On the multilevel internal structure of the asymptotic distribution of resonances

Abstract

We prove that the asymptotic distribution of resonances has a multilevel internal structure for the following classes of Hamiltonians H: Schrödinger operators with point interactions in ${\mathbb{R}}^3$, quantum graphs, and 1-D photonic crystals. In the case of $N\ge 2$ point interactions, the set of resonances $\Sigma (H)$ essentially consists of a finite number of sequences with logarithmic asymptotics. We show how the leading parameters $\mu$ of these sequences are connected with the geometry of the set $Y=\{y_j{\}}_{j=1}^N$ of interaction centers $y_j\in {\mathbb{R}}^3$. The minimal parameter ${\mu }^{min}$ corresponds to the sequences with `more narrow' and so more observable resonances. The asymptotic density of such narrow resonances is described by the multiplicity of ${\mu }^{\min }$, which occurs to be connected with the symmetries of Y and naturally introduces a finite number of classes of configurations of $Y$. In the case of quantum graphs and 1-D photonic crystals, the decomposition of $\Sigma (H)$ into a finite number of asymptotic sequences is proved under additional commensurability conditions. To address the case of a general quantum graph, we introduce families of special counting and asymptotic density functions for two types of curved complex strips. The obtained results and effects are compared with those of obstacle scattering.

32 pages. Typos are corrected. Minor inaccuracies are corrected in condition (C1) of Section 4.2 and in Example 4.7. References are updated