Nondegenerate stationary perturbation of a simple gapped label: first- and second-order coefficients and an explicit cubic remainder

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 in the simple-label case: $H_0$ is self-adjoint on $D(H_0)$, $V=V^{\ast }\in \mathcal{B}(\mathfrak{H})$ with $v=\mathrm{\parallel }V\mathrm{\parallel }\ge 0$, $H(\lambda )=H_0+\lambda V$ on $D(H_0)$ for real $\lambda$, and $E$ is an isolated eigenvalue of $H_0$ of multiplicity one with gap at least $g>0$, $\mathrm{dist}(E,\sigma (H_0)\setminus \{E\})\ge g$. Let ${\psi }_0$ be a normalized eigenvector, $P_0{\psi }_0={\psi }_0$, let $S:={\iota }_{Q_0}(H_0{\mathrm{\mid }}_{D(H_0)\cap Q_0\mathfrak{H}}-E{)}^{-1}{Q}_0,Q_0=I-P_0,$ be the reduced inverse of the model clause (c)(iii). In the branch $v>0$ put $R:=\frac{g}{8v}\mathrm{.}$ Then clauses (a)-(e) hold for every real $\lambda$ with $\mathrm{\mid }\lambda \mathrm{\mid }<R$ in this positive-norm branch; clause (f) covers $v=0$ independently of $R$:

(a) $H(\lambda )$ is self-adjoint on $D(H_0)$; the model's followed label $E_{\lambda }$ of the contour $\mathrm{\mid }z-E\mathrm{\mid }=g\mathrm{/}2$ exists, is the unique spectral value of $H(\lambda )$ inside that contour, is real, and its spectral projection is the orthogonal rank-one projection $P_{\lambda }=E_{H(\lambda )}(\{E_{\lambda }\})$;

(b) the gauge-fixed eigenvector $\psi (\lambda ):=\frac{P_{\lambda }{\psi }_0}{⟨P_{\lambda }{\psi }_0,{\psi }_0⟩}$ is analytic in $\lambda$, satisfies $⟨\psi (\lambda ),{\psi }_0⟩=1$ and $(H_0+\lambda V)\psi (\lambda )=E_{\lambda }\psi (\lambda )$, and $q:=2\mathrm{\mid }\lambda \mathrm{\mid }v\mathrm{/}g\le \frac{1}{4},\mathrm{\parallel }{P}_{\lambda }-P_0\mathrm{\parallel }\le \frac{1}{3},\mathrm{\parallel }\psi (\lambda )\mathrm{\parallel }\le 2;$

(c) the model's label identity $E_{\lambda }=⟨(H_0+\lambda V)\psi (\lambda ),{\psi }_0⟩=E+\lambda ⟨V\psi (\lambda ),{\psi }_0⟩$ holds, so the deviation $\lambda \mapsto E_{\lambda }-E$ of the model's energy label is controlled by the perturbation alone;

(d) the exact coefficients are $E^{\mathrm{\prime }}(0)=⟨V{\psi }_0,{\psi }_0⟩,{\psi }^{\mathrm{\prime }}(0)=-SV{\psi }_0,E^{\mathrm{\prime }\mathrm{\prime }}(0)=-2⟨VSV{\psi }_0,{\psi }_0⟩,$ with $\mathrm{\mid }{E}^{\mathrm{\prime }}(0)\mathrm{\mid }\le v$, $\mathrm{\parallel }{\psi }^{\mathrm{\prime }}(0)\mathrm{\parallel }\le v\mathrm{/}g$ and $\mathrm{\mid }{E}^{\mathrm{\prime }\mathrm{\prime }}(0)\mathrm{\mid }\le 2v^2\mathrm{/}g$; the second-order coefficient involves the whole complementary spectral subspace of $H_0$ through $S$, continuum included;

(e) the cubic remainder obeys $\mid E_{\lambda }-E-\lambda E^{\mathrm{\prime }}(0)-\frac{{\lambda }^2}{2}{E}^{\mathrm{\prime }\mathrm{\prime }}(0)\mid \le \frac{Rv(\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}R{)}^3}{1-\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}R},$ so at $\mathrm{\mid }\lambda \mathrm{\mid }=R\mathrm{/}2$ the bound is $g\mathrm{/}32$ and for $\mathrm{\mid }\lambda \mathrm{\mid }\le R\mathrm{/}2$ it is at most $128v^3\mathrm{\mid }\lambda {\mathrm{\mid }}^3\mathrm{/}{g}^2$;

(f) if $v=0$, that is $V=0$, then $P_{\lambda }=P_0$, $E_{\lambda }=E$ and $\psi (\lambda )={\psi }_0$ for every real $\lambda$, and no radius involving division by $v$ is used in this branch;

(g) the theorem is conditional on the strict positive gap, on the multiplicity one of $E$ and on the boundedness of $V$; it asserts no unbounded or degenerate perturbation theorem, no time-dependent perturbation, and no statement about which branch of the family a preparation occupies.

• Where the continuum enters. The reduced inverse $S$ acts on the whole complementary subspace $Q_0\mathfrak{H}$, so the second-order coefficient $-2⟨VSV{\psi }_0,{\psi }_0⟩$ automatically includes the contribution of the continuous spectrum of $H_0$; the theorem needs no discrete-eigenbasis expansion and no sum over bound states.
• Determinant of the remainder. The bound is a genuine cubic bound on the disk of radius $R=g\mathrm{/}(8v)$; no uniformity in $\lambda$ beyond that disk is claimed, and the gap enters both the radius and the constant.