bibkey: kellerpinchoverpogorzelski2021rellich authors: Matthias Keller; Yehuda Pinchover; Felix Pogorzelski year: 2021 title: From Hardy to Rellich inequalities on graphs doi: 10.1112/plms.12376 url: https://arxiv.org/abs/1909.02286v1 claim: The positive-function ground-state transform rewrites a graph Schrodinger quadratic form using transformed conductances and a pointwise potential term; positivity of the transforming function does not imply square summability or attainment of a normalized spectral minimum. strata_touched: [] license: citation-only triage: anchor

Positive-function graph transform and its normalization boundary

The primary preprint is Keller–Pinchover–Pogorzelski, From Hardy to Rellich inequalities on graphs, arXiv:1909.02286v1, §6, proof of Theorem 6.1, displayed ground-state-transform identity. The journal article is Proceedings of the London Mathematical Society 122(3), 458–477, DOI:10.1112/plms.12376. The note cites the source and does not redistribute its text or PDF.

The paper uses a graph with symmetric nonnegative weights $b(x,y)$, a full-support vertex measure $m$ and a real potential $q$, with $H=m^{-1}\Delta +q$. For a positive function $u$ in the local operator domain and a finite-support real test $\phi$, the identity in that proof reads, in the corresponding quadratic-form notation,

$$Q_H(u\phi )=\frac{1}{2}\sum _{x,y}b(x,y)u(x)u(y)|\phi (x)-\phi (y){|}^2+\sum _{x}m(x)u(x)(Hu)(x)|\phi (x){|}^2.$$

Real and imaginary parts give the complex finite-support version. A finite Dirichlet restriction includes its exterior killing in $\Delta$ and hence in $Hu$; omitting it changes the operator. The transforming function is a positive function on vertices, not a normalized eigenvector by definition. The paper’s use of a positive supersolution and a Hardy weight in Theorem 6.1 has additional hypotheses. The algebraic identity alone does not supply such a Hardy weight for an arbitrary attractive operator.

The FIB boundary geometry volume §52 consumes this identity with actual occurrence weights and $u=1$. The shifted operator $H+gI=aL$ is nonnegative; the unshifted pointwise term is $-g$, with the declared Dirichlet killing added at the outgoing occurrence. Its uniform lower certificate is therefore explicit. The volume’s source-specific nonattainment argument uses the infinite number of actual occurrences: $u=1$ is not in their counting-measure ${\ell }^2$ space. Neither the graph transform nor the paper’s Rellich inequality supplies the missing normalized compactness, native conductance realization or physical field law.