The Black-Scholes operator is given by
$$L_{BS}u(x) = \frac{1}{2}\sigma^2x^2\frac{\partial^2}{\partial x^2}u(x) + rx\frac{\partial}{\partial x}u(x) - ru(x)$$
I want to prove that this operator has no eigenvalue with positive real part.
Unfortunately, I am not entirely sure what the domain of this operator is but I am inclined to believe that it is something like
$$\mathcal{D} = \{u \in C^2: u \geq 0, u(x) \leq Me^{\alpha x^2} \text{ for some }M, \alpha \}$$
Writing $L_{BS}u = \lambda u$, I do get eigenvalues with positive real part unless I am making an error somewhere. It is also possible that I am not in the right domain. Any help/suggestion appreciated.