A lower bound for the small-time cost of boundary control of the one-dimensional Schrödinger equation

We consider the one-dimensional Schrödinger equation on an interval of length $L$, driven by an $L^2$ Dirichlet control at one endpoint, with initial data measured in $H^{-1}$. We prove that its null-control norm cost satisfies $C_S(T,L)\ge c_L T^{7/2}\exp(L^2/(\pi T))$ for all sufficiently small $T$. Thus $\liminf_{T\downarrow0}T\log C_S(T,L)\ge L^2/\pi$, excluding the proposed coefficient $L^2/4$. The proof uses one sine initial state: its exact control moments prescribe the zeros of an entire function, and Jensen's formula turns those zeros into the lower bound.