These last years there has been growing interest in studying the discontinuous piecewise differential systems, manely due to their wide range of applications in distinct natural phenomena. One of the main ingredients for understanding the dynamics of this class of differential systems are the limit cycles. In this paper we study the maximum number of limit cycles of planar discontinuous piecewise linear Hamiltonian systems separated by a curve \(\Sigma\) , formed by either by the boundary of a sector with angle \(\alpha \in (0, \pi )\) with vertex at the origin, or by a singular irreducible cubic curve. As it is known and here we show again, the shape of the discontinuity curve separating the distinct pieces of the discontinuous piecewise differential systems formed by linear Hamiltonian systems plays a main role on the number of limit cycles that such differential systems can exhibit. Thus our objective is to solve the extension of the 16th Hilbert problem to the classes of piecewise differential systems here considered, i.e. we provide an upper bound for the maximum number of limit cycles of planar discontinuous piecewise linear Hamiltonian systems separated by a curve \(\Sigma\) . Moreover, we show that such upper bounds are reached.
