Finite-lattice polynomial Hamiltonian data

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). A finite-mode regulator in this item is a fixed finite nonempty label set $J$ with $N:=\mathrm{\mid }J\mathrm{\mid }\ge 1$, a fixed enumeration of $J$ by $\{1,\dots ,N\}$, and the real coordinate vector $q=(q_1,\dots ,q_N)\in {\mathbb{R}}^N$. The labels may name sites of a finite lattice or finitely many modes; their geometry enters only through the chosen polynomial below. No limit in $N$ is part of this definition.

Choose real coefficients $a_1,\dots ,a_N>0$, constants $c>0$ and $C_0\ge 0$, and a real polynomial $V\in \mathbb{R}[q_1,\dots ,q_N]$ such that $V(q)\ge c\mathrm{\mid }q{\mathrm{\mid }}^2-C_0(q\in {\mathbb{R}}^N)\mathrm{.}$ Let $H_N:=L^2({\mathbb{R}}^N,dq;\mathbb{C})$, with vectors identified up to almost everywhere equality and inner product $⟨u,v⟩:={\int }_{{\mathbb{R}}^N}u(q)\overline{v(q)}dq,$ linear in the first variable. Under Countable Choice this is a complex Hilbert space by $L^2$ with the integral pairing is a Hilbert space. Let $D_0:=C_c^{\mathrm{\infty }}({\mathbb{R}}^N;\mathbb{C})$, the smooth compactly supported complex-valued functions. The associated polynomial differential expression is the map $h_0:D_0\to H_N,h_{0}u:=-{\sum }_{j=1}^N{a}_j{\mathrm{\partial }}_j^{2}u+Vu\mathrm{.}$ It is well defined because each derivative of $u$ and the product $Vu$ are smooth and compactly supported. Set $W:=V+C_0+1$, so $W(q)\ge 1+c\mathrm{\mid }q{\mathrm{\mid }}^2$. This definition specifies finite mathematical data and a differential expression; it does not choose a self-adjoint operator domain or assert a continuum or infinite-volume limit.