## 1 Introduction  
During the last fifteen years, hybrid nanostructures which are composed of semiconductor quantum dots (SQDs) and metal nanoparticles have attracted scientific interest [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28] since they exhibit unique optical effects. These novel properties owe their presence to the formation of hybrid excitons and have been extensively investigated in systems where a metal nanosphere is coupled to an SQD [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. The origin of this type of exciton lies in the long-range Coulomb interaction that couples the excitons with the plasmons. The use of nanoparticles with a more complex structure and adjustable characteristics may further reinforce the optical effects that have already been explored in such hybrid systems; it may also give rise to even more interesting phenomena. For instance, the coupling of a metal nanoshell-dielectric core complex with an SQD provides the advantage of tuning the optical properties of the nanosystem more efficiently. This is achieved by properly adjusting the size and the material of the core, as shown in a series of studies where the population inversion [29], the energy absorption rate [29], and the two-photon resonance [30] are examined. It is also known that composite electrochemical coatings can be used in the field of nanostructured metal systems' formation [31; 32]. The coupling of such composite nanoparticles with quantum emitters are ideal candidates for applications in sensing and light harvesting. The linear absorption coefficient has already been investigated thoroughly in the case of an SQD-metal nanosphere hybrid system [6; 9; 10; 28].  
This study investigates the novel properties which are associated with the absorption spectra of the coupled nanoparticles when introducing a dielectric core at the center of the metal nanosphere. First, we obtain the density matrix equations and apply a first-order series expansion of the components of the density matrix, with respect to the incident field. Then, we solve the derived equations, in the steady state, and calculate the absorption coefficient for the SQD and the metal nanoshell (MNS). We also present analytical formulas for the calculation of the positions, the widths and the magnitudes of the minima and the maxima of the SQD absorption function. When the dielectric constant of the core is approximately equal to unity, the increase of its radius leads to the enhancement of gain without inversion; it also induces the suppression of both the absorption peak and the exciton lifetime. However, above a specific value of the core radius, the hybrid structure starts exhibiting the opposite trends; the same behavior is detected due to the enhancement of the dielectric constant of the core. For any fixed value of the core radius, the increase of the dielectric constant of the core is found to be responsible for the suppression of gain without inversion, the magnification of the absorption peak and the increase of the exciton lifetime. In the case of an MNS with low thickness, a gain dip appears on the absorption spectrum of the MNS, providing that the dielectric constant of the core tends to unity.