Boundary-Generated Overlap Measures under Principal Jacobi Extensions

Abstract

Let $H_n$ be a finite real symmetric Jacobi matrix, let $H_nu=\lambda u$, and embed $u$ by zeros into a larger principal Jacobi extension $\widehat H$. The squared overlaps of the embedded vector with the eigenvectors of $\widehat H$ define a probability measure $\nu$ on shifted energies. For a nonzero bridge residual, we show that this overlap measure is generated exactly by the local spectral measure $\rho$ at the first added coordinate: if $\beta$ is the squared residual norm, then $x^2,d\nu(x)=\beta,d\rho(x)$. The possible atom of $\nu$ at $x=0$ is recovered as the residual probability mass required by normalization; in the zero-residual case $\nu=\delta_0$. Consequently, every centered overlap moment of order at least two is a bridge residual factor times a rooted return moment of the extended Jacobi matrix. This yields a finite-radius moment ladder: the moment of order $\ell+2$ depends only on the radius-$\lfloor\ell/2\rfloor$ neighborhood of the bridge, and on a path the first appearances of a distant edge and diagonal coefficient occur at exactly determined orders. We then combine these locally generated moments with a finite truncated-moment linear program to obtain optimal, independently checkable lower bounds for the overlap mass carried by selected nearby eigenvalues. The optimization formalism itself is classical; the contribution is the deterministic mechanism that supplies its moments from a principal Jacobi extension. We include a closed-form four-site example, degeneracy and zero-residual cases, an alternative Stieltjes-transform proof, and a numerical pilot on 255 embedded eigenvectors. The results are finite-dimensional and deterministic; no localization or large-volume limit is asserted.