Semiclassical Spectrum of the Schrödinger Operator on a Geometric Graph

We study how to construct asymptotic solutions of the spectral problem for the Schrödinger equation on a geometric graph. Differential equations on sets of this type arise in the study of processes in systems that can be represented as a collection of one-dimensional continua interacting only via their endpoints (e.g., vibrations of networks formed by strings or rods, steady states of electrons in molecules, or acoustical systems). The interest in Schrödinger equations on networks has increased, in particular, owing to the fact that nanotechnology objects can be described by thin manifolds that can in the limit shrink to graphs (see [1]). The main result of the present paper is an algorithm for constructing quantization rules (generalizing the well-known Bohr–Sommerfeld quantization rules). We illustrate it with a number of examples. We also consider the problem of describing the kernels of the Laplace operator acting on $k$-forms defined on a network. Finally, we find the asymptotic eigenvalues corresponding to eigenfunctions localized at a vertex of the graph.