The one-dimensional continuity identity and the constant flux of stationary solutions

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $m,\mathrm{\hslash }>0$, let $\mathrm{\Omega }\subseteq \mathbb{R}$ and $I\subseteq \mathbb{R}$ be nonempty open intervals (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let $V:\mathrm{\Omega }\to \mathbb{R}$ be a function. Let $u,v:\mathrm{\Omega }\times I\to \mathbb{R}$ be functions whose partial derivatives ${\mathrm{\partial }}_{t}u,{\mathrm{\partial }}_{t}v,{\mathrm{\partial }}_{x}u,{\mathrm{\partial }}_{x}v,{\mathrm{\partial }}_x^{2}u,{\mathrm{\partial }}_x^{2}v$ exist and are continuous on $\mathrm{\Omega }\times I$ (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point, The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set) and satisfy the pair of equations $\mathrm{\hslash }{\mathrm{\partial }}_{t}u=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\partial }}_x^{2}v+Vv,-\mathrm{\hslash }{\mathrm{\partial }}_{t}v=-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\partial }}_x^{2}u+Vu$ pointwise on $\mathrm{\Omega }\times I$. Write $\psi :=u+iv$ for the corresponding $\mathbb{C}$-valued function with component functions $u$ and $v$ (Real and imaginary parts, complex conjugation, and modulus), so that the two displayed equations are the componentwise reading of $i\mathrm{\hslash }{\mathrm{\partial }}_t\psi =-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\partial }}_x^2\psi +V\psi$. Put $n(x,t):=\mathrm{\mid }\psi (x,t){\mathrm{\mid }}^2=u(x,t{)}^2+v(x,t{)}^2,j(x,t):=\frac{\mathrm{\hslash }}{m}\mathrm{Im}\text{\hspace{0.17em}⁣}(\overline{\psi (x,t)}{\mathrm{\partial }}_x\psi (x,t))=\frac{\mathrm{\hslash }}{m}(u{\mathrm{\partial }}_{x}v-({\mathrm{\partial }}_{x}u)v)(x,t)\mathrm{.}$ Then:

(a) Continuity identity. ${\mathrm{\partial }}_{t}n(x,t)+{\mathrm{\partial }}_{x}j(x,t)=0$ for every $(x,t)\in \mathrm{\Omega }\times I$.

(b) Constant flux of stationary solutions. Separately, let $u_{\mathrm{s}},v_{\mathrm{s}}:\mathrm{\Omega }\to \mathbb{R}$ have continuous second derivatives, and let $E\in \mathbb{R}$ and $V:\mathrm{\Omega }\to \mathbb{R}$ satisfy $-\frac{{\mathrm{\hslash }}^2}{2m}{u}_{\mathrm{s}}^{\mathrm{\prime }\mathrm{\prime }}+V{u}_{\mathrm{s}}=E{u}_{\mathrm{s}}$ and $-\frac{{\mathrm{\hslash }}^2}{2m}{v}_{\mathrm{s}}^{\mathrm{\prime }\mathrm{\prime }}+V{v}_{\mathrm{s}}=E{v}_{\mathrm{s}}$ pointwise on $\mathrm{\Omega }$. Then $j_{\mathrm{s}}(x):=\frac{\mathrm{\hslash }}{m}(u_{\mathrm{s}}{v}_{\mathrm{s}}^{\mathrm{\prime }}-u_{\mathrm{s}}^{\mathrm{\prime }}{v}_{\mathrm{s}})(x)$ is constant on $\mathrm{\Omega }$.

The assertion is an identity for the declared differentiable functions and the declared real pair $V$; it constructs no probability law, no state and no physical interpretation.

• Meaning of the hypotheses. The two real equations are exactly the real and imaginary parts of the complex equation displayed in the statement, and no differentiability of a $\mathbb{C}$-valued function beyond the stated continuous partial derivatives of $u$ and $v$ is used. The function $V$ need not be continuous for the identity (a); it is used only through the given pointwise equations, and in (b) only through the two stationary equations.
• What is not asserted. The lemma produces no normalized state, no probability, no boundary condition and no scattering or transmission coefficient. In particular the constancy in (b) is a statement about a differentiable size on the open interval $\mathrm{\Omega }$ and assumes nothing about the behaviour of $j$ at the endpoints of $\mathrm{\Omega }$ or at infinity.