Finding a matrix representation for operator A

[Msilva]

I need to find a matrix representation for operator $A=x\frac{d}{dx}$ using Legendre polinomials as base.
I would use $$a_{mn}=\int^{-1}_{-1}P_m(x)\,x\frac{d}{dx}\,P_n(x)\,dx$$, but I have the problem that Legendre polinomials aren't orthonormal $$\langle P_{i}|P_{l}\rangle=\delta_{il}\frac{2}{2i+1}$$.

I don't know if I must mutiply the integral of the $a_{mn}$ terms by the norm, because I have terms in n and m order.

 

[S.G. Janssens]

I have some remarks, to kick things off:

• To let other help you better, it may be a good idea to specify the Hilbert space $H$ (including a definition of the inner product) on which $A$ acts and the domain $D(A)$ of $A$.
• If, for example, $H = L^2(-1,1)$ with the standard inner product, then the (normalized) Legendre polynomials that I know form an orthonormal basis of $H$.
• It has always remained somewhat of a mystery to me why physicists (I presume you are one?) insist on representing unbounded operators as matrices.

 

[Mark44]

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I need to find a matrix representation for operator $A=x\frac{d}{dx}$ using Legendre polinomials as base.
I would use $$a_{mn}=\int^{-1}_{-1}P_m(x)\,x\frac{d}{dx}\,P_n(x)\,dx$$,

Did you mean $a_{mn}=\int^{+1}_{-1}P_m(x)\,x\frac{d}{dx}\,P_n(x)\,dx$?

but I have the problem that Legendre polinomials aren't orthonormal $$\langle P_{i}|P_{l}\rangle=\delta_{il}\frac{2}{2i+1}$$.

I don't know if I must mutiply the integral of the $a_{mn}$ terms by the norm, because I have terms in n and m order.