Fredholm determinant and the Sturm-Liouville problems in quantum mechanics

The Fredholm determinant of a nonrelativistic Hamiltonian defined on a compact one-dimensional space is evaluated exactly. The Schroedinger equation is rewritten as a first-order differential equation, which is integrated formally. Then a 2 x 2 eigenvalue equation is proved to be proportional to the Fredholm determinant. Our method turns out to be a powerful tool to solve eigenvalue problems in quantum mechanics. Finally several applications of our method are given. (orig.)