Excellent example! This is indeed what I am worried about. Thanks a lot!

Thank you so much! So we truncate the infinite matrix and find the eigenvalues, then we take limits. If the limits exist, then we regard the limit as the eigenvalue of infinite matrices. Do you think it is a legitimate treatment of eigenvalues of infinite matrices? Please do not advise me to read the general theory of linear operator in Hilbert space, seriously I know those stuff. But I just don't know how should we deal with infinite matrices. Do you think infinite sparse matrices are easier to deal with? Thank you so much!

Thanks a lot! Benjamin, I am really like the references! It is very helpful!

I am talking about the infinite matrix in Hilbert space.

I have the similar idea, but i just don't know how to verify it. Thank you so much for your answer!

Thanks for your comments. Yes, you are right. Probably it is not a universal case!