Distributional equation convention for complex Schrödinger solutions

Let $Q\subseteq {\mathbb{R}}^{1+n}$ be open, with $n\ge 1$; its first coordinate is time $t$ and the remaining coordinates are space $x=(x_1,\dots ,x_n)$. A test function on $Q$ is a complex-valued smooth function with compact support in $Q$; write $\mathcal{D}(Q)=C_c^{\mathrm{\infty }}(Q;\mathbb{C})$ and equip it with the test-function topology. A complex distribution on $Q$ is a continuous complex-linear functional on $\mathcal{D}(Q)$; its value on $\eta$ is written $⟨u,\eta ⟩$. The pairing is linear in both entries, with no conjugation of the test function. The space of these functionals is ${\mathcal{D}}^{\mathrm{\prime }}(Q)$.

For a multi-index $\alpha$ in the space-time coordinates, the distributional derivative is defined by $⟨{\mathrm{\partial }}^{\alpha }u,\eta ⟩=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}⟨u,{\mathrm{\partial }}^{\alpha }\eta ⟩$. Here $\mathrm{\mid }\alpha \mathrm{\mid }$ is the sum of the entries of $\alpha$. The spatial Laplacian is ${\mathrm{\Delta }}_{x}u={\sum }_{j=1}^n{\mathrm{\partial }}_{x_j}^{2}u$.

If $u,F\in {\mathcal{D}}^{\mathrm{\prime }}(Q)$, the equation $i{\mathrm{\partial }}_{t}u+{\mathrm{\Delta }}_{x}u=F$ holds in distributions on $Q$ precisely when $⟨i{\mathrm{\partial }}_{t}u+{\mathrm{\Delta }}_{x}u,\eta ⟩=⟨F,\eta ⟩$ for every $\eta \in \mathcal{D}(Q)$. The same convention applies to an equation with additional terms, provided each term has independently been specified as a distribution. In particular, a nonlinear expression is included only when the chosen solution class makes it a distribution.

This interior space-time identity asserts neither existence of an initial value or boundary trace nor any regularity or local integrability of a nonlinear expression. Such properties and conditions require separate hypotheses. The bilinear distribution pairing is distinct from any conjugated Hilbert-space or antidual pairing.

Test function space d of an open set

Test function topology

Distribution

Distributional derivative

The test functions and their topology give the domain of the continuous linear functionals in Distribution. By Distributional derivative, each specified derivative is again a distribution; finite sums and scalar multiples of distributions are distributions. Consequently both sides of the displayed equation are functionals on the same test space, and equality on every test function is a well-defined equality in ${\mathcal{D}}^{\mathrm{\prime }}(Q)$. This construction does not establish that a proposed nonlinear expression defines a distribution.