  In a series of papers in the last ten years, various aspects of the
mathematical foundations of the quantum theory of atoms in molecules have been
considered by this author and his coworkers in some details. Although these
considerations answered part of the questions raised by some critics on the
mathematical foundations of the quantum theory of atoms in molecules, however,
new mathematical problems also emerged during these studies that were reviewed
elsewhere [Int. J. Quantum Chem. 111, 4497 (2011)]. Beyond mathematical
subtleties of the formalism that were the original motivation for initial
exchanges and disputes, the questions raised by critics had a constructive
effect and prompted the author to propose a novel extension of the theory, now
called the multi-component quantum theory of atoms in molecules [Theor. Chem.
Acc. 132, 1365 (2013)]. Taking this background into account, in this paper a
new set of open problems is put forward that the author believes proper answers
to these questions, may open new doors for future theoretical developments of
the quantum theory of atoms in molecules. Accordingly, rather than emphasizing
on the rigorous mathematical formulation, the practical motivations behind
proposing these questions are discussed in detail and the relevant literature
are reviewed while when possible, evidence and routes to answers are also
provided. The author hopes that proposing these open questions as a compact
package may motivate more mathematically oriented people to participate in
future developments of the quantum theory of atoms in molecules and its
multi-component version.
