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I am trying to diagonalise a (2n+1)x(2n+1) matrix which has diagonal terms A_ll = (-n+l)^2 and other non vanishing terms are A_l(l+1) = A_(l+1)l = constant.

Is there any way I can solve it for general n without having to use any numerical methods.

I remember once a professor diagonalised such a matrix for a fixed value of n using some tricks, but I can't remember how he did that. Can anyone help me out here?

Any help is appreciated. Cheers!