Level spacing distribution I have started learning level spacing distribution and it says that level spacing distribution for classically regular system show poison curve. But is it valid for the integrable system as well, because assuming simple harmonic oscillator, there the energy levels are just $E_n=(n+\frac{1}{2})\hbar \omega$ hence level spacing are all same which is $s = \hbar \omega $? Now in this case level spacing distribution would be $P(s) = \delta (s-\hbar \omega)$ not Poissonian.
 A: The level spacing distribution is essentially a statistical characterization, aimed at studying intrinsically complex or perturbed systems - so it's not surprising it doesn't work so well on the harmonic oscillator.
In the paper The problem of quantum chaos in a kicked harmonic oscillator (1991), Berman et al. are clear about that in their conclusions:

In the zero approximation (no perturbation: x = 0) the system (2.3) is linear, and even in the classical limit it should be investigated out of the framework of KAM theory [20]. There is no quantum consideration of these systems in the region of parameters of quantum chaos at present.

Though it should be mentioned that Gulisashvili and MacCluer calculations lead them to conclude that (e-print):

The annihilation operator of the unforced quantum harmonic oscillator is chaotic.

And there are exceptions to this correspondence even among systems where it should/could work. An example is described in the paper Core-induced chaos in diamagnetic lithium:

We present a theoretical study of the connections between quantum and classical descriptions of lithium in a magnetic field. We find that the localized nature of the ionic core causes a breakdown in the generic connections between energy-level statistics and classical motion: classical chaos is observed in a regime where the energy-level distribution is Poissonian.

A: It is been amazingly described here at page 176:

Generally, for classical integrable nonlinear systems, i.e. $det( \frac{\partial^2H}{\partial I_i\partial I_j}) \ne 0$, with
respect to the action variables $I_i$, with at least two independent degrees of freedom, Berry and Tabor convincingly argued in favor of the following conjecture (Berry-Tabor conjecture): In the limit of large energies (semiclassical limit), the statistical properties of the quantum spectra of classically integrable systems correspond
to the prediction for randomly distributed energy levels.

And of course, the harmonic oscillator does not have "at least two" independent degrees of freedom, and Berry-Tabor conjecture does not apply to it.
