= Solution
Set $\xi_j=\sqrt{m\omega/\hbar}\,x_j$. Separation into three one-dimensional oscillators gives
$$
\Psi_{n_1n_2n_3}=h_{n_1}(\xi_1)h_{n_2}(\xi_2)h_{n_3}(\xi_3)e^{-m\omega r^2/(2\hbar)},
$$
$$
E_N=\hbar\omega\left(N+\frac32\right),\qquad N=n_1+n_2+n_3.
$$
The ground state is
$$
E_0=\frac32\hbar\omega,\qquad \Psi_0=e^{-m\omega r^2/(2\hbar)}.
$$
The number of triples of nonnegative integers summing to $N$ is
$$
\binom{N+2}{2},
$$
which is the degeneracy of $E_N$.
Finally,
$$
\chi=(x_1+ix_2)^2e^{-m\omega r^2/(2\hbar)}
$$
is a pure level-$N=2$ oscillator state: the constant terms in the two degree-two Hermite contributions cancel. Part (i) gives $L_3\chi=2\hbar\chi$, while $N=2$ gives $H\chi=(7/2)\hbar\omega\chi$.
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