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We present a theoretical description of Bogolyubov-type excitations ofexciton-polariton Bose-Einstein condensates (BECs) in semiconductormicrocavities. For a typical two dimensional (2D) BEC we focus on two limitingcases, the weak- and strong-coupling regimes, where a perturbation theory andthe Thomas-Fermi approximation, respectively, are valid. We calculateintegrated scattering intensity spectra for probing the collective excitationsof the condensate in both considered limits. Moreover, in relation to recentexperiments on optical modulation allowing localization of condensates in atrap with well controlled shape and dimensions, we study the quasi-onedimensional (1D) motion of the BEC in microwires and report the correspondingBogolyubov's excitation spectrum. We show that in 1D case the characteristicpolariton-polariton interaction constant is expressed as $g_{1}=3\lambda\mathcal{N}/(2L_{y})$ ($\lambda $ is the 2D polariton-polaritons interactionparameter in the cavity, $\mathcal{N}$ the number of the particles, and $L_{y}$the wirecavity width). We reveal some interesting features for 2D and 1DBogolyubov spectra for both repulsive $(\lambda >0)$ and attractive $(\lambda<0)$ interaction.

Excited states of exciton-polariton condensates in 2D and 1D harmonic traps