Singular Legendre operators on graphs: trace formulas and Fourier singular support
Abstract
We study the singular Legendre operator on the edges of a finite connected graph, subject to renormalized continuity and Kirchhoff conditions at the vertices. An exact scattering trace formula and a reduction to the normalized adjacency matrix describe its spectrum. For the symmetrized spectral measure in r = λ + 1/4, we determine the singular support of its Fourier transform. Let L_G be the set of lengths kπ of nonempty closed edge walks with k steps (including immediate reversals). We prove the exact closed-walk length-spectrum identity sing supp µ (r)
where g is the odd girth (the minimum length of an odd closed walk, with loops counted as one step), and g = ∞ for a bipartite graph. Every nonzero even period carries a Dirac mass and, except for the single-edge graph, a non-classical conormal principal-value term. At the surviving odd periods, the leading logarithmic conormal singularity is determined by g and the spectral invariant tr(A_G^g ) of the normalized adjacency matrix. The singular Legendre circle and equilateral stars illustrate these phenomena.
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BibTeX
@misc{falbel2026singular,
title = {{Singular Legendre operators on graphs: trace formulas and Fourier singular support}},
author = {Elisha Falbel},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05805676v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Singular Legendre operators on graphs: trace formulas and Fourier singular support
AU - Elisha Falbel
PY - 2026
DA - 2026/10/19
PB - HAL
UR - https://hal.science/hal-05805676v1
AB - We study the singular Legendre operator on the edges of a finite connected graph, subject to renormalized continuity and Kirchhoff conditions at the vertices. An exact scattering trace formula and a reduction to the normalized adjacency matrix describe its spectrum. For the symmetrized spectral measure in r = λ + 1/4, we determine the singular support of its Fourier transform. Let L_G be the set of lengths kπ of nonempty closed edge walks with k steps (including immediate reversals). We prove the exact closed-walk length-spectrum identity sing supp µ (r) where g is the odd girth (the minimum length of an odd closed walk, with loops counted as one step), and g = ∞ for a bipartite graph. Every nonzero even period carries a Dirac mass and, except for the single-edge graph, a non-classical conormal principal-value term. At the surviving odd periods, the leading logarithmic conormal singularity is determined by g and the spectral invariant tr(A_G^g ) of the normalized adjacency matrix. The singular Legendre circle and equilateral stars illustrate these phenomena.
ER -