We study a one-dimensional Helmholtz equation of the form (-\Delta_\mu + k(x)^2)u = \lambda u, where \mu is a finite Borel measure on [0,1], possibly singular or fractal, and k(x) is a continuous potential. This framework extends the theory of the fractal Laplacians studied by Bird, Ngai, and Teplyaev to include spatially varying potentials.

We derive a Volterra–Stieltjes integral formulation for the eigenvalue problem and prove existence, uniqueness, and differentiability of solutions under minimal assumptions on k and \mu.