A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation

This chapter examines the mathematical properties of the time-independent one-dimensional Schrödinger equation as they relate to Sturm-Liouville problems. The regular Sturm-Liouville theory was generalized in 1908 by the German mathematician Hermann Weyl on a finite closed interval to second-order differential operators with singularities at the endpoints of the interval. Unlike the classical case, the spectrum may contain both a countable set of eigenvalues and a continuous part. The chapter first considers the one-dimensional Schrödinger equation in the standard dimensionless form (with independent variable x) and various relevant theorems, along with the proofs, before discussing bound states, taking into account bound-state theorems and complex eigenvalues. It also describes Weyl's theorem, given the Sturm-Liouville equation, and looks at two cases: the limit point and limit circle. Four examples are presented: an “eigensimple” equation, Bessel's equation of order ? greater than or equal to 0, Hermite's equation, and Legendre's equation.