Solvability of eigenvalue problem with Schwartz data

Question

Fix $a\in\mathbb{R}$ and define the operator $T$ acting on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ by sending $\phi$ to $\Delta\phi-a^2\phi$. Then $T$ is clearly a bounded operator.Question is whether $T$ is surjective or not?

Answer 1

Taking the Fourier transform, up to constants depending on the definition used, $$ \mathcal{F}(T\phi)(\xi)=-(\xi^2+a^2)\mathcal{F}(\phi)(\xi). $$ Let $\psi\in\mathcal{S}$. Then, if $a\ne0$, $$ -\frac{\mathcal{F}(\psi)(\xi)}{\xi^2+a^2}\in\mathcal{S}. $$ Let $$ \phi=\mathcal{F}^{-1}\Bigl(-\frac{\mathcal{F}(\psi)(\xi)}{\xi^2+a^2}\Bigr). $$ Then $T\phi=\psi$.

I leave to you the case $a=0$.