# Can I expect reasonable results when expanding in scaled Legendre polynomials?

Imagine I want to compute the eigenvalues of an operator $\hat O$ defined on $L^2(\mathbb{R})$, however using a properly scaled N-dimensional polynomial basis of $L^2([-a,a])$ which fulfills Dirichlet boundary conditions: $P_n(\pm a)=0$ (e.g. scaled Legendre polynomials), i.e. $O_{i,j}=\langle P_i|\hat O|P_j\rangle$.

Can I expect convergence to the exact result for $N, a\to\infty$?

• You're not showing the form of your operator, but I would expect convergence. – nicoguaro Mar 14 '18 at 12:27
• If the answer depends on the form of the operator this would be quite interesting. For now, I consider e.g. the harmonic oscillator, i.e. -Δ+x². – Jodocus Mar 14 '18 at 12:46
• I don't think that it will be different for each operator, but it is good to know. Regarding the convergence I am dubious now, since Legendre polynomial would not be orthogonal in the limit. Nevertheless, I think that the answer will get better for increasing N. – nicoguaro Mar 14 '18 at 14:16