The declared variational trial reading: assigned energy observable, form domain, trial value, bottom and gap

(a) Framework data. Adopt the state and composition postulate P1, the observable and probability postulate P2 and the autonomous-dynamics postulate P4 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) for a declared system, in the state reading of The state space of the framework: preparations, pure rays and density operators, the sharp observable reading of Assigned sharp observables and the outcome law of the framework and the evolution reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution; units are those of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Assigned energy observable. As an additional model assumption of the pair, fix a self-adjoint operator $H:D(H)\subseteq \mathfrak{H}\to \mathfrak{H}$ on the declared state space $\mathfrak{H}$ and declare the pair $(H,\mathrm{J})$ to be the system's sharp energy observable in the sense of Assigned sharp observables and the outcome law of the framework clause (a); this identification of the operator with the energy of the system is an assumption of the model, not a consequence of P1, P2 or P4. Assume $H$ is bounded below: there is a real $c$ with $⟨Hu,u⟩\ge c\mathrm{\parallel }u{\mathrm{\parallel }}^2$ for every $u\in D(H)$. The spectrum $\sigma (H)$ is the set of the cited spectral theorem, and the model bottom is $E_0:=\inf \sigma (H)\in [c,+\mathrm{\infty }),$ read in joules. An assignment of a concrete material system to the pair $(H,\mathfrak{H})$ is a tested identification in the sense of the preparation and apparatus discipline P7 and is not made here.

(c) Form domain and trial value. Let $Q(H):=D((H-cI{)}^{1\mathrm{/}2})$ and $q_H$ be the form domain and the closed semibounded form of Spectral form domain and core of a semibounded operator for the lower bound $c$ of clause (b), with $q_H[u]={\int }_{\mathbb{R}}\lambda d{E}_u(\lambda )(u\in Q(H)),$ $E$ the projection-valued measure of $H$ and $E_u(B)=⟨E(B)u,u⟩$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure). The definition uses the facts, proved by that lemma, that $q_H$ does not depend on the choice of the lower bound $c$, that $D(H)$ is dense in $Q(H)$, and that $q_H[u]=⟨Hu,u⟩$ for $u\in D(H)$. A trial vector of the model is a vector $u\in Q(H)$ with $\mathrm{\parallel }u\mathrm{\parallel }=1$; its trial value is $q_H[u]$, read in joules. For a trial vector that lies in $D(H)$, the trial value is the mean energy of the pure preparation $[u]$ under the observable $(H,\mathrm{J})$ in the moment reading of Mean, second moment and variance of an assigned observable in a preparation; for a general trial vector in $Q(H)\setminus D(H)$ the trial value is the form value and is not asserted to be the mean of any outcome law.

(d) Bottom gap and discrete ladder. If the bottom is an isolated spectral value, declare a gap to be a real $g>0$ with $\sigma (H)\cap (E_0,E_0+g)=\mathrm{\varnothing },$ and put $P:=E(\{E_0\})$ for the bottom spectral projection. The discrete spectrum below the essential spectrum is the part of ${\sigma }_{\mathrm{d}}(H)$ of Discrete and essential spectrum of a self-adjoint operator below $\mathrm{\Lambda }:=\inf {\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(H)$, listed with multiplicity in nondecreasing order as $E_1\le E_2\le \cdots$, with the convention $E_n:=\mathrm{\Lambda }$ once the list is exhausted, $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. The definition asserts neither that $E_0$ is an eigenvalue, nor that any eigenvector exists. The ladder convention does give $E_0=E_1$: the nonempty closed spectrum of the bounded-below operator contains its bottom; if the bottom is discrete, it is the first listed eigenvalue, and otherwise it lies in the essential spectrum, so $\mathrm{\Lambda }=E_0$ and the empty below-threshold list is padded by $E_1=\mathrm{\Lambda }$. This equality does not imply that the bottom is attained by an eigenvector.

(e) Scope. Clauses (a)-(d) are a reading of the framework's state, observable and dynamics postulates together with the declared operator $H$; they assert nothing about whether a preparation attains the trial value, about the behaviour of any preparation under time evolution beyond P4, about equilibration, environmental coupling, measurement or apparatus, and about any measured energy. All spectral statements are statements about the declared operator.