The need to understand job scheduling on devices with intermittent availability is of significant interest today because of the use of battery-powered devices - including electric vehicles - that rely on recharging intervals or energy harvesting. In some recent work by Islam and Nirjon, effective heuristics were proposed for scheduling recurring tasks with deadlines on such intermittently available devices. The broader computational complexity of job scheduling has not been explored in this setting where there is a relationship between job durations and energy consumption. We provide a richer understanding of this problem space. We consider two recharging approaches, one where the battery has to be fully charged during a recharging interval (sometimes considered better for extending battery lifetime) and another where the battery can be partially charged, and we study different scheduling objectives: minimizing the sum of completion times, minimizing the maximum tardiness, and minimizing the number of tardy jobs. We also consider four different relationships between job duration and energy consumption: (i) energy consumption is equal for all jobs irrespective of job length; (ii) job length is equal for all jobs irrespective of energy consumption; (iii) energy consumption is directly proportional to job length; and (iv) there is an arbitrary relationship between job length and energy consumption. In effect, we consider 24 different scheduling problems, and establish that most problems subject to a complete recharging requirement are NP-Hard but that most problems can be solved in polynomial time when partial recharging is permitted. Interestingly, we have been unable to resolve the computational complexity for the one case of minimizing the sum of completion times subject to partial recharging.