  It has been recently shown numerically that the transition from integrability
to chaos in quantum systems and the corresponding spectral fluctuations are
characterized by $\frac{1}{f^{\alpha}}$ noise with $1\leq\alpha\leq 2$. The
system of interacting trapped bosons is inhomogeneous and a complex system. The
presence of external harmonic trap makes it more interesting as in the atomic
trap the bosons occupy partly degenerate single-particle states. Earlier
theoretical and experimental results show that at zero temperature the
low-lying levels are of collective nature and high-lying excitations are of
single particle nature. We observe that for few bosons, $P(s)$ distribution
shows the Shnirelman peak which exhibits a large number of quasi-degenerate
states. For large number of bosons the low-lying levels are strongly affected
by the interatomic interaction and the corresponding level fluctuation shows a
transition to Wigner with increase in particle number. It does not follow GOE
(Gaussian Orthogonal Ensemble) Random Matrix predictions. For high-lying levels
we observe the uncorrelated Poisson distribution. Thus it may be a very
realistic system to prove that $\frac{1}{f^{\alpha}}$ noise is ubiquitous in
nature.
