A deep analytic analysis of the stability for the one-dimensional linearized compressible Navier-Stokes system (LNSS) is presented. A very detailed study of the spectrum of the unbounded linear differential operator associated with LNSS has been completed. A Fourier analysis method and an explicit calculation using the eigenvalues of the linear operator make it possible to obtain exponential stability results with a decay rate determined solely according to the parameters of LNSS. Consequently, some exponential stability results are proved in different Hilbert spaces, (in -norm and -norm). Finally, numerical experiments are presented to verify the theoretical results.
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