# Eigenvalues of an integral operators

Dear all, I want to find the first eigenvalues (and even the eigenvectors if possible) of a difference operator

$$G(y) = \int_0^1 V( t - y ) F(t) dt$$

The function $V$ is given by some infinite series and is easily computable to any decent degree of accuracy. This is a very classical problem, but I wouldn't want to rediscover the wheel if there is a shop that sells some :) Many thanks in advance, Olivier

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Do you mean $G(F)(y)$? Or what is $F$?

( 2023-02-23 14:15:38 +0200 )edit

Yes, G(F)(y). A basic answer consists in approximating the integral. Wiith Simpson's formula, this gives

def GetEigenvalues (bigK):
ak = [1, 4] + [2 for k in range(3-1, bigK-1)] + [4, 1]
coefs = [0] + [ Wstar(k/bigK) / 3/bigK for k in range(1, bigK + 1)]
myMat = matrix([[ ak[k]*coefs[abs(k-ell) + 1]
for ell in range(1, bigK + 1)]
for k in range(1, bigK + 1)])
myEigen = myMat.eigenvalues()
myOrderedEigen = sorted(myEigen, key = lambda x: -abs(x))
return(myOrderedEigen)


where Wstar(y) is my kernel function (which is even and vanishes at 0). This gives an idea, but the above has no control of approximation. I would like to find something more refined. Best, Olivier

( 2023-02-24 09:46:44 +0200 )edit