Emergence of new category of continued fractions from the Sturm–Liouville problem and the Schrödinger equation

On the present work it is presented a new category of continued fraction functions (CFF) of a real variable $$\lambda$$ , named “the $$\epsilon$$ class”, such that $$\lambda$$ enters in the CFF through small or infinitesimal contributions in the partial quotients. These CFF emerge on the secular equation of the Sturm–Liouville equation (SLE) by a finite difference treatment. Also, it was obtained the eigenvalues of the one-dimensional Schrodinger equation (a particular case of SLE) from the CFF secular equation. The case of the two-dimensional Schrodinger equation was partially studied.