The pole-free Weil quadratic form: a simple even ground state

Fix (\lambda>1) and let (QW_{\lambda}) be the Weil quadratic form on (L^{2}([\lambda^{-1},\lambda],d^{*}u)) in the normalization of Connes–Consani–Moscovici. We study the pole-free form (QW_{\lambda}^{\circ}) obtained by deleting the rank-two contribution (W_{0,2}) associated with the poles at (0) and (1). We prove that the lower-bounded self-adjoint operator (A_{\lambda}^{\circ}) associated with (QW_{\lambda}^{\circ}) has compact resolvent and a simple lowest eigenvalue, whose normalized eigenfunction is real, strictly positive almost everywhere, and invariant under (u\mapsto u^{-1}). The proof is organized around a Lévy–Khintchine representation of the archimedean term, irreducibility of the corresponding Dirichlet form, preservation of positivity under the bounded prime-power perturbation, and the Perron–Frobenius–Faris theorem. The distinction between the pole-free operator and the full Weil operator is maintained throughout.