Eigendecomposition of the Laplace-Beltrami operator is instrumental for a variety of applications from physics to data science. We develop a numerical method of computation of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a smooth bounded domain based on the relaxation to the Schr\"odinger operator with finite potential on a Riemannian manifold and projection in a special basis. We prove spectral exactness of the method and provide examples of calculated results and applications, particularly, in quantum billiards on manifolds.

翻译：Laplace-Beltrami操作员的Eigendecomfication有助于从物理学到数据科学的各种应用,我们根据对Schr\'doninger操作员的放松和特别的投射潜力有限的情况,在平滑的封闭域内开发一种计算Laplace-Beltrami操作员的电源值和元功能的数字方法。我们证明了该方法的光谱精确性,并提供了计算结果和应用的例子,特别是元件的量子板。