In this work we focus on the characterization of the space L 2 .SI C/ on Riemannian 2-manifolds S induced by a xed magnetic vector potential A 0 in the nonlinear Ginzburg-Landau (GL) superconductivity model. e linear di erential operator governing the GL model is the surface Schrödinger operator .ir C A 0 / 2 on S. We obtain a complete orthonormal system in L 2 .SI C/ from a collection of nontrivial solutions of the weak-form of the spectral problem associated with .ir CA 0 / 2 . en, after proving that any member of this basis satis es a higher regularity condition, we conclude that each is also an eigenfunction of the strong-form of the surface Schrödinger operator, and must satisfy a natural Neumann condition over any nonempty component of the manifold boundary @S.ese results form the theoretical foundations used to develop e cient computational tools for simulating the Langevin version of the surface GL model.

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