= Solution
Arrange the interior values into a vector $U$. Represent the <five-point Dirichlet Laplacian as a Kronecker sum> $L_h=h^{-2}(D\otimes I+I\otimes D)$, where $D$ is the <tridiagonal matrix> with diagonal $-2$ and adjacent entries one. This <Kronecker sum> is real symmetric; the real sampled potential has a real diagonal matrix $D_V$. Thus
$$
iU'=H_hU,\qquad H_h=L_h-D_V=H_h^*.
$$
The generator $-iH_h$ is a <skew-Hermitian matrix>. Consequently
$$
\frac{d}{dt}(U^*U)
=2\operatorname{Re}(-iU^*H_hU)=0,
\qquad U(t)=e^{-itH_h}U(0).
$$
The <matrix exponential> is a <unitary matrix>, either by differentiating its product with its adjoint or by <unitary diagonalization of a normal matrix>. For the two-dimensional <discrete L2 norm> $\|U\|_h^2=h^2\sum_{m,n}|U_{m,n}|^2$, this gives
$$
\boxed{\|U(t)-\widetilde U(t)\|_h
=\|U(0)-\widetilde U(0)\|_h\quad(t\geq0).}
$$
Applying the same equation to a difference proves <stability of a numerical method> with a mesh-independent constant one. This is <norm conservation of a semidiscrete Schrödinger equation>. It concerns continuous time after spatial discretization; an arbitrary subsequent time integrator need not preserve this stability.
\b[The printed coordinates do not discretize the stated square.] For $[-1,1]^2$ with $M$ interior points in each direction, use $h=2/(M+1)$, $x_m=-1+mh$, $y_n=-1+nh$, and sample $V$ there. The printed $h=1/(M+1)$ and unshifted coordinates instead describe a grid on $[0,1]^2$. The matrix proof is valid for either geometry with its corresponding boundary values, so this transcription-independent statement flaw does not alter the stability conclusion.
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