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We design distributed spectrum sensing and access strategies for opportunistic spectrum access (OSA) under an energy constraint on secondary users. Both the continuous and the bursty traffic models are considered for different applications of the secondary network. In each slot, a secondary user sequentially decides whether to sense, where in the spectrum to sense, and whether to access. By casting this sequential decision-making problem in the framework of partially observable Markov decision processes, we obtain stationary optimal spectrum sensing and access policies that maximize the throughput of the secondary user during its battery lifetime. We also establish threshold structures of the optimal policies and study the fundamental tradeoffs involved in the energy-constrained OSA design. Numerical results are provided to investigate the impact of the secondary user's residual energy on the optimal spectrum sensing and access decisions.