  In this study, we have obtained the exact solutions of the Schr\"{o}dinger
equation for a multi-layered quantum antidot (MLQAD) within the effective mass
approximation and dielectric continuum model for the spherical symmetry. The
MLQAD is nano-structured semiconductor system that consists of a spherical core
(e.g. $Ga_{1-x}Al_{x}As $) and a coated spherical shell (e.g. $Ga_{1-y}Al_{y}As
$) as the whole anti-dot is embedded inside a bulk material (e.g. $GaAs $). The
dependence of the electron energy spectrum and its radial probability density
on nano-system radius are studied. The numeric calculations and analysis of
oscillator strength of intersubband quantum transition from the ground state
into two first allowed excited states at the varying radius, for both the
finite and infinite confining potential (CP) as well as constant shell
thickness, are performed. It is shown that, in particular, the binding energy
and the oscillator strength of the hydrogenic impurity of a MLQAD behave
differently from that of a single-layered quantum antidot (SLQAD). For a MLQAD
with finite core and shell CPs, the state energies and the oscillator strengths
of the impurity are found to be dependent on the shell thickness. At the large
core radius and very small shell thickness, our results are closer to
respective values for a SLQAD that previously reported.
