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Variational trial bounds, gap-overlap control and the min-max characterisation of the declared model's discrete eigenvalues

Physical scope

The variational and min-max statements of the declared trial model: every trial value is at least the model bottom E0, a trial value within epsilon of an isolated bottom with gap g forces ground-projection content at least 1 - epsilon/g, and the model's discrete eigenvalues below the essential spectrum are exactly the min-max values over trial subspaces of the form domain. All quantities are readings of the declared operator; no measurement, apparatus, sampling model or laboratory identification is asserted.

Statement

Adopt the framework data and the declared variational trial model of The declared variational trial reading: assigned energy observable, form domain, trial value, bottom and gap: a declared system with self-adjoint assigned energy observable $H$ on $H$ , bounded below by a real $c$ , with model bottom $E_{0} = \inf σ (H)$ , form domain $Q (H)$ , form $q_{H}$ and spectral projection-valued measure $E$ ; a trial vector is a unit vector $u \in Q (H)$ and its trial value is $q_{H} [u]$ . Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice) as the choice hypotheses of the cited spectral suppliers. Then:

(a) Trial lower bound. For every trial vector $u \in Q (H)$ , $q_{H} [u] \geq E_{0},$ so every trial value is an upper bound for the model bottom; consequently $E_{0} = \inf {q_{H} [u] : u \in Q (H), ∥ u ∥ = 1}$ .

(b) Gap-overlap control. Suppose in addition that the bottom is isolated with gap $g > 0$ and let $P : = E ({E_{0}})$ (clause (d) of the model). Then for every trial vector $u$ with $q_{H} [u] \leq E_{0} + ε$ , $ε \geq 0$ , $∥ (I - P) u ∥^{2} \leq \frac{ε}{g};$ in particular, if $ε < g$ , then $P u \neq 0$ .

(c) Min-max. Let $Λ : = \inf σ_{e s s} (H)$ and let $E_{1} \leq E_{2} \leq \dots$ be the discrete eigenvalue ladder of the model clause (d), with $E_{n} : = Λ$ after exhaustion and $\inf \emptyset = + \infty$ . Then for every integer $n \geq 1$ , $E_{n} = \inf_{\begin{array}{c} L \subseteq Q (H) linear \\ \dim L = n \end{array}} \sup_{\begin{array}{c} u \in L \\ ∥ u ∥ = 1 \end{array}} q_{H} [u] = \sup_{\begin{array}{c} F \subseteq Q (H) linear \\ \dim F \leq n - 1 \end{array}} \inf_{\begin{array}{c} u \in Q (H) \cap F^{⊥} \\ ∥ u ∥ = 1 \end{array}} q_{H} [u],$ with the infimum over an empty set read as $+ \infty$ ; moreover $E_{1} = E_{0}$ , and when $E_{n} < Λ$ both outer values are attained, respectively by a span of the first $n$ and a span of the first $n - 1$ orthonormal eigenvectors.

(d) Consequence for trial subspaces. For every linear subspace $L \subseteq Q (H)$ of dimension $n \geq 1$ whose supremum is finite, $E_{n} \leq \sup {q_{H} [u] : u \in L, ∥ u ∥ = 1}$ ; for $n = 1$ this is clause (a) with $u$ a unit vector of $L$ .

(e) Reading and scope. All numbers $E_{0}$ , $E_{n}$ , $q_{H} [u]$ are read in joules. For a trial vector in $D (H)$ the trial value is the mean energy of the pure preparation in the moment reading of Mean, second moment and variance of an assigned observable in a preparation; for a trial vector in $Q (H) ∖ D (H)$ it is a form value and no outcome-law mean is asserted. The theorem asserts nothing about whether the infimum in (a) is attained, about a ground eigenvector, about time evolution, equilibration or environmental coupling, and about any measured energy; it is a statement about the declared operator $H$ .

Facts & Assumptions

Given: the declared trial model of The declared variational trial reading: assigned energy observable, form domain, trial value, bottom and gap with its data $H, c, E_{0}, Q (H), q_{H}, E$ ; a gap $g > 0$ and $P = E ({E_{0}})$ in clause (b); $Λ$ and the ladder $(E_{n})$ in clause (c); the choice hypotheses.

[F1]

Spectral data of the model. $H$ is self-adjoint with dense domain and $E$ its regular projection-valued measure; $σ (H)$ is the set of real $λ$ every neighbourhood of which has nonzero spectral projection, the spectrum is contained in $[E_{0}, + \infty)$ because $E_{0} = \inf σ (H)$ , and $D (H) = {u : \int λ^{2} d E_{u} (λ) < \infty}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, Discrete and essential spectrum of a self-adjoint operator).

[F2]

Form and moments. $Q (H)$ and $q_{H} [u] = \int_{R} λ d E_{u} (λ)$ are independent of the lower bound chosen in the model, $D (H)$ is dense in $Q (H)$ for the form norm, $q_{H} [u] \geq c ∥ u ∥^{2}$ , and $q_{H} [u] = ⟨ H u, u ⟩$ for $u \in D (H)$ (Spectral form domain and core of a semibounded operator). $E_{u} (B) = ∥ E (B) u ∥^{2}$ is a measure of total mass $∥ u ∥^{2}$ (Projection valued measure).

[F3]

Supplied trial and gap estimates. Let $H^{'}$ be a self-adjoint operator on a nonzero complex Hilbert space, bounded below with bottom $E_{0}^{'}$ , let $g^{'} > 0$ satisfy $σ (H^{'}) \cap (E_{0}^{'}, E_{0}^{'} + g^{'}) = \emptyset$ , put $P^{'} : = E_{H^{'}} ({E_{0}^{'}})$ , and let $q_{H^{'}}$ be the form of Spectral form domain and core of a semibounded operator. On the spectral support $[E_{0}^{'}, \infty)$ , $∣ t ∣ \leq (t - c) + ∣ c ∣$ and $t - c \leq ∣ t ∣ + ∣ c ∣$ for any lower bound $c$ ; since $E_{u} (R) = ∥ u ∥^{2} < \infty$ , these inequalities show $Q (H^{'}) = D (∣ H^{'} ∣^{1 / 2})$ . Thus for every unit $u \in Q (H^{'})$ , the supplied theorem gives $q_{H^{'}} [u] - E_{0}^{'} \geq g^{'} ∥ (I - P^{'}) u ∥^{2}$ ; and if $q_{H^{'}} [u] \leq E_{0}^{'} + ε$ then $∥ (I - P^{'}) u ∥^{2} \leq ε / g^{'}$ , with $P^{'} u \neq 0$ when $ε < g^{'}$ (Spectral form domain and core of a semibounded operator, Spectral theorem for unbounded self-adjoint operators (PVM form), Variational ground gap overlap clauses (a), (b)).

[F4]

Supplied min-max theorem. Under the Axiom of Choice, for a self-adjoint operator $A$ bounded below on a complex Hilbert space, with form domain $Q (A)$ , form $q_{A}$ , $Λ_{A} = \inf σ_{e s s} (A)$ and the eigenvalue ladder $E_{1}^{A} \leq E_{2}^{A} \leq \dots$ below $Λ_{A}$ (equal to $Λ_{A}$ after exhaustion), the two min-max formulas of clause (c) hold for every $n \geq 1$ , with infima over empty sets $+ \infty$ ; if $E_{n}^{A} < Λ_{A}$ both outer values are attained by spans of the first $n$ and first $n - 1$ orthonormal eigenvectors (Min-max principle below the essential spectrum).

Proof

technique · direct; read the trial value as a spectral integral over the carrier $[E_{0},+\infty)$ for (a), invoke the supplied gap estimate for (b), instantiate the supplied min-max theorem for (c), and compare trial families for (d)

Given: the data of the statement and [F1]-[F4].

1.1F1F2givenchoosealgebra

[Clause (a)] By [F1] the spectrum lies in $[E_{0}, + \infty)$ , so every real $t < E_{0}$ has a neighbourhood, namely $(2 t - E_{0}, E_{0})$ , with zero spectral projection, and countable subadditivity of the finite measure $E_{u}$ over the rational intervals contained in $(- \infty, E_{0})$ gives $E_{u} ((- \infty, E_{0})) = 0$ . Hence $E_{u}$ is carried by $[E_{0}, + \infty)$ and, for $u \in Q (H)$ with the finite integral of [F2], $q_{H} [u] = \int_{R} λ d E_{u} (λ) = \int_{[E_{0}, \infty)} λ d E_{u} (λ) \geq E_{0} \int_{[E_{0}, \infty)} d E_{u} (λ) = E_{0} ∥ u ∥^{2} = E_{0},$ since $u$ is a trial vector. For the reverse infimum inequality, let $η > 0$ . Since $E_{0} = \inf σ (H)$ , choose $t \in σ (H)$ with $E_{0} \leq t < E_{0} + η / 2$ . Choose a bounded open interval $J$ containing $t$ and contained in $(- \infty, E_{0} + η)$ . The spectral-support characterization in [F1] gives $E (J) \neq 0$ ; choose a unit vector $u$ in its range. Its spectral measure is supported in the bounded set $J$ , hence $u \in Q (H)$ and $q_{H} [u] \leq E_{0} + η$ . Thus the infimum of trial values is exactly $E_{0}$ , as claimed in (a).

1.2F2F3given

[Clause (b)] The model's clause (b) supplies a self-adjoint $H$ bounded below with bottom $E_{0} = \inf σ (H)$ , and clause (b) of the statement supplies the gap $σ (H) \cap (E_{0}, E_{0} + g) = \emptyset$ and $P = E ({E_{0}})$ ; the model's $Q (H)$ and $q_{H}$ are those of [F2], which are the ones used by the supplied estimate [F3] with $H^{'} = H$ , $E_{0}^{'} = E_{0}$ , $g^{'} = g$ , $P^{'} = P$ . Applying [F3] gives $q_{H} [u] - E_{0} \geq g ∥ (I - P) u ∥^{2}$ , hence $∥ (I - P) u ∥^{2} \leq ε / g$ whenever $q_{H} [u] \leq E_{0} + ε$ , and $P u \neq 0$ when $ε < g$ .

1.3F2F4given

[Clause (c)] The operator $H$ is self-adjoint and bounded below, so [F4] applies with $A = H$ ; its form domain and form are $Q (H)$ and $q_{H}$ by [F2], its essential spectrum is $σ_{e s s} (H)$ with $Λ = \inf σ_{e s s} (H)$ , and its eigenvalue ladder below $Λ$ , with the after-exhaustion convention, is exactly the ladder $(E_{n})$ of the model clause (d) by Discrete and essential spectrum of a self-adjoint operator. Hence the two displayed min-max identities and the attainment statement for $E_{n} < Λ$ hold.

2.1F2F4step 1.1step 1.3givenalgebra

[The identification $E_{1} = E_{0}$ ] For $n = 1$ the second min-max formula in [F4] has only $F = {0}$ , so it gives $E_{1} = \inf {q_{H} [u] : u \in Q (H), ∥ u ∥ = 1}$ . Step 1.1 identifies this infimum as $E_{0}$ , hence $E_{1} = E_{0}$ and the additional assertion of (c) follows. This argument does not assume that the infimum is attained or that a bottom eigenvector exists.

3.1step 1.3step 2.1F2algebra

[Clause (d)] Let $L \subseteq Q (H)$ have dimension $n$ and finite supremum $s = \sup {q_{H} [u] : u \in L, ∥ u ∥ = 1}$ . This particular $L$ belongs to the first min-max infimum of step 1.3, so $E_{n} \leq s$ directly; attainment of the inner supremum is unnecessary. If $n = 1$ and $u$ is any unit spanning vector, the unit vectors of the complex line are $α u$ with $∣ α ∣ = 1$ , and the quadratic form satisfies $q_{H} [α u] = ∣ α ∣^{2} q_{H} [u] = q_{H} [u]$ . Thus $s = q_{H} [u]$ and $E_{1} = E_{0}$ reduces the assertion to (a).

4.1F2F3given∎

[Clause (e)] The units are those of the model: $H$ and $q_{H}$ in joules, so $E_{0}$ and $E_{n}$ are read in joules. For $u \in D (H)$ , $q_{H} [u] = ⟨ H u, u ⟩$ by [F2], which is the mean of the assigned sharp energy observable in the pure preparation $[u]$ by Mean, second moment and variance of an assigned observable in a preparation clause (a); for $u \in Q (H) ∖ D (H)$ the value $q_{H} [u]$ is the form value of [F2] and no outcome-law mean is asserted. No measurement, apparatus or sampling model entered the argument, and every object is a spectral datum of the declared operator $H$ .

Remarks

No attained-ground-state hypothesis. Clauses (a) and (c) hold whether or not $E_{0}$ is an eigenvalue and whether or not the infimum of the trial values is attained; clause (b) states precisely what an isolated bottom with gap $g$ forces. This is the form-domain reading required of the variational bound: the trial vector need not lie in $D (H)$ .
Choice. The min-max clause cites the published min-max theorem, which assumes the Axiom of Choice; the trial lower bound of clause (a) and the gap estimate of clause (b) are, through their suppliers, Countable-Choice statements in their other dependencies. Consumers that use clause (c) carry the Axiom of Choice.

Variational trial bounds, gap-overlap control and the min-max characterisation of the declared model's discrete eigenvalues

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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