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2022•Communications in Mathematics and ApplicationsOpen access

Some Aspects of Theory of Schrödinger Operators on Riemannian Manifold

Farah Diyab, B. Surender Reddy

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Abstract

This paper deals with a given Riemannian manifold \(\mathcal{M}\). One of the main tasks is description of spectrum of several classes of Schrödinger operator \(P=\frac{-h^{2}}{2}\Delta _{g}+V\) where \(\Delta _{g}\) is Laplace Beltrami operator and \(V\) is potential on manifold. We illustrate the inverse and direct problems of \(\Delta _{g}\) and the way to discover the geometry of Riemannian manifold from spectral data.

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What this paper is about

This paper deals with a given Riemannian manifold \(\mathcal{M}\). One of the main tasks is description of spectrum of several classes of Schrödinger operator \(P=\frac{-h^{2}}{2}\Delta _{g}+V\) where \(\Delta _{g}\) is Laplace Beltrami operator and \(V\) is potential on manifold. We illustrate the inverse and direct problems of \(\Delta _{g}\) and the way to discover the geometry of Riemannian manifold from spectral data.

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Available abstract

This paper deals with a given Riemannian manifold \(\mathcal{M}\). One of the main tasks is description of spectrum of several classes of Schrödinger operator \(P=\frac{-h^{2}}{2}\Delta _{g}+V\) where \(\Delta _{g}\) is Laplace Beltrami operator and \(V\) is potential on manifold. We illustrate the inverse and direct problems of \(\Delta _{g}\) and the way to discover the geometry of Riemannian manifold from spectral data.

Key concepts: Mathematics, Laplace–Beltrami operator, Riemannian manifold, Spectral geometry, Manifold (fluid mechanics), Operator (biology), Riemannian geometry, Laplace operator

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