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We present an exact analytic solution to describe the quantum dynamics of the vacuum field coupled with an oscillating boundary (mirror) in a one-dimensional cavity. We show that the finite part of the field energy density grows in the form of two traveling wave packets which become narrower as time increases. We also discover a "sub-Casimir" region where the energy density can be even smaller than the corresponding static vacuum value.
C. K. Law (Mon,) studied this question.