Constructed one-coordinate positive-quartic example on full $L^2$

The parameters satisfy $a>0$, $r>0$, and $\lambda >0$, and $V(q)=q^2\mathrm{/}2+q^4\ge q^2\mathrm{/}2$, so the coercivity hypotheses hold with $c=1\mathrm{/}2$ and $C_0=0$. The coordinate variable $q$ ranges over $\mathbb{R}$; vectors in $H_1$ are square-integrable complex functions of $q$, identified up to equality almost everywhere by [F3]. The characteristic functions ${\mathbf{1}}_{[3n,3n+1)}$, $n\in \mathbb{N}$, are nonzero pairwise orthogonal vectors, so this Hilbert space is infinite-dimensional.

By [A1], the Countable Choice hypothesis of [F2] holds. Substituting $a=r=\lambda =1$ into it gives $0\le E_g^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(1)-E_g^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(0)\le 3E_{\ast }\mathrm{/}2$. This is an exact bound for the selected model, not an exact value of either ground energy.

By [A1], the Countable Choice hypothesis of [F1] holds. Applying [F1] to the same selected model gives a global unitary group $U_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\tau )$ for every $\tau \in \mathbb{R}$, norm preservation for all states, and differentiability precisely on the stated Hamiltonian domain. The quartic coefficient is positive, and the Hilbert space is infinite-dimensional by step 1.1; thus this is a one-coordinate polynomial operator model on the full Hilbert space, not a finite matrix replacement. No observation or continuum extrapolation is asserted.