Principal Investigator: Alejandro Uribe
The principal investigator will study aspects of the relationship between the semiclassical behavior (i.e. as Planck's constant tends to zero) of quantum-mechanical objects with their geometric counterpart in phase space. More specifically, problems to be investigated include: Direct and inverse spectral problems for Schrodinger operators in various semi-classical settings, rigorous analyses of semiclassical approximations commonly used in the physics and quantum chemistry communities (Bohr-Sommerfeld conditions, approximate propagators and computation of correlation functions), as well as quantum-mechanical analogies of constructions in symplectic geometry (like symplectic cutting). The settings will be both classical (when the phase space is Euclidean) and more geometric (e.g. when the phase space is a compact Kahler manifold or the cotangent bundle of a configuration space).
The relationship between mathematics and physics is profoundly important for both the theory and applications of these subjects. In one direction, mathematics is the language of physics and its methods can have powerful applications; in the other, physical problems often give rise to new mathematical problems that can be extremely fruitful. The proposed research will study problems at the interface between quantum mechanics and geometry. The problems are generally of two kinds: spectral, meaning related to the energy of a quantum system, and dynamical, comparing the evolution of a quantum system with its classical counterpart. The spectral problems seek to understand the relationship between the possible energy levels of a quantum system and the geometry of the corresponding classical system. The dynamic problems should result in a better understanding of the underpinnings of some popular computational methods in quantum chemistry.
