A Direct Proof for an Eigenvalue Problem by Counting Lagrangian Submanifolds

摘要

We focus on one of the Schr?dinger operators called the Bochner-Laplacian. Using Jensen's Formula and Vandermonde convolution, we show directly that for each k = 0, 1, 2,..., the number of Lagrangian submanifolds which satisfy the Maslov quantization condition is just equal to the multiplicity of the kth eigenvalue of the operator.