Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$:
$T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial y}-y\frac{\partial}{\partial x})$
Let $\alpha \not\in\sigma(T)$ and let $(T-\alpha I)^{-1}u(x,y)=f(x,y)$.
Now, let $F$ be the Fourier transform $(\zeta,\eta)$. We have
$\Big[(\zeta^{2} +\eta^{2})-\partial^{2}_{\zeta} -\partial^{2}_{\eta}-2i(\eta \frac{\partial}{\partial \zeta}-\zeta\frac{\partial}{\partial \eta}) -\alpha I)\Big]\widehat{f}(\zeta,\eta)=\widehat{u}(\zeta,\eta)$
So, I need to calculate the resolvent of this operator. Does anybody have any ideas how to complete this.