= Solution
For a sufficiently regular solution with zero <Dirichlet boundary conditions>, differentiate its squared <L2 norm>. Since $u_t=-i(\Delta u-Vu)$,
$$
\frac{d}{dt}\|u(t)\|_{L^2}^2
=2\operatorname{Re}\int_\Omega\overline u\,u_t
=2\operatorname{Re}\left[-i\left(\int_\Omega\overline u\,\Delta u
-\int_\Omega V|u|^2\right)\right].
$$
<Integration by parts> and the zero boundary trace give $\int\overline u\Delta u=-\int|\nabla u|^2$. The potential contribution is real because $V$ is real. The quantity in parentheses is consequently real, and its product with $-i$ has zero real part. Hence
$$
\boxed{\|u(t)\|_{L^2(\Omega)}=\|u(0)\|_{L^2(\Omega)}.}
$$
This is the <energy method> for a <skew-adjoint> evolution. For general <L2 space> data, the same result follows from the <strongly continuous unitary group> generated by a self-adjoint realization of $\Delta-V$. A sufficient assumption is a bounded real potential with the Dirichlet Laplacian; arbitrary real potentials require appropriate domain and self-adjointness assumptions. The differentiated proof presumes the regularity needed for the displayed integrals, rather than asserting it for every pointwise real function $V$.
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