Eigenvalues of an integral operators

asked 2023-02-23 10:59:14 +0200

Olivier R. gravatar image

updated 2023-02-23 14:13:52 +0200

Max Alekseyev gravatar image

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 $F$?

Max Alekseyev gravatar imageMax Alekseyev ( 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

Olivier R. gravatar imageOlivier R. ( 2023-02-24 09:46:44 +0200 )edit
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