I Particle on a cylinder with harmonic oscillator along z-axis

[Salmone]

I need to know if I have solved the following problem well:

A spin-less particle of mass m is confined to move on the surface of a cylinder of infinite height with a harmonic potential on the z-axis and Hamiltonian $H=\frac{p_z^2}{2m}+\frac{L_z^2}{2mR^2}+\frac{1}{2}m\omega^2z^2$ and I need to calculate its eigenvalues and eigenvectors.
I thought of separating the Hamiltonian as it is written in the text, and find the Hamiltonian of a particle on a ring and a harmonic oscillator along the z-axis with eigenfunctions of the type $|\psi\rangle=|m\rangle|n\rangle$ where $m$ are the eigenvalues of $L_z$ equal to $\frac{1}{\sqrt{2\pi}}e^{im\phi}$ and $n$ are the eigenvalues of the harmonic oscillator. For the eigenvalues I had thought they were $E=\frac{n^2\hbar^2}{2mR^2}+\hbar\omega(n+1/2)$. Is this right?
Also, if I had a perturbation like $\epsilon cos(\phi)$ to calculate on the fundamental level I could say by eye that its effect is zero since in the fundamental level there is no angular momentum?

 

[vanhees71]

You have to write $\mu$ for the mass to distnguish if from the eigenvalue of $\hat{L}_z$, $m \hbar$. Then $E_{mn}=m^2\hbar^2/(2\mu R^2)+\hbar \omega (n+1/2)$, where $m \in \mathbb{Z}$, $n \in \mathbb{N}_0$.

 

[Salmone]

Thank you @vanhees71 for the answer, so the eigenvalues are correct, what about the total Hamiltonian eigenvectors? Are they correct? And the perturbation?