Consider the wave equation $$u_{tt}-\Delta u+qu = 0 \qquad (x, t) \in \Omega \times (0, T)$$ $$u=u_t=0 \qquad x \in \Omega, \ t=0$$ $$\frac{\partial u}{\partial \nu} = f, \qquad (x, t) \in \partial \Omega \times (0, T)$$ where $\Omega$ is a open bounded set in $\mathbb{R}^n$ with a smooth boubdary and $n \geq 2$, $q=q(x)$ is a potential function and $T >$ diameter$(\Omega)$.
The Neumann to Dirchlet map associated with above problem is $$\Delta_q : f \in L^2(\partial \Omega \times (0, T))\to u\big|_{ \partial \Omega \times (0, T)} \in H^1(\partial \Omega \times (0, T))$$ $\Lambda_q$ is a bounded linear operator.
Now if our map is $F$ is such that $$F(q) = \Delta_q$$
Then my question is how to check the continuity of map $F$.
Please give any hint or reference if possible.