Eigenvalues problem for the Laplacian operator on a free domain

Question

I want to ask a question about the eigenvalues and eigenvectors if they exist for the following question: $$ - \Delta u = \lambda u{\text{ x}} \in \mathbb{R}$$ I have tried to do the Fourier transform of both side of the equation, I got $$({\xi ^2} - \lambda )\hat u = 0{\text{ }}\xi \in \mathbb{R}$$ then, I can not go further than that. Any help would be appreciated.

Answer 1

You mention "Dirichlet" in the title but haven't said anything about boundary conditions... in case, certainly $$-u''(x) = \lambda u(x)$$ can be solved for $u$ on $\mathbb{R}$ for any $\lambda$. This is a standard linear second-order homogeneous ODE. For background on how to solve these, see for instance this chapter of Stewart.

For $\lambda\geq 0$ the solutions are $$u(x) = Ae^{ix\sqrt{\lambda}} + Be^{-ix\sqrt{\lambda}}$$ where $i$ is the imaginary unit and $A,B$ are arbitrary constants. You can apply Euler's formula to extract the real solutions, $$u(x) = C\cos\left(x\sqrt{\lambda}\right) + D\sin\left(x\sqrt{\lambda}\right)$$ for real constants $C,D$.

For $\lambda <0$ instead the real solutions are $$u(x) = Ce^{x\sqrt{-\lambda}} + De^{-x\sqrt{-\lambda}}.$$