Invertibility of parabolic pseudodifferential operators with rapidly increasing symbols

We consider pseudodifferential operators with rapidly increasing double symbols analytic with respect to the variable dual to the time on the lower complex half-plane. We construct invertibility theory for these operators in weighted Sobolev spaces with weights related to growths of symbols and give applications to heat equations with potentials of power, exponential, and superexponential growths. Dedicated to the memory of Professor Leonid Romanovich Volevich