The kernel of Laplace-Beltrami operators

A. A. Tolchennikov

Sbornik. Mathematics, Tome 199 (2008) no. 7, pp. 1071-1087

Résumé

An isomorphism is described for the kernel of the Laplace operator $\Delta^{\!\Lambda}$ (determined by a Lagrangian plane $\Lambda\subset\mathbb C^k\oplus\mathbb C^k$) with potential $\sum_{j=1}^kc_j\delta_{q_j}(x)$ on a manifold. The isomorphism is given by $\Gamma\colon\ker\Delta^{\!\Lambda}\to\Lambda\cap\nobreak L$, where $L$ is an (explicitly calculated) Lagrangian plane. A similar isomorphism also holds for the Laplace operator on a decorated graph. The inequality $1\le\dim\ker\Delta^{\!\Lambda_0}\le n-v+2$ is established for the Laplace operator $\Delta^{\!\Lambda_0}$ on a decorated graph (obtained by decorating a connected finite graph with $n$ edges and $v$ vertices) with ‘continuity’ conditions. It is also shown that the quantity $n-v+1-\dim\ker\Delta^{\!\Lambda_0}$ does not decrease when new edges or manifolds are added. Bibliography: 12 titles.