The domain of this thesis is included in the general theory of discrete one dimensional random operators. In Chapter 1, we provide interesting elements of the dynamical system which give rise to operators having the same spectrum. Moreover, we give an exposition of the spectral properties of a Schrödinger operator and we describe special cases of potentials. In particular case of periodic potential, the integrated density of states is explicitly provided. In Chapter 2, we introduce a new type of potential, associated with the 2-odometer and called odometric potential. This potential is limit periodic and of Gordon type. An approximation of the Lebesgue measure of the spectrum is obtained. In Chapter 3, we study spectral properties of new operators, called sparse operators, defined by Hp = Sp + S-p + V (where S is the shift operator on l2(Z), p is a non-negative integer dans V is a potential). In some particular cases, we prove that the nature of the spectrum does not change with p. Applications include some classes of periodic, random and substitutional potentials.
