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The currently available unlicensed spectrum is reaching its limit and various demands for applications and data rates in wireless communications requires additional spectrum which imposes limits on spectrum access. These requirements demand for efficient and intelligent use of spectrum. The system model and the problem of the optimum spectrum allocation in cognitive radios are introduced and formulated in this paper. We derived a lower limit of the demanded allocations for all the users in the system and obtain the optimal solution which attains this lower bound. An algorithm which finds these optimum values of the demanded allocations is introduced to find the optimum spectrum allocation in time domain. The convergence of this algorithm is shown using the eigen values of the Gram matrices. Simulations are provided to support the mathematical formalism and the convergence of the algorithm to optimal solution.