Electron. J. Differential Equations, Vol. 2020 (2020), No. 76, pp. 123.
Space versus energy oscillations of Prufer phases
for matrix SturmLiouville and Jacobi operators
Hermann SchulzBaldes, Liam Urban
Abstract:
This note considers Sturm oscillation theory for regular matrix SturmLiouville
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: SturmLiouville operators; Jacobi operators; oscillation theory;
matrix Prufer phases.
DOI: 10.58997/ejde.2020.76
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Hermann SchulzBaldes
Department Mathematik
FriedrichAlexanderUniversität ErlangenNürnberg
Germany
email:schuba@mi.unierlangen.de


Liam Urban
Department Mathematik
FriedrichAlexanderUniversität ErlangenNürnberg
Germany
email: liam.urban@tutanota.com

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