Some Further Insight into the Sturm–Liouville Theory

Salvatore De Gregorio (), Lamberto Lamberti and Paolo De Gregorio
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Salvatore De Gregorio: Independent Researcher, 00143 Rome, Italy
Lamberto Lamberti: Independent Researcher, 00162 Rome, Italy
Paolo De Gregorio: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Mathematics, 2025, vol. 13, issue 15, 1-33

Abstract: Some classical texts on the Sturm–Liouville equation ( p ( x ) y ′ ) ′ − q ( x ) y + λ ρ ( x ) y = 0 are revised to highlight further properties of its solutions. Often, in the treatment of the ensuing integral equations, ρ = c o n s t is assumed (and, further, ρ = 1 ). Instead, here we preserve ρ ( x ) and make a simple change only of the independent variable that reduces the Sturm–Liouville equation to y ″ − q ( x ) y + λ ρ ( x ) y = 0 . We show that many results are identical with those with λ ρ − q = c o n s t . This is true in particular for the mean value of the oscillations and for the analog of the Riemann–Lebesgue Theorem. From a mechanical point of view, what is now the total energy is not a constant of the motion, and nevertheless, the equipartition of the energy is still verified and, at least approximately, it does so also for a class of complex λ . We provide here many detailed properties of the solutions of the above equation, with ρ = ρ ( x ) . The conclusion, as we may easily infer, is that, for large enough λ , locally, the solutions are trigonometric functions. We give the proof for the closure of the set of solutions through the Phragmén–Lindelöf Theorem, and show the separate dependence of the solutions from the real and imaginary components of λ . The particular case of q ( x ) = α ρ ( x ) is also considered. A direct proof of the uniform convergence of the Fourier series is given, with a statement identical to the classical theorem. Finally, the proof of J. von Neumann of the completeness of the Laguerre and Hermite polynomials in non-compact sets is revisited, without referring to generating functions and to the Weierstrass Theorem for compact sets. The possibility of the existence of a general integral transform is then investigated.

Keywords: anharmonic oscillators; equipartition; fourier series; integral transforms; completeness of Laguerre and Hermite polynomials (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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