We study the large eigenvalue asymptotics for the Schrödinger operator HV=-1/2Δ+V on the real and the complex hyperbolic n-spaces. Here Δ 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 HV 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].
All Science Journal Classification (ASJC) codes
- Applied Mathematics