bibkey: jacoblangertrunk2016variational
authors: Birgit Jacob, Matthias Langer, and Carsten Trunk
year: 2016
title: Variational principles for self-adjoint operator functions arising from second-order systems
doi: 10.7153/oam-10-29
url: https://files.ele-math.com/articles/oam-10-29.pdf
claim: Theorem 2.4 supplies a generalized Rayleigh variational principle for a holomorphic self-adjoint operator function with monotone form crossings. On the actual confined Weil domain it supplies negative excess-quotient attainment without assuming a uniform lower bound on the excess relative to the ambient norm.
strata_touched: []
license: bibliographic-reference-only
triage: anchor

Generalized Rayleigh attainment on the actual excess domain

The inspected primary is the publisher’s complete article, Operators and Matrices 10(3) (2016), 501–531, DOI:10.7153/oam-10-29. The 31-page PDF has SHA-256 1e118a97d5f29abfa71e15a5a3f1c92e4a30acbe9f7ec0033aed9d44170339f6. Definition 2.1 is on printed p.505; Theorem 2.4 and formulas (2.5)–(2.8) are on pp.507–508. Its general Section 2 is used here; the positivity hypotheses of the later second-order-system model are not imposed on the Weil energy.

Published supplier and exact parameter correspondence

The literature-attested supplier assumes a holomorphic type-(B) operator family, self-adjoint on a real parameter interval, a finite initial negative spectral index, and scalar forms that cross zero strictly downwards. Before the operator-function essential spectrum, Theorem 2.4 characterizes isolated eigenvalues by a generalized Rayleigh functional. Its empty-spectrum alternative (2.8) is part of the result; existence cannot be inferred merely by naming (2.5).

Keep W317, W322 and W326 and MR1–MR13 on their actual domain. Write

$$\mathcal{H}=L^2((\pi ,\infty ),dx),\mathcal{V}=\{F:M(F)+\mathcal{P}(F)<\infty \},D=M-H>0(F\ne 0).$$

The last inequality is strict on every nonzero vector, but is not a uniform inequality $D\ge \delta H$ on the entire domain. Use

$$t_z=\mathfrak{E}-zD=q_{-z}+zH,\Re z<0,p(F)=\mathfrak{E}(F)/D(F).$$

W317’s infinitesimal moment bound gives closed sectorial forms on the common domain $\mathcal{V}$: locally in $\Re z<0$ their shifted real part controls $\mathcal{P}+M$, and their imaginary part is bounded by that same shifted form. Their affine coefficients give holomorphic type (B). On real $z<0$, W322 gives self-adjointness and compact resolvent, so $0$ is outside the essential spectrum of each represented operator. Thus the operator-function essential spectrum misses the chosen real interval. The crossing condition is exactly ${\partial }_z{t}_z[F]=-D(F)<0$.

If a negative source exists, MR5 and W317 already give $-\infty <\rho :=\inf p<0$. Choose $a<\rho$, take the paper’s interval $\Delta =(a,0)$, and choose $a^{\prime }\in (a,\rho )$. Then $t_{a^{\prime }}[F]>0$ for every nonzero $F$, hence its negative spectral index is $\kappa =0$. The function $p$ is a generalized Rayleigh functional under Definition 2.1 even when its value lies at or above the right endpoint zero.

If the operator function had empty spectrum in $\Delta$, (2.8) with $n=1$ would give $\inf p\ge 0$, contradicting $\rho <0$. Its first eigenvalue therefore exists. Formula (2.5), with $\kappa =0,n=1$, gives $\rho =\min p$ and a nonzero actual vector at which it is attained. The source-specific consequence is the already recorded MR11:

$$\lambda =-\rho >0,Q_{\lambda }=\mathfrak{E}+\lambda (M-H)\ge 0,Q_{\lambda }(F_{\ast })=0.$$

This is a direct supplier application, not a new variational principle. It requires neither a $D^{-1/2}$ change of domain nor a separately invented $D$-completion. Real and complex infima agree by the existing real/imaginary splitting in MR6. The true unit-$H$ cap remains MR13’s parameter-dependent $K_{\lambda }$; the theorem does not turn it into cap four, identify every older constrained minimizer, or supply a lower sign bound for the complete arithmetic energy. This source correspondence has not been compiled in Lean and does not prove RH or strict Robin.