The one-dimensional free Schr"odinger operator as the weak second-derivative operator

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Let $m,\mathrm{\hslash }>0$, put $c:=2{\pi }^2{\mathrm{\hslash }}^2\mathrm{/}m$, and let $H_0$ be the free one-dimensional operator of Real multiplication operators and free plus bounded-potential Schrödinger operators clause (b) with $d=1$, this $c$ and $b=0$: $H_0:={\mathcal{F}}_2^{\ast }{M}_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}{\mathcal{F}}_2,D(H_0):=\{f\in L^2(\mathbb{R};\mathbb{C}):\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f\in L^2(\mathbb{R};\mathbb{C})\},$ where ${\mathcal{F}}_2$ is the unitary Plancherel transform of that theorem and $M_{c\mathrm{\mid }\xi {\mathrm{\mid }}^2}$ is multiplication by the function $\xi \mapsto c\mathrm{\mid }\xi {\mathrm{\mid }}^2$. Let $D^2$ denote the second weak derivative, defined componentwise on the locally integrable classes of $\mathbb{R}$ (Weak derivative of a locally integrable function), and let $W^{2,2}(\mathbb{R};\mathbb{C})$ be the Sobolev space (Integer-order Sobolev spaces and their norms). Then:

(a) $D(H_0)=W^{2,2}(\mathbb{R};\mathbb{C})$ as sets and $H_{0}f=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f\text{for every }f\in D(H_0);$ equivalently, ${\mathcal{F}}_2(H_{0}f)(\xi )=c\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f(\xi )$ and ${\mathcal{F}}_2(-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f)(\xi )=c\mathrm{\mid }\xi {\mathrm{\mid }}^2{\mathcal{F}}_{2}f(\xi )$ for almost every $\xi$, and the two $L^2$ classes agree.

(b) For every real bounded measurable $V:\mathbb{R}\to \mathbb{R}$, the operator $H_0+M_V$ of clause (c) of the cited theorem, with domain $D(H_0)$, coincides with the operator $f\mapsto -\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f+Vf$ on the same domain; in particular, for $E\in \mathbb{R}$ and $\psi \in W^{2,2}(\mathbb{R};\mathbb{C})$ the eigenvalue equation $(H_0+M_V)\psi =E\psi$ is equivalent to the weak equation $-\frac{{\mathrm{\hslash }}^2}{2m}{D}^2\psi +V\psi =E\psi$ elementwise in $L^2(\mathbb{R};\mathbb{C})$.

The statement identifies operators on the same domain; it asserts neither smoothness of the classes in $W^{2,2}$ nor any physical interpretation.

• Why the factor $c$. With the Plancherel convention $\widehat{f}(\xi )=\int f(x)e^{-2\pi ix\xi }dx$ the weak Laplacian has multiplier $-4{\pi }^2\mathrm{\mid }\xi {\mathrm{\mid }}^2$, so $-\frac{{\mathrm{\hslash }}^2}{2m}{D}^2$ has multiplier $\frac{2{\pi }^2{\mathrm{\hslash }}^2}{m}\mathrm{\mid }\xi {\mathrm{\mid }}^2$: the constant $c$ of the cited theorem is fixed by this convention and not a convention-free physical statement.
• What is not asserted. The lemma compares two mathematical operators. It asserts no smoothness of $W^{2,2}$ classes, no boundary values, no self-adjointness statement beyond that of the cited theorem, and no identification with any measured Hamiltonian.