Eigenvalues for a non self adjoint operator

[qtm912]

TL;DR Summary

Trying to obtain eigenvalues for a non self adjoint linear second order differential operator , but unsure about how (if) to use the weight function

Hi all- I am trying to obtain eigenvalues for an equation that has a very simple second order linear differential operator L acting on function y - so it looks like :
L[y(n)] = Lambda (n) * y(n)

Where y(n) can be written as a sum of terms in powers of x up to x^n

but I find L is non self adjoint. As indicated the question involves assuming that the form of y is a polynomial say in x and use "the equating of coefficients of equal powers method" to work out the coefficients of the x powers and the nth eigenvalue for the expansion. That should be manageable but unsure if it is necessary first to use a weight function w (to convert L to self adjoint form) and if so at what stage. Initial thought was that the eigenvalues are as given by this calculation but that the eigenfunction expansion would have to be calculated as above and then dividing it by the weight function - not sure really if any of this is on the right track.

Guidance appreciated.

 

[fresh_42]

Do what you said: write $y(n)$ as polynomial, apply $L$ and compare the terms. No weight functions or self adjoint needed. The equation $Ly=\lambda y$ is the same for any linear function $L$.

 

[qtm912]

Thanks fresh_42 - thinking about it some more it lines up with what I had in mind - for the eigenvalues anyway. But we have not applied any weight function; now the operator by assumption was not self adjoint so does this not mean that the polynomial terms I later derive in the y expansion that is expressed as a sum of polynomial eigenfunctions in x, will no longer be orthogonal. To make them orthogonal the weight function would play a role would it? If not how would one proceed to obtain an expansion of polynomials in terms of an orthogonal basis.

 

[fresh_42]

Any basis can be made orthogonal by the Gram-Schmidt algorithm.

 

[qtm912]

Noted - and after thinking about it further I realized that orthogonalisation in this case was not necessary using this method of deriving the coefficients. Your comments were very helpful in clearing some doubts thanks - the topic has been fully addressed and can be closed..