The general problem of microwave breakdown of a gas subjected to a nonuniform field over a semi‐infinite spatial region is treated. This problem is particularly applicable to high‐power‐handling antennas radiating into free space or into a plasma gas, in which case there is no outer surface for the electrons to diffuse to. Basically, solutions to the electron continuity equation determine an eigenvalue spectrum which governs the breakdown criteria. When the domain is finite, the spectrum must be discrete. This case has been previously treated. For the infinite domain, we demonstrate that for a realistic form of the net ionization rate, a continuous spectrum arises along with a bounded discrete spectrum which, when it exists, dominates the breakdown criteria. Two specific examples are included and the results are applied to breakdown predictions of a rectangular slot. A variational calculation is also treated.