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qtm912

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- 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.

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.