Degenerate stationary perturbation: the perturbed eigenvalues stay within 4 lambda^2 v^2/g of the first-order shifts, and no branch relaxation is asserted

Adopt the framework data and the declared perturbed-family model of The declared perturbed-family model: bounded self-adjoint perturbation, isolated gapped label and followed spectral cluster with $H_0$ self-adjoint on $D(H_0)$, $V=V^{\ast }\in \mathcal{B}(\mathfrak{H})$, $v=\mathrm{\parallel }V\mathrm{\parallel }>0$, $H(\lambda )=H_0+\lambda V$ on $D(H_0)$ for real $\lambda$, and an isolated eigenvalue $E$ of $H_0$ of finite multiplicity $r\ge 1$ with gap at least $g>0$, $\mathrm{dist}(E,\sigma (H_0)\setminus \{E\})\ge g$. Let $P_0=E_{H_0}(\{E\})$, $Q_0=I-P_0$, and $V_E:=\left(P_{0}V{P}_0\right){\mid }_{P_0\mathfrak{H}}$ the first-order operator of the model clause (c)(ii). Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the suppliers. Then for every real $\lambda$ with $\mathrm{\mid }\lambda \mathrm{\mid }v<g\mathrm{/}4$:

(a) $H(\lambda )$ is self-adjoint on $D(H_0)$;

(b) for every real $z$ with $\mathrm{\mid }z-E\mathrm{\mid }+\mathrm{\mid }\lambda \mathrm{\mid }v\le g\mathrm{/}2$, the restricted operator $T_{\lambda }(z):=Q_0(H(\lambda )-z)Q_0$ on $Q_{0}D(H_0)=D(H_0)\cap Q_0\mathfrak{H}$ has a bounded two-sided inverse $B(z):Q_0\mathfrak{H}\to Q_{0}D(H_0)$ with $\mathrm{\parallel }B(z)\mathrm{\parallel }\le 4\mathrm{/}g$;

(c) every real eigenvalue $z$ of $H(\lambda )$ with eigenvector $\psi \mathrm{\ne }0$ and $\mathrm{\mid }z-E\mathrm{\mid }+\mathrm{\mid }\lambda \mathrm{\mid }v\le g\mathrm{/}2$ lies within the first-order shift set: $\mathrm{dist}(z,E+\lambda \mathrm{spec}(V_E))\le \frac{4{\lambda }^2{v}^2}{g},$ where $\mathrm{spec}(V_E)\subseteq [-v,v]$ is the spectrum of the Hermitian operator $V_E$ on the $r$-dimensional eigenspace; moreover $P_0\psi \mathrm{\ne }0$ for every such eigenvector. Every eigenvalue in the model's followed cluster inside $\mathrm{\mid }z-E\mathrm{\mid }<g\mathrm{/}2$ lies in this controlled window and therefore obeys the same bound;

(d) first-order content. The eigenvalues of $V_E$ are the model's first-order shifts: the set $E+\lambda \mathrm{spec}(V_E)$ has at most $r$ points, all within $\mathrm{\mid }\lambda \mathrm{\mid }v$ of $E$; the entire followed cluster, and every other eigenvalue in the window of (c), are localised at distance $O({\lambda }^2{v}^2\mathrm{/}g)$ from this set, and no ordering or one-to-one correspondence between the $r$ spectral branches and the points of $\mathrm{spec}(V_E)$ is asserted;

(e) No relaxation or selection. The theorem is a statement about the spectra of the declared family. It asserts no instantaneous transition of any preparation to the branch of the smallest value of $\mathrm{spec}(V_E)$, no relaxation to a lowest branch, no equilibration and no measurement; which spectral subspace a preparation occupies is fixed by the preparation and by the evolution postulate P4 of the declared model, not by the label $E_{\lambda }$;

(f) the theorem is conditional on the strict positive gap, on the finite multiplicity of $E$ and on the boundedness of $V$; it asserts nothing about degenerate essential-spectrum labels, gap-closing families, unbounded perturbations or time-dependent couplings.

• Why the first-order operator, not the full perturbation. The supplied reduction shows that the perturbed labels are localised at the spectrum of the compression $P_{0}V{P}_0$, which is the model's first-order shift data; the second-order remainder is bounded uniformly by $4{\lambda }^2{v}^2\mathrm{/}g$, independently of the structure of the perturbation inside the eigenspace.
• No branch selection. The statement is the corrected form of the textbook assertion that a degenerate level "immediately" moves to its lowest branch; only the spectral splitting is deduced here. Population of branches is a question about the preparation and its dynamics, not about the eigenvalue asymptotics.