Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 341 / 4

For a sufficiently regular solution with zero Dirichlet boundary conditions, differentiate its squared L2 norm. Since $u_t=-i(\Delta u-Vu)$,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. HenceThis 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 $V$.

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 $D_V$. ThusThe generator $-i{H}_h$ is a skew-Hermitian matrix. ConsequentlyThe 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 $\Vert U{\Vert }_h^2=h^2{\sum }_{m,n}|U_{m,n}{|}^2$, this givesApplying 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.

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.