| Abstract: |
| Quantum graphs have long served as effective models for thin structures. In various applications, however, such as photonic crystals and dynamical systems, one encounters stratified structures that naturally motivate higher-dimensional analogs of graph-like models. This leads to the study of ``open book`` type geometries, where multiple smooth ``pages`` meet transversely along a common ``binding``. Such structures can be viewed as higher-dimensional stratified varieties equipped with differential operators.
This talk develops a mathematical framework for quantum open books and defines the associated differential operators. After introducing metric open book structures, we formulate natural junction conditions and identify a subclass of gluing data that ensures the Fredholm property under suitable ellipticity assumptions. We also determine conditions guaranteeing self-adjointness of the corresponding operator, in relation to well-known results from quantum graph theory.
From a broader perspective, these structures provide a multiscale modeling framework, in which lower-dimensional singular sets may appear as geometric interfaces between higher-dimensional components. This viewpoint connects the analysis of quantum open books with current developments in multiscale systems, particularly in settings where geometric, spectral, and analytic features interact across different dimensions. We also discuss several directions for future research. |
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