= Solution
Write $D=d/dx$. For rapidly decaying smooth functions, <integration by parts> gives $D^*=-D$, while multiplication by the real functions $p$ and $q$ is self-adjoint. Therefore
$$
(iD^3)^*=iD^3,
\qquad
\bigl[-i(qD+Dq)\bigr]^*=-i(qD+Dq),
\qquad
p^*=p,
$$
and hence
$$
\boxed{(L\psi,\phi)=(\psi,L\phi).}
$$
Likewise $D^2$ and multiplication by $q$ are self-adjoint, so their purely imaginary multiples in $A=3iD^2-4iq$ are anti-self-adjoint:
$$
\boxed{(A\psi,\phi)=-(\psi,A\phi).}
$$
If $L\psi=\lambda\psi$ with $\psi\ne0$, self-adjointness gives
$$
\overline\lambda(\psi,\psi)
=(L\psi,\psi)
=(\psi,L\psi)
=\lambda(\psi,\psi).
$$
Consequently \b[every eigenvalue of $L$ is real].
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