  We discuss reachable states for the Hermite heat equation on a segment with
boundary $L^2$-controls. The Hermite heat equation corresponds to the heat
equation to which a quadratic potential is added. We will discuss two
situations: when one endpoint of the segment is the origin and when the segment
is symmetric with respect to the origin. One of the main results is that
reachable states extend to functions in a Bergman space on a square one
diagonal of which is the segment under consideration, and that functions
holomorphic in a neighborhood of this square are reachable.
