Magnetic quadratic-form data

Assume Countable Choice and let $d\ge 1$ be an integer. Work in $H=L^2({\mathbb{R}}^d;\mathbb{C})$ with $⟨u,v⟩={\int }_{{\mathbb{R}}^d}u\bar{v}$, linear in the first variable, using the quotient and complex-function conventions of the linked Euclidean $L^2$ definitions. Fix real scalars $m>0$, $\mathrm{\hslash }>0$, and $q$. Let $A=(A_1,\dots ,A_d)\in C^{\mathrm{\infty }}({\mathbb{R}}^d;{\mathbb{R}}^d)$ be real and let $V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d;\mathbb{R})$ satisfy $V\ge -v_0$ almost everywhere for some finite $v_0\ge 0$. Put $\mathcal{D}=C_c^{\mathrm{\infty }}({\mathbb{R}}^d;\mathbb{C})$ and, for $u\in \mathcal{D}$, define ${\mathrm{\Pi }}_j^{A}u=-i\mathrm{\hslash }{\mathrm{\partial }}_{j}u-q{A}_{j}u(1\le j\le d)\mathrm{.}$ For $u,v\in \mathcal{D}$ define the sesquilinear form $q_{A,V}(u,v)=\frac{1}{2m}{\sum }_{j=1}^d⟨{\mathrm{\Pi }}_j^{A}u,{\mathrm{\Pi }}_j^{A}v⟩+{\int }_{{\mathbb{R}}^d}V(x)u(x)\overline{v(x)}dx,$ and write $q_{A,V}[u]=q_{A,V}(u,u)$. Each integral is finite: the test functions and their magnetic derivatives are bounded with compact support, and $V$ is integrable on compact sets. The form is Hermitian, densely defined, and bounded below by $-v_0\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$.

For ${\epsilon }_0>0$ define its shifted form norm by $\mathrm{\parallel }u{\mathrm{\parallel }}_{q,v_0,{\epsilon }_0}^2=q_{A,V}[u]+(v_0+{\epsilon }_0)\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\mathrm{.}$ The form is closable when every sequence $(u_n)\subset \mathcal{D}$ with $\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_2\to 0$ and $\mathrm{\parallel }{u}_n-u_m{\mathrm{\parallel }}_{q,v_0,{\epsilon }_0}\to 0$ as $n,m\to \mathrm{\infty }$ also satisfies $\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_{q,v_0,{\epsilon }_0}\to 0$. This condition is independent of the choice of ${\epsilon }_0>0$. When it holds, the closed form closure is the completion of $\mathcal{D}$ in this norm, identified with its $L^2$ limits; the corresponding closed form is denoted ${\bar{q}}_{A,V}$.

Position and momentum operators on Euclidean L2

Complex Lp classes and Euclidean test-function conventions

The space $L^p(\mu )$ as the quotient by null functions

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)