Annales Academić Scientiarum Fennicć
Mathematica
Volumen 33, 2008, 319-336

UNIVERSAL BOUNDS FOR EIGENVALUES OF SCHRÖDINGER OPERATOR ON RIEMANNIAN MANIFOLDS

Qiaoling Wang and Changyu Xia

Universidade de Brasília, Departamento de Matemática
70910-900 Brasília-DF, Brazil; wang 'at' mat.unb.br

Universidade de Brasília, Departamento de Matemática
70910-900 Brasília-DF, Brazil; xia 'at' mat.unb.br

Abstract. In this paper we consider eigenvalues of Schrödinger operator with a weight on compact Riemannian manifolds with boundary (possibly empty) and prove a general inequality for them. By using this inequality, we study eigenvalues of Schrödinger operator with a weight on compact domains in a unit sphere, a complex projective space and a minimal submanifold in a Euclidean space. We also study the same problem on closed minimal submanifolds in a sphere, compact homogeneous space and closed complex hypersurfaces in a complex projective space. We give explict bound for the (k + 1)-th eigenvalue of the Schrödinger operator on such objects in terms of its first k eigenvalues. Our results generalize many previous estimates on eigenvalues of the Laplacian.

2000 Mathematics Subject Classification: Primary 35P15, 53C20, 53C42, 58G25.

Key words: Universal bounds, eigenvalues, Schrödinger Operator with weight, spherical domains, minimal submanifolds, sphere, homogeneous space, complex projective space, complex hypersurfaces.

Reference to this article: Q. Wang and C. Xia: Universal bounds for eigenvalues of Schrödinger operator on Riemannian manifolds. Ann. Acad. Sci. Fenn. Math. 33 (2008), 319-336.

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