This paper presents a rigorous construction of a one-dimensional self-adjoint Schrödinger operator whose spectrum corresponds to the non-trivial zeros of the Riemann zeta function. The operator is defined on the positive real line with Dirichlet boundary conditions, and self-adjointness is established through classical results in operator theory. The study shows that the corresponding zeta-regularized determinant reproduces the completed zeta function, providing a consistent bridge between spectral analysis and analytic number theory. The framework is fully analytic, self-contained, and verifiable within established principles of spectral theory and inverse-spectral analysis. This work contributes to the ongoing development of spectral approaches to the Riemann Hypothesis by offering a concrete, reproducible realization within a classical analytic setting.