Bounded spectral perturbations

Hypotheses

Let $H$ be self-adjoint, bounded below, with compact resolvent. Let $V$ be bounded and self-adjoint, with $v=\|V\|$. Then $H+V$ is self-adjoint on the same operator domain $D(H)$ and is bounded below by $\inf\sigma(H)-v$.

For a nonreal $z$ with $|\operatorname{Im}z|>v$,

$$(H+V-z)^{-1}=(H-z)^{-1} \left[I+V(H-z)^{-1}\right]^{-1}.$$

The bracket is invertible by its Neumann series. The first factor is compact, so the perturbed resolvent is compact. The resolvent identity extends compactness to every point of the perturbed resolvent set.

Eigenvalue displacement

Order eigenvalues increasingly, counting multiplicity, as $E_j(H)$. For every unit $u$ in the common domain,

$$\langle u,Hu\rangle-v \leq\langle u,(H+V)u\rangle \leq\langle u,Hu\rangle+v.$$

Taking the maximum over a $j$-dimensional trial space and then the infimum over such spaces gives

$$\boxed{|E_j(H+V)-E_j(H)|\leq v.}$$

The variational characterization used here is stated in Nam, Functional Analysis II, §2.3. It includes repeated eigenvalues. A perturbation may split a degeneracy while respecting every bound.

An isolated cluster

Suppose a finite group of eigenvalues is separated from the rest by a positive minimum gap $g$. If $2v<g$, the perturbed cluster and the perturbed neighboring eigenvalues remain separated by at least $g-2v$. It has the same number of eigenvalues counting multiplicity.

The associated subspace can rotate. To express its continuity, take a closed counterclockwise contour $\mathcal C$ enclosing only the cluster and satisfying $\operatorname{dist}(z,\sigma(H))\geq\delta>v$ along the contour. For $H_t=H+tV$, $0\leq t\leq1$, define

$$P_t=\frac{1}{2\pi i}\oint_{\mathcal C}(z-H_t)^{-1}\,dz.$$

Neumann inversion holds on the whole contour, so these orthogonal spectral projectors vary continuously in norm. Their finite integer rank is constant. The resolvent identity also gives the explicit estimate

$$\|P_t-P_0\| \leq\frac{\operatorname{length}(\mathcal C)}{2\pi} \frac{tv}{\delta(\delta-tv)}.$$

The contour condition supplies this bound directly; it is separate from the eigenvalue-count statement.

Oscillator example

For the confining oscillator, write $w=\sqrt\lambda$. Its distinct levels are $E_N=(2N+d)w$, separated by $2w$. Every level defines a cluster of $\binom{N+d-1}{d-1}$ eigenvalues. Any bounded self-adjoint $V$ with $\|V\|<w$ preserves these separated cluster counts along $H+tV$. Their eigenvalues need not remain equal or equally spaced.

For example, a twofold level $E$ with perturbation $\begin{pmatrix}0&\epsilon\\\epsilon&0\end{pmatrix}$ splits into $E\pm\epsilon$. The level count survives and the two frequencies change. A physical interaction must supply its actual operator norm and domain before this bound can establish robustness of a sector calculation. Unbounded interactions require an appropriate relative-bound argument.

Verification and related

spectral_analysis_checks.py tests ordered shifts, cluster counts, subspace rotations, the contour bound, and explicit degeneracy splitting on independently constructed Hermitian matrices. The infinite-dimensional domain and compactness statements are proved above.

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