I have a problem: Given the Hamiltonian $ H_n= -\frac{d|}{d^2}+V_n $ where $ V_n= a n e^{-n|x|}; a\in\Bbb R, n=1,2,....$
1)Domain in which the operators are self adjoint;
2)study and list the properties of the spectrum;
3)same as last point for $n\to \infty$
I think I know the answer to the first point: the second derivative is self adjoint in $H^2={f\in L^2(\Bbb R)}: \int x^2\lvert f(x)\rvert^2< \infty$ and $V_n $ is symmetric with $D(-\frac{d|}{d^2})\subset D(V_n)$ so, thanks to Kato-Rellich theorem I know $H$ is self adjoint in $H^2$.
But I'm really unsure about the other questions.
Thank you in advance.