Logarithmic Deformations of Spectral Systems —Part I: Spectral Analysis and Classification of Logarithmically Perturbed Harmonic Systems

In this first paper of a series, we develop the complete spectral and classification structure of the half-line Schrödinger operator O0 = − d2 /dx2 + μ(μ+1) /x2 + ω2 x2 (x > 0, μ > 1/2, ω > 0) deformed by a logarithmic potential γ log x. Building on the established spectral properties of O0, we introduce a spectral-transcendental framework and prove: 1 Spectral Zeta: A closed form for the logarithmic spectral zeta S(s; ω) = ∑n≥0 Vlog nn /Asn (Re(s) > 1) in terms of the Hurwitz zeta function. We establish its meromorphic continuation to C, featuring a double pole at s = 1, a simple pole at s = 0, and simple poles at negative integers. 2 Spectral Sum Classification: A complete classification of {Sk(N) k,N≥1: Sk(N) ∈ Q + Q log 2 for N even; Sk(N) ∈ Q for N ≥ 2k + 1 odd; and Sk(N) ∈ Q + Qπ2 for N ≤ 2k − 1 odd. 3 Exact Constants: Closed-form evaluations for key constants, including B0(1, 1) = 23π4/46080 − γE ζ(3) 32 + γ2E π2 768 ≈ 0.03122, C(1) = 1, the MZV constant c, and the Borel-2 constant CBorel 1 . 4 Commutator Trace Identity: The analytical identity T(α) = 1/2 ζH (4, (α + 1)/2) for α ∈ {0, 1, 2}, its general conjecture for all α > 0, and the Pochhammer-ratio obstruction at non-integer values.