I know a few solutions of the equation $\Delta \phi + \phi =0$ on $\mathbb{R}^2$. They can be described as the Fourier transform of any finite measure $\mu$ on $S^1$. In particular take $\mu$ a finite, positive measure on $S^1$ and consider $G$ defined on $\mathbb{R}^2$ by $G(x,y) = \int_{0}^{2\pi} e^{i\langle(x,y),(\cos\theta,\sin \theta)\rangle} d\mu(\theta)$. It is easy to check that $\Delta G+G=0$ on $\mathbb{R}^2$ and also $G$ is bounded on $\mathbb{R}^2$. I have the following two questions:

1. Does this equation have any unbounded solutions? Is it possible to describe all solutions of this equation?

2. Suppose $H$ is a bounded solution of this equation, then can we conclude that it necessarily has to be the Fourier transform of a finite measure on $S^1$.

Any help or reference will be very helpful.

Thanks!

I know a few solutions of the equation $\Delta \phi + \phi =0$ on $\mathbb{R}^2$. They can be described as the Fourier transform of any finite measure $\mu$ on $S^1$. In particular take $\mu$ a finite, positive measure on $S^1$ and consider $G$ defined on $\mathbb{R}^2$ by $$G(x,y) = \int_{0}^{2\pi} e^{i\langle(x,y),(\cos\theta,\sin \theta)\rangle} \mathrm d \mu(\theta)$$ It is easy to check that $$\Delta G+G=0$$ on $\mathbb{R}^2$ and also $G$ is bounded on $\mathbb{R}^2$. I have the following two questions:

1. Does this equation have any unbounded solutions? Is it possible to describe all solutions of this equation?

2. Suppose $H$ is a bounded solution of this equation, then can we conclude that it necessarily has to be the Fourier transform of a finite measure on $S^1$.

Any help or reference will be very helpful.

Thanks!