Rigidity, separability, and cusp conditions of a wave function

We introduce in quantum mechanics a concept of \textit{rigidity} and a concept of a \textit{pinned point} of a wave function. The concept of a pinned point is a generalization of a familiar concept in the description of a vibrating string, while the concept of rigidity is introduced to describe the sensitivity of a wave function to changes in energy, potential, and/or external perturbation. Through these concepts and their mathematical implications, we introduce and formulate cusp conditions and cusp functions as fundamental properties of an arbitrary $N$-body quantum system with $N\ge 2$, greatly expanding their relevance beyond the Coulombic systems. The theory provides rigorous constraints on an arbitrary $N$-body quantum system, specifically on its short-range pair correlation that is essential to a better understanding of strongly correlated systems. More broadly, the theory and the derivations presented here are part of a reconstruction of the mathematical and conceptual foundation of an $N$-body quantum theory, incorporating previously hidden properties and insights revealed through the concepts of rigidity and pinned points. It includes general analytic properties of a 2-body wave function versus energy and their relations to cusp conditions and cusp functions. It includes a rigorous derivation and an understanding, in terms of an emergent length scale, of the 2-particle separability in an $(N>2)$-body quantum system and its relations to cusp conditions. It also includes a classification of quantum systems, both 2-body and $N$-body, based on the universal behaviors in their short-range correlation.