Eigenvalue asymptotics for the Schrödinger operators on the real and the complex hyperbolic spaces

✍ Scribed by Yuzuru Inahama; Shin-Ichi Shirai

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 329 KB
• Volume: 83
• Category: Article
• ISSN: 0021-7824
• DOI: 10.1016/j.matpur.2004.01.005

✦ Synopsis

We study the large eigenvalue asymptotics for the Schrödinger operator H V = -1 2 + V on the real and the complex hyperbolic n-spaces. Here is the Laplace-Beltrami operator and V is a scalar potential. We assume that V is real-valued, continuous, semi-bounded from below and diverges at infinity in an appropriate sense. Then it is proven that the number of eigenvalues of H V less than λ behaves semi-classically as λ ∞. This is a natural generalization of the result obtained by Inahama and Shirai [Eigenvalue asymptotics for the Schrödinger operators on the hyperbolic plane, J. Funct. Anal., submitted for publication].