Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation $$\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u,$$ but much less on those of the Hermite heat equation $$\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U$$ due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).