The declared perturbed-family model: bounded self-adjoint perturbation, isolated gapped label and followed spectral cluster

(a) Framework data. Adopt P1, P2 and 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, with the state, sharp-observable and evolution readings of Assigned sharp observables and the outcome law of the framework and 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 (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice) as the choice hypotheses of the cited mathematical suppliers.

(b) The perturbed family. As an additional model assumption of the pair, fix a self-adjoint operator $H_0:D(H_0)\subseteq \mathfrak{H}\to \mathfrak{H}$ on the declared state space, declare $(H_0,\mathrm{J})$ to be the unperturbed sharp energy observable, fix a bounded self-adjoint operator $V\in \mathcal{B}(\mathfrak{H})$ (A bounded linear operator between normed spaces) with $v:=\mathrm{\parallel }V\mathrm{\parallel }<\mathrm{\infty }$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum) and fix a real coupling $\lambda$ (dimensionless: $\lambda V$ is read in joules). Declare the perturbed family $H(\lambda ):=H_0+\lambda V,D(H(\lambda )):=D(H_0),$ to be the model's sharp energy observable at coupling $\lambda$. Since $V$ is $H_0$-bounded with $H_0$-bound $0\le a<1$ (the bound inequality holds with $a=0$ and $b=v$), the Kato-Rellich theorem applies and $H(\lambda )$ is self-adjoint on $D(H_0)$ for every real $\lambda$ (Kato-Rellich theorem, Symmetric, self-adjoint and essentially self-adjoint operators). This self-adjointness is a supplied theorem about the declared operators; the reading of the family as the system's energy is the model assumption.

(c) The gapped label. Fix an isolated eigenvalue $E\in \mathbb{R}$ of $H_0$ of finite multiplicity $r\ge 1$ with gap $\mathrm{dist}(E,\sigma (H_0)\setminus \{E\})\ge g>0,$ and put $P_0:=E_{H_0}(\{E\})$ and $Q_0:=I-P_0$ (Discrete and essential spectrum of a self-adjoint operator, Projection valued measure, Resolvent and spectrum of an unbounded operator). The definitions used by the pair are those of the cited suppliers:

(i) for real $\lambda$ with $2\mathrm{\mid }\lambda \mathrm{\mid }v\mathrm{/}g<1$, the followed spectral cluster is the spectrum of $H(\lambda )$ inside $\mathrm{\mid }z-E\mathrm{\mid }<g\mathrm{/}2$, and $P_{\lambda }:=\frac{1}{2\pi i}{\oint }_{\mathrm{\mid }z-E\mathrm{\mid }=g\mathrm{/}2}(z-H(\lambda ){)}^{-1}dz$ (counterclockwise) is its rank-$r$ orthogonal spectral projection, by Finite-rank gapped spectral cluster under a bounded perturbation. The cluster is a nonempty finite set and is exactly the spectrum of the restriction to $P_{\lambda }\mathfrak{H}$. In the simple case $r=1$ only, write $E_{\lambda }$ for its unique spectral value; this scalar is the followed perturbed label of Contour perturbation projection and is analytic on the valid coupling disk. For $r>1$ no unique scalar label, branch ordering or individual scalar holomorphy is presumed. At $\lambda =0$ the cluster is $\{E\}$ and its projection is $P_0$. If $v=0$, these data are constant for all real couplings, with no radius involving division by $v$.

(ii) the first-order operator is the Hermitian operator $V_E:=\left(P_{0}V{P}_0\right){\mid }_{P_0\mathfrak{H}}$ on the finite-dimensional eigenspace $P_0\mathfrak{H}$, whose spectrum $\mathrm{spec}(V_E)\subseteq [-v,v]$ is the model's first-order shift data (joules per unit coupling);

(iii) the complementary reduced inverse $S:={\iota }_{Q_0}(H_0{\mathrm{\mid }}_{D(H_0)\cap Q_0\mathfrak{H}}-E{)}^{-1}{Q}_0$ is a bounded operator with the displayed exact complementary-domain inverse. For every finite multiplicity $r\ge 1$, including $v=0$, its existence follows by applying part (c) of Finite-rank gapped spectral cluster under a bounded perturbation at coupling $\lambda =0$ and $w=E$: since the cluster projection is then $P_0$, the restriction of $E-H_0$ to $D(H_0)\cap Q_0\mathfrak{H}$ has a bounded two-sided inverse $R_Q(E):Q_0\mathfrak{H}\to D(H_0)\cap Q_0\mathfrak{H}$, and $S=-{\iota }_{Q_0}{R}_Q(E)Q_0$. For $r=1$ the same operator is supplied by Simple-level reduced-resolvent coefficients. Its role is to record that the whole complementary spectral subspace, continuum included, contributes to the second-order data.

(d) Scope. Clauses (a)-(c) cover bounded perturbations of a self-adjoint operator and a fixed isolated finite-multiplicity label only. The definition asserts no time-dependent perturbation, no adiabatic ramping of $\lambda$, no statement that a preparation occupies the branch of $E_{\lambda }$ or of any eigenvalue of $V_E$, no branch-population or relaxation statement, no measurement and no identification of the model with a laboratory system; the labels $E_{\lambda }$ are spectral data of the declared family.