Abstract: In this paper we present a comprehensive description of the mathematical foundations of quantum mechanics. The relevant topics include linear inner product, the completeness of Hilbert space, the functional space and continuous bases representations. We also discuss the self-adjoint operator, the eigen-value problem of the Sturm-Liouville system and the natural boundary conditions. We employ quantum problems as examples to discuss natural boundary condition under the non-regular singularity cases. These conditions are not externally applied but contain in the Schrdinger equation implicitly.

Key words: Hilbert space, self-adjoint operator, eigen-value theory, natural boundary condition, non-regular singularity