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Zbl 0909.58056
Gurarie, David
Closed geodesics and flat tori in spectral theory on symmetric spaces. (English)
[A] Séminaire de théorie spectrale et géométrie. Année 1992-1993. Chambéry: Univ. de Savoie, Fac. des Sciences, Service de Math. Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 11, 85-103 (1993).

Summary: Closed geodesics are known to play an important role in the spectral theory of Laplacians on Riemannian manifolds ${\cal M}$. They also contribute to the spectral theory of Schrödinger operators $\Delta+V$, typically in the form of higher-order corrections to the principal (Laplace) eigenvalue.\par We give a brief survey of the classical ``shape of metric'' and ``shape of potential'' problems of spectral theory, and explore the role of ``length spectrum'' (the length of all closed path/geodesics), and the related ``Radon transform of $V$''. Then we outline some recent progress in special cases: the $n$-space theory, and Schrödinger operators on higher rank symmetric spaces. The latter case brings in new players: the flat tori. They naturally appear in higher rank compact symmetric spaces and play the role of closed geodesics here. We conclude by a list of open problems.

MSC 2000:: *58J50 Spectral problems; spectral geometry; scattering theory
53C22 Geodesics

Keywords: length spectrum; Radon transform; spectral theory; flat tori; closed geodesic

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