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Rays, Waves, and ScatteringTopics in Classical Mathematical Physics$
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John A. Adam

Print publication date: 2017

Print ISBN-13: 9780691148373

Published to Princeton Scholarship Online: May 2018

DOI: 10.23943/princeton/9780691148373.001.0001

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

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

(p.459) Chapter Twenty-Three A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation
Rays, Waves, and Scattering

John A. Adam

Princeton University Press

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.

Keywords:   bound states, Schrödinger equation, bound-state theorems, eigenvalues, Weyl's theorem, Sturm-Liouville equation, limit point, limit circle, Hermite's equation, Legendre's equation

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