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# memo_has_moved_text();On some aspects of oscillation theory and geometry

Bruno Bianchini, Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Via Trieste 63, I-35121 Padova, Italy, Luciano Mari, Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy and Marco Rigoli, Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, I-20133 Milano, Italy

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 225, Number 1056
ISBNs: 978-0-8218-8799-8 (print); 978-1-4704-1056-8 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00681-2
Published electronically: December 13, 2012
Keywords:Oscillation, spectral theory, index, Schrodinger operator, uncertainty principle, compactness, immersions, comparison
MSC: Primary 34K11, 58C40, 35J15; Secondary 35J10, 53C21, 57R42

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Chapters

• Chapter 1. Introduction
• Chapter 2. The Geometric setting
• Chapter 3. Some geometric examples related to oscillation theory
• Chapter 4. On the solutions of the ODE $(vz’)’ + Avz = 0$
• Chapter 5. Below the critical curve
• Chapter 6. Exceeding the critical curve
• Chapter 7. Much above the critical curve

### Abstract

The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation we prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE's that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep our investigation basically self-contained we also collect some, more or less known, material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view.