; -\; \backslash lambda\; \backslash int\; K(x,y)\; \backslash varphi(y)\backslash ,dy$

and the resolvent is written in the alternate form as

$R(\backslash lambda)=\; \backslash frac\{1\}\{I-\backslash lambda\; K\}.$

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Fredholm determinant The Fredholm determinant is commonly defined as

$\backslash det(I-\backslash lambda\; K)\; =\; \backslash exp\; \backslash left[$

-\sum_n \frac{\lambda^n}{n} \operatorname{Tr}\, K^n \right]

where

$\backslash operatorname\{Tr\}\backslash ,\; K\; =\; \backslash int\; K(x,x)\backslash ,dx$

and

$\backslash operatorname\{Tr\}\backslash ,\; K^2\; =\; \backslash iint\; K(x,y)\; K(y,x)\; \backslash ,dx\backslash ,dy$

and so on. The corresponding zeta function is

$\backslash zeta(s)\; =\; \backslash frac\{1\}\{\backslash det(I-s\; K)\}.$

The zeta function can be thought of as the determinant of the resolvent.

The zeta function plays an important role in studying dynamical systems. Note that this is the same general type of zeta function as the Riemann zeta function; however, in this case, the corresponding kernel is not known. The existence of such a kernel is known as the Hilbert–Pólya conjecture.

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Main results The classical results of the theory are Fredholm's theorems, one of which is the Fredholm alternative.

One of the important results from the general theory is that the kernel is a compact operator when the space of functions are equicontinuous.

A related celebrated result is the Atiyah–Singer index theorem, pertaining to index (dim ker – dim coker) of elliptic operators on .

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History Fredholm's 1903 paper in Acta Mathematica is considered to be one of the major landmarks in the establishment of operator theory. David Hilbert developed the abstraction of Hilbert space in association with research on integral equations prompted by Fredholm's (amongst other things).

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See also

• Green's functions
• Spectral theory
• Fredholm alternative

• D. E. Edmunds (1987). 9780198535423, Oxford University Press. ISBN 9780198535423
• Jon Mathews (1970). 9780805370027, W. A. Benjamin. ISBN 9780805370027

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