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This is part of a proof from Sidney Coleman's "Aspects of Symmetry," page 340.

We start with the equation

$$(\partial_{t}^2 - W(t))\psi = \lambda \psi$$

where W is a bounded function of time, and the operator acts on the space of functions vanishing at $\pm T/2$. We define the determinant of the operator as the product of its eigenvalues:

$$\det(\partial_{t}^2 - W) = \prod \lambda_n $$.

Now, the ratio

$$\frac{ \left(\det{\partial_{t}^2 - W^{(1)} - \lambda}\right)}{ \left(\det{\partial_{t}^2 - W^{(2)} - \lambda}\right)}$$

is a meromorphic function of $\lambda$ with a simple zero at each $\lambda_n^{(1)}$ and a simple pole at each $\lambda_n^{(2)}$. The proof then claims that by elementary Fredholm theory, the ratio goes to one as $\lambda \to \infty$. What does Coleman mean here by Fredholm theory? Most of what I have found on Fredholm is about finding solutions to certain integral equations, which doesn't seem relevant here.

I am open to either proofs of this statement using the relevant theory, or reference suggestions for where to learn about the appropriate theory. Thanks!

This is part of a proof from Sidney Coleman's "Aspects of Symmetry," page 340.

We start with the equation

$$(\partial_{t}^2 - W(t))\psi = \lambda \psi$$

where W is a bounded function of time, and the operator acts on the space of functions vanishing at $\pm T/2$. We define the determinant of the operator as the product of its eigenvalues:

$$\det(\partial_{t}^2 - W) = \prod \lambda_n $$.

Now, the ratio

$$\frac{ \det\left({\partial_{t}^2 - W^{(1)} - \lambda}\right)}{ \det\left({\partial_{t}^2 - W^{(2)} - \lambda}\right)}$$

is a meromorphic function of $\lambda$ with a simple zero at each $\lambda_n^{(1)}$ and a simple pole at each $\lambda_n^{(2)}$. The proof then claims that by elementary Fredholm theory, the ratio goes to one as $\lambda \to \infty$. What does Coleman mean here by Fredholm theory? Most of what I have found on Fredholm is about finding solutions to certain integral equations, which doesn't seem relevant here.

I am open to either proofs of this statement using the relevant theory, or reference suggestions for where to learn about the appropriate theory. Thanks!

This is part of a proof from Sidney Coleman's "Aspects of Symmetry," page 340.

We start with the equation

$$(\partial_{t}^2 - W(t))\psi = \lambda \psi$$

and define the determinant of the operator as the product of its eigenvalues:

$$\det(\partial_{t}^2 - W) = \prod \lambda_n $$.

Now, the ratio

$$\frac{ \left(\det{\partial_{t}^2 - W^{(1)} - \lambda}\right)}{ \left(\det{\partial_{t}^2 - W^{(2)} - \lambda}\right)}$$

is a meromorphic function of $\lambda$ with a simple zero at each $\lambda_n^{(1)}$ and a simple pole at each $\lambda_n^{(2)}$. The proof then claims that by elementary Fredholm theory, the ratio goes to one as $\lambda \to \infty$. What does Coleman mean here by Fredholm theory? Most of what I have found on Fredholm is about finding solutions to certain integral equations, which doesn't seem relevant here.

I am open to either proofs of this statement using the relevant theory, or reference suggestions for where to learn about the appropriate theory. Thanks!

This is part of a proof from Sidney Coleman's "Aspects of Symmetry," page 340.

We start with the equation

$$(\partial_{t}^2 - W(t))\psi = \lambda \psi$$

where W is a bounded function of time, and the operator acts on the space of functions vanishing at $\pm T/2$. We define the determinant of the operator as the product of its eigenvalues:

$$\det(\partial_{t}^2 - W) = \prod \lambda_n $$.

Now, the ratio

$$\frac{ \left(\det{\partial_{t}^2 - W^{(1)} - \lambda}\right)}{ \left(\det{\partial_{t}^2 - W^{(2)} - \lambda}\right)}$$

is a meromorphic function of $\lambda$ with a simple zero at each $\lambda_n^{(1)}$ and a simple pole at each $\lambda_n^{(2)}$. The proof then claims that by elementary Fredholm theory, the ratio goes to one as $\lambda \to \infty$. What does Coleman mean here by Fredholm theory? Most of what I have found on Fredholm is about finding solutions to certain integral equations, which doesn't seem relevant here.

I am open to either proofs of this statement using the relevant theory, or reference suggestions for where to learn about the appropriate theory. Thanks!

This is part of a proof from Sidney Coleman's "Aspects of Symmetry," page 340.

We start with the equation

$(\partial_{t}^2 - W(t))\psi = \lambda \psi$

and define the determinant of the operator as the product of its eigenvalues:

$\det(\partial_{t}^2 - W) = \prod \lambda_n $.

Now, the ratio

$\frac{ \left(\det{\partial_{t}^2 - W^{(1)} - \lambda}\right)}{ \left(\det{\partial_{t}^2 - W^{(2)} - \lambda}\right)}$

is a meromorphic function of $\lambda$ with a simple zero at each $\lambda_n^{(1)}$ and a simple pole at each $\lambda_n^{(2)}$. The proof then claims that by elementary Fredholm theory, the ratio goes to one as $\lambda \to \infty$. What does Coleman mean here by Fredholm theory? Most of what I have found on Fredholm is about finding solutions to certain integral equations, which doesn't seem relevant here.

I am open to either proofs of this statement using the relevant theory, or reference suggestions for where to learn about the appropriate theory. Thanks!

This is part of a proof from Sidney Coleman's "Aspects of Symmetry," page 340.

We start with the equation

$$(\partial_{t}^2 - W(t))\psi = \lambda \psi$$

and define the determinant of the operator as the product of its eigenvalues:

$$\det(\partial_{t}^2 - W) = \prod \lambda_n $$.

Now, the ratio

$$\frac{ \left(\det{\partial_{t}^2 - W^{(1)} - \lambda}\right)}{ \left(\det{\partial_{t}^2 - W^{(2)} - \lambda}\right)}$$

is a meromorphic function of $\lambda$ with a simple zero at each $\lambda_n^{(1)}$ and a simple pole at each $\lambda_n^{(2)}$. The proof then claims that by elementary Fredholm theory, the ratio goes to one as $\lambda \to \infty$. What does Coleman mean here by Fredholm theory? Most of what I have found on Fredholm is about finding solutions to certain integral equations, which doesn't seem relevant here.

I am open to either proofs of this statement using the relevant theory, or reference suggestions for where to learn about the appropriate theory. Thanks!