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!



Solutions of $\Delta \phi + \phi =0$ on $\mathbb{R}^2$
Every solution of $\Delta\phi+\phi=0$ on $\mathbf R^2$ can be written as the “Poisson integral” (where for short $(u,v)=w$, $(x,y)=z$)
$$
\phi(z)= \left\langle T, e^{i\langle z,\cdot\rangle}\right\rangle=\int\_{\mathrm S^1}e^{i\langle z,w\rangle}dT(w)
$$
for a unique ***entire functional*** $\,T$ on the circle $\mathrm S^1$; these include Radon measures, Schwartz distributions, hyperfunctions, and more. Details and proofs see Hashizume et al. ([1972](//ams.org/mathscinet-getitem?mr=55:3168), p. 543), Helgason ([1974](//ams.org/mathscinet-getitem?mr=51:3353), p. 348; [1984](//ams.org/mathscinet-getitem?mr=86c:22017), p. 5), or Agmon ([1999](//ams.org/mathscinet-getitem?mr=2000k:35043)). For example, if I am not mistaken,


* $T\_1=$ Dirac measure at $(-i,\sqrt2)\in (\mathrm S^1)^{\mathbf C}$ gives $\phi\_1(z)=e^{x+\sqrt2iy}$ (found by Maple);
* $T\_2=$ derivative of $T\_1$ in the direction $(\sqrt2,i)$ gives $\phi\_2(z)=(\sqrt2x+iy)e^{x+\sqrt2iy}$ (new).


