Welcome to Zhennan Zhou's Homepage
My publications can be found here.
Semiclassical Schrödinger Equations: Analysis and computations.
The Schrödinger equation is a partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. After nondimensionalization, the Schrödinger equation is parameterized by the socalled semiclassical parameter, ε, which is reminiscent of the recaled Planck constant. As ε → 0, the quantum dynamics is converging to the classical dynamics in a rather weak sense, such that many interesting analysis questions and computational questions arise. I am interested in various questions in such a field: the spectral method, the wave packet approach, the surface hopping method, the Schrödinger equation with vector potentials or random coefficients, etc.
Ensemble average calculations in Quantum Chemistry.
Simulation of complex chemical system at the quantum level has always been a central and challenging task in the theoretical and computational chemistry. In a canonical ensemble of quantum particles, direct simulation of statistical properties, such as the thermal averages, are not numerically infeasible. The prevailing numerical methods for thermal average calculation are based on the ring polymer representation: with the imaginary time path integral techniques, it maps a quantum particle to a ring polymer consists of its replica on the classical phase space. I am interested in different mathematical aspects of the quantum thermal averages, such as the nonadiabatic problem, the continuum limit, the preconditioning techniques, the Bayesian inversion problem, etc.
Transport Equations in Biology.
In the microscopic level, various mathematical models in biology are associated with stochastic processes, and in the macroscopic level, these models are linked to the partial differential equations which exhibit distinct transport phenomenon. I take special interest in the chemotaxis models, the tumor growth models, the integrate-and-fire models for neuron networks, etc.
I am also interested in quite a few other mathematical problems, which are more or less connected to the trending problems in science and engineering: micromagnetics, the mean field games theory, waves in poro-viscoelastic media, etc.