Rigorous Spectral Analysis of a Schrödinger Operator with Fractional and Logarithmic Perturbations: Fixed-Point Formulation and Explicit Constants
We present a complete spectral analysis of the Schrödinger operator
O0 = − d2/dx2 + μ(μ + 1)/x2 + ω2 x2
on the half-line L2(0, ∞), perturbed by two unbounded terms: a fractional power Oσ0 with 0 < σ < 1, and a logarithmic term γ log x. We first prove the complete spectral analysis of O0 via reduction to Laguerre’s equation, and establish that the minimal, maximal, and Friedrichs operators coincide (the limit-point property at both endpoints). We then prove relative boundedness of both perturbations with relative bound zero, and apply the Kato–Rellich theorem to obtain self-adjointness of the full operator, with purely discrete spectrum.
We provide an explicit closed-form formula for the logarithmic matrix elements Vlogmn = ⟨ψm, log x ψn⟩ in terms of the digamma function, prove a uniform bound |Vlogn,n+r | ≤ 1/(2|r|), and derive the complete second-order expansion of the eigenvalues, which includes both a logarithmic term −α(log n + γE − log 2)/(16ωH∗n (n + 1)2) with αμ + 1/2 and an algebraic term −λπ2σ(1 − σ)(4ωn)σ−2/(24H2n).
Next we develop a fixed-point formulation on the weighted Banach space Z′ 2, where we prove contraction of the operator Bn(F) with constant 0.032, and obtain an exact implicit equation
En = en + γVlognn + Fn, Fn = Φn(Fn), with Lip(Φn) ≤ 2.95 × 10−6 and |Fn| ≤ 1.26 × 10−5ω. We further establish: (i) analyticity of En(λ, γ) in a neighborhood of (0, 0) with radius ρ = min(Λgap/2, ω/175); (ii) a rigorous second-order expansion with both algebraic and logarithmic corrections; (iii) generalization of the composite gap to all σ ∈ (0, 1).
All constants are explicit and numerically usable. The case n = 0 is treated separately due to the index constraint s ≥ 1.
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