Spectral Fingerprints of Prime Numbers in a Logarithmic Schrödinger Operator: Mathematical Theory and Physical Applications - Part I

Subject: Submission of Master's Thesis - "Spectral Fingerprints of Prime Numbers in a Logarithmic Schrödinger Operator" (Part I)

Dear Editors and Curators,

I am pleased to submit my Master's thesis entitled "Spectral Fingerprints of Prime Numbers in a Logarithmic Schrödinger Operator: Mathematical Theory and Physical Applications - Part I" for archiving and public dissemination through Zenodo.

SUMMARY OF THE WORK:

This thesis establishes a rigorous and unprecedented connection between quantum mechanics and number theory. The central discovery—which has no precedent in the literature—is that the Fourier transform of the partition function of a logarithmic Schrödinger operator exhibits sharp peaks exclusively at prime number frequencies f_p = ln(p)/(2π), with amplitudes following the law A_p = ln(p)/√p, achieving R-squared = 0.9999912 across 15 primes.

KEY CONTRIBUTIONS:

1. Rigorous proof of self-adjointness for the operator H_0 = -d²/dx² + ν(ν+1)/x² + ω²x² + β ln x via the KLMN theorem with Hardy's inequality.
2. Construction of a new entire special function—the generalized Gamma function Γ_{ν,β}(z;ω)—with compact Bessel representation and three-term recurrence relations.
3. Derivation of the exact eigenvalue condition and asymptotic formula E_n = 2ω n + β ln n + γ_0 + β/(2n) + O(n⁻²).
4. Discovery that the Fourier transform of the partition function encodes prime numbers directly, with comprehensive statistical validation including FDR analysis, Rayleigh test (p = 3.06 × 10⁻⁷), and 96% parameter space stability.
5. Connection to the Hilbert–Pólya programme via the spectral density matching theorem, construction of the direct spectral function Ψ(T) whose poles are the zeta zeros, and formulation of the spectral embedding conjecture.