we define the operator P with domain $C_0^\infty(R^{2n})$ by $P=y.\partial _x -\partial _x V(x).\partial_y-\Delta_y+\frac{|y|^2}{4}-\frac{n}{2}$
I want to prove that $P=X_0 +\sum _{i=1}{^n}X_j^*X_j$ where
$X_0=y.\partial_x-\partial_x V(x).\partial_y$
and $X_j=\partial _{y_j}+ \frac{y_j}{2}$ for $j=1,..,n$ this is what i wrote :
$(\sum _{i=1}{^n}X_j^*X_j)u=X_j=\sum _{i=1}{^n} ( -\partial _{y_j}+ \frac{y_j}{2})(\partial _{y_j}+ \frac{y_j}{2})=\sum _{i=1}{^n} -\partial _{y_j}^2-\frac{1}{2}+\frac{y_j}{2} \partial _{y_j}+\frac{{y_j}^2}{4}=-\Delta _y -\frac{n}{2}+\frac{y}{2}\partial _{y}+\frac{{|y|}^2}{4}$ But we haven't the terme $\frac{y}{2}\partial _{y}$ in the first formula?can someone tell me where is my misstake?Thanks.