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I have an integral equation which is not exactly an eigenvalue type equation, but similar:

$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$

Here $\lambda$ can be thought of as an eigenvalue, so it is expected that only for a discrete set of $\lambda$ we have a solution. $K(x,y,\lambda)$ is some fairly simple expression involving ratios of polynomials and also $\lambda$ appears in a rational fashion.

What would be the typical numerical methods to find the lowest eigenvalue $\lambda$ and eigenfunction $f(x)$?

Actually, we can formulate the question generally: how does one find the eigenfunction corresponding to zero eigenvalue of a linear integral operator that depends on some $\lambda$? The task involves finding the particular $\lambda$ for which the operator has a zero eigenvalue to begin with.

$${\cal O}(\lambda) v = 0$$

Generally ${\cal O}(\lambda)$ doesn't have zero eigenvalues, but for a discrete set of $\lambda$ it does. How can I find it numerically?

I have an integral equation which is not exactly an eigenvalue type equation, but similar:

$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$

Here $\lambda$ can be thought of as an eigenvalue, so it is expected that only for a discrete set of $\lambda$ we have a solution. $K(x,y,\lambda)$ is some fairly simple expression involving ratios of polynomials and also $\lambda$ appears in a rational fashion.

What would be the typical numerical methods to find the lowest eigenvalue $\lambda$ and eigenfunction $f(x)$?

Actually, we can formulate the question generally: how does one find the eigenfunction corresponding to zero eigenvalue of a linear integral operator that depends on some $\lambda$? The task involves finding the particular $\lambda$ for which the operator has a zero eigenvalue to begin with.

$${\cal O}(\lambda) v = 0$$

Generally ${\cal O}(\lambda)$ doesn't have zero eigenvalues, but for a discrete set of $\lambda$ it does. How can I find it numerically? Probably an iterative procedure would be most cost effective. Is this true?

I have an integral equation which is not exactly an eigenvalue type equation, but similar:

$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$

Here $\lambda$ can be thought of as an eigenvalue, so it expected that only for a discrete set of $\lambda$ we have a solution. $K(x,y,\lambda)$ is some fairly simple expression involving ratios of polynomials and also $\lambda$ appears in a rational fashion.

What would be the typical numerical methods to find the lowest eigenvalue $\lambda$ and eigenfunction $f(x)$?

Actually, we can formulate the question generally: how does one find the eigenfunction corresponding to zero eigenvalue of a linear integral operator that depends on some $\lambda$? The task involves finding the particular $\lambda$ for which the operator has a zero eigenvalue to begin with.

$${\cal O}(\lambda) v = 0$$

Generally ${\cal O}(\lambda)$ doesn't have zero eigenvalues, but for a discrete set of $\lambda$ it does. How can I find it numerically?

I have an integral equation which is not exactly an eigenvalue type equation, but similar:

$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$

Here $\lambda$ can be thought of as an eigenvalue, so it is expected that only for a discrete set of $\lambda$ we have a solution. $K(x,y,\lambda)$ is some fairly simple expression involving ratios of polynomials and also $\lambda$ appears in a rational fashion.

What would be the typical numerical methods to find the lowest eigenvalue $\lambda$ and eigenfunction $f(x)$?

Actually, we can formulate the question generally: how does one find the eigenfunction corresponding to zero eigenvalue of a linear integral operator that depends on some $\lambda$? The task involves finding the particular $\lambda$ for which the operator has a zero eigenvalue to begin with.

$${\cal O}(\lambda) v = 0$$

Generally ${\cal O}(\lambda)$ doesn't have zero eigenvalues, but for a discrete set of $\lambda$ it does. How can I find it numerically?

I have an integral equation which is not exactly an eigenvalue type equation, but similar:

$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$

Here $\lambda$ can be thought of as an eigenvalue, so it expected that only for a discrete set of $\lambda$ we have a solution. $K(x,y,\lambda)$ is some fairly simple expression involving ratios of polynomials and also $\lambda$ appears in a rational fashion.

What would be the typical numerical methods to find the lowest eigenvalue $\lambda$ and eigenfunction $f(x)$?

I have an integral equation which is not exactly an eigenvalue type equation, but similar:

$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$

Here $\lambda$ can be thought of as an eigenvalue, so it expected that only for a discrete set of $\lambda$ we have a solution. $K(x,y,\lambda)$ is some fairly simple expression involving ratios of polynomials and also $\lambda$ appears in a rational fashion.

What would be the typical numerical methods to find the lowest eigenvalue $\lambda$ and eigenfunction $f(x)$?

Actually, we can formulate the question generally: how does one find the eigenfunction corresponding to zero eigenvalue of a linear integral operator that depends on some $\lambda$? The task involves finding the particular $\lambda$ for which the operator has a zero eigenvalue to begin with.

$${\cal O}(\lambda) v = 0$$

Generally ${\cal O}(\lambda)$ doesn't have zero eigenvalues, but for a discrete set of $\lambda$ it does. How can I find it numerically?