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I first constructed a symmetric matrix as the Laplacian operator, and its eigenvectors are a series of harmonics functions as expected. I programmed it and convinced myself. The matrix looks like: $$ \left(\begin{array}{ccc} 1& -1 & 0 & 0 & 0 \\ -1& 2 & -1 & 0 & 0 \\ 0& -1 & 2 & -1 & 0 \\ 0& 0 & -1 & 2 & -1 \\ 0& 0 & 0 & -1 & 1\end{array}\right) $$ multiplying this matrix with $\{x_0, x_1, x_2, x_3, x_4\}$ leads to $\{x_0-x_1, -x_0+2x_1-x_2,-x_1+2x_2-x_3,..\}$; The term $-x_{i-1}+2x_i-x_{i+1}$ is equivalent to the second order derivative (in the Laplacian) on a discrete 1-dimensional domain: $(x_i-x_{i-1})-(x_{i+1}-x_i)$.

Everything works fine until I try to construct the matrix for Hermite polynomials. Wiki says the operator is $$ L[u]=u''-xu'=-\lambda u.\tag1 $$ I think the matrix for the first derivative should be $$ \left( \begin{array}{ccc} 1& -1 & 0 & 0 & 0 \\ 0& 1 & -1 & 0 & 0 \\ 0& 0 & 1 & -1 & 0 \\ 0& 0 & 0 & 1 & -1 \\ 0& 0 & 0 & 0 & 0\end{array}\right) $$ however, combining this new matrix with the Laplacian matrix (according to $(1)$) does not produce the expected eigenvectors.

Another source derives polynomial from the operator $$ H=-\frac12\frac{d^2}{dx^2}+\frac12x^2, $$ but what are the matrix entries for $x^2$?

Any help? Thanks!

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    $\begingroup$ Isn't this a pure mathematics question? Please make a connection to a compute science topic! (Also, please come up with a more descriptive title.) $\endgroup$ – Raphael Sep 21 '15 at 7:45
  • $\begingroup$ Alternatively, if this should have been on Math.SE, you can click "flag" to flag it for moderator attention and ask the moderators to migrate it to Math.SE. $\endgroup$ – D.W. Sep 22 '15 at 7:31
  • $\begingroup$ Ahh, I see: This was also posted on Math.SE. Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted. $\endgroup$ – D.W. Sep 22 '15 at 7:32