Electron. J. Differential Equations, Vol. 2020 (2020), No. 76, pp. 1-23.

Space versus energy oscillations of Prufer phases for matrix Sturm-Liouville and Jacobi operators

Hermann Schulz-Baldes, Liam Urban

Abstract:
This note considers Sturm oscillation theory for regular matrix Sturm-Liouville operators on finite intervals and for matrix Jacobi operators. The number of space oscillations of the eigenvalues of the matrix Prufer phases at a given energy, defined by a suitable lift in the Jacobi case, is shown to be equal to the number of eigenvalues below that energy. This results from a positivity property of the Prufer phases, namely they cannot cross -1 in the negative direction, and is also shown to be closely linked to the positivity of the matrix Prufer phase in the energy variable. The theory is illustrated by numerical calculations for an explicit example.

Submitted August 11, 2019. Published July 17, 2020.
Math Subject Classifications: 34B24, 34C10.
Key Words: Sturm-Liouville operators; Jacobi operators; oscillation theory; matrix Prufer phases.
DOI: 10.58997/ejde.2020.76

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Hermann Schulz-Baldes
Department Mathematik
Friedrich-Alexander-Universität Erlangen-Nürnberg
Germany
email:schuba@mi.uni-erlangen.de
Liam Urban
Department Mathematik
Friedrich-Alexander-Universität Erlangen-Nürnberg
Germany
email: liam.urban@tutanota.com

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