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Finding a matrix representation for operator A

  1. Nov 10, 2015 #1
    I need to find a matrix representation for operator [itex]A=x\frac{d}{dx}[/itex] using Legendre polinomials as base.
    I would use [tex]a_{mn}=\int^{-1}_{-1}P_m(x)\,x\frac{d}{dx}\,P_n(x)\,dx[/tex], but I have the problem that Legendre polinomials aren't orthonormal [tex]\langle P_{i}|P_{l}\rangle=\delta_{il}\frac{2}{2i+1}[/tex].

    I don't know if I must mutiply the integral of the [itex]a_{mn}[/itex] terms by the norm, because I have terms in n and m order.
     
  2. jcsd
  3. Nov 10, 2015 #2

    Krylov

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    Science Advisor
    Education Advisor

    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.
     
  4. Nov 10, 2015 #3

    Mark44

    Staff: Mentor

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