I am stuck with the following question and I am hoping that someone could help me with it:
Let $D$ be the line $y=0$ in $\mathbb{R}^2$. I define the Schwartz space $\mathcal{S}(\mathbb{R}^2\backslash D)$ of $\mathbb{R}^2\backslash D$ to be the subspace of functions $\varphi\in \mathcal{S}(\mathbb{R}^2)$ (usual Schwartz space) such that $\varphi$ and all its derivatives vanish identically on $D$. This space is easily seen to be dense in $L^2(\mathbb{R}^2)$. Denote by $\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$ the usual Laplacian. Then, I would like to know if we can find a finite number of positive real numbers $\lambda_1,\ldots,\lambda_n$ such that
$$\displaystyle \overline{\left(\Delta-\lambda_1\right)\mathcal{S}(\mathbb{R}^2\backslash D)}+\ldots+\overline{\left(\Delta-\lambda_n\right)\mathcal{S}(\mathbb{R}^2\backslash D)}=L^2(\mathbb{R}^2)$$
where $\overline{V}$ denotes the closure of a subspace $V$ in $L^2(\mathbb{R}^2)$. Also, if it is true, what happens if I replace $\mathcal{S}(\mathbb{R}^2\backslash D)$ by say $\mathcal{S}\left(\mathbb{R}^2\backslash (D\cup D')\right)$ where $D'$ is the line $x=0$.