Integral equations Fredholm equations and Fredholm theory Theorem

Fredholm series for the resolvent

Builds on Continuous kernel Fredholm determinant Fredholm minors Fredholm equation of the second kind Resolvent kernel Characteristic value of a kernel

Statement

Let $a < b$ and let $K \colon [a, b] \times [a, b] \to \C$ be a continuous kernel. Let $D(\lambda)$ be its Fredholm determinant and $D(x, y; \lambda)$ its first Fredholm minor. Let $\lambda \in \C$ with $D(\lambda) \ne 0$ , and put $R(x, y; \lambda) = \frac{D(x, y; \lambda)}{D(\lambda)}, \qquad x, y \in [a, b].$ Then:

$R(\cdot, \cdot; \lambda)$ is continuous on $[a, b]^2$ and satisfies $R(x, y; \lambda) - K(x, y) = \lambda \int_a^b K(x, t) R(t, y; \lambda) \, dt = \lambda \int_a^b R(x, t; \lambda) K(t, y) \, dt;$
for every continuous $f \colon [a, b] \to \C$ , the Fredholm equation of the second kind $\varphi(x) - \lambda \int_a^b K(x, y) \varphi(y) \, dy = f(x)$ has exactly one continuous solution, namely $\varphi(x) = f(x) + \lambda \int_a^b R(x, y; \lambda) f(y) \, dy;$
$\lambda$ is not a characteristic value of $K$ .

So $R$ is the resolvent kernel of $K$ , and it is a meromorphic function of $\lambda$ whose poles lie among the zeros of $D$ .

Remarks

The proof rests on two identities between entire functions of $\lambda$ . They come from expanding the determinants in the series along the first row or the first column: $D(x, y; \lambda) = K(x, y) D(\lambda) + \lambda \int_a^b K(x, t) D(t, y; \lambda) \, dt,$ and the same with $K$ and $D$ exchanged in the integral. Dividing by $D(\lambda)$ gives part 1, and part 1 shows by direct substitution that the formula in part 2 solves the equation. Uniqueness comes from the same identities applied to a solution of the homogeneous equation, which must then be $0$ .

Example: for $K(x, y) = xy$ on $[0, 1]$ , $D(\lambda) = 1 - \lambda/3$ and $D(x, y; \lambda) = xy$ . So $R(x, y; \lambda) = \frac{3xy}{3 - \lambda}$ for $\lambda \ne 3$ . For $|\lambda| < 3$ this agrees with the sum $\sum_{n \ge 0} \lambda^n xy / 3^n$ of the Liouville–Neumann series.

The Neumann series converges only for small $\abs{\lambda}$ , while Fredholm's formula is valid for every $\lambda$ with $D(\lambda) \ne 0$ .

Going deeper

For a student

The result turns the Fredholm equation of the second kind into an explicit computation whenever $D(\lambda) \ne 0$ . It extends the resolvent kernel beyond the disc where the Neumann series converges, as a meromorphic function of $\lambda$ . At a zero $\lambda_0$ of $D$ , the homogeneous equation has nonzero solutions, and these can be read off from the first Fredholm minor that does not vanish identically at $\lambda_0$ . In operator language, $R$ is the kernel of $\lambda^{-1}((I - \lambda T_K)^{-1} - I)$ , where $T_K$ is the integral operator with kernel $K$ .

Reading path

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Characteristic value of a kernel concept
Continuous kernel concept
Fredholm determinant concept
Fredholm equation of the second kind concept
Fredholm minors concept
Resolvent kernel concept
Fredholm series for the resolvent statement

Tag	`4W0EG94RZ5`
URI	`m2://4W0EG94RZ5`
Kind	statement, theorem
Writing	W0 stub
MSC	45B05
Status	This copy of the site has no build data. Run `m2 status 4W0EG94RZ5`.
About	Resolvent kernel, Fredholm determinant, Fredholm minors
Uses	Continuous kernel, Fredholm determinant, Fredholm minors, Fredholm equation of the second kind, Resolvent kernel, Characteristic value of a kernel