Georgia Institute of Technology College of Sciences

Energetic stability of matter in quantum mechanics, which refers to the ques-
tion of whether the ground state energy of a many-body quantum mechanical
system is finite, has long been a deep question of mathematical physics. For a
system of many non-relativistic electrons interacting with many nuclei in the
absence of electromagnetic fields this question traces back to the seminal work
of Tosio Kato in 1951 and Freeman Dyson and Andrew Lenard in 1967/1968.
In particular, Dyson and Lenard showed the ground state energy of the many-
body Schrödinger Hamiltonian is bounded below by a constant times the total
particle number, regardless of the size of the nuclear charges. This says such a
system is energetically stable (of the second kind). This situation changes dra-
matically when electromagnetic fields and spin interactions are present in the
problem. Even for a single electron with spin interacting with a single nucleus
of charge $Z > 0$ in an external magnetic field, Jurg Fröhlich, Elliot Lieb, and
Michael Loss in 1986 showed that there is no ground state energy if $Z$ exceeds
a critical charge $Z_c$ and the ground state energy exists if $Z < Z_c$ . In other
words, if the nuclear charge is too large, the one-electron atom is energetically
unstable.

Another notion of stability in quantum mechanics is that of dynamic stabil-
ity, which refers to the question of global well-posedness for a system of partial
differential equations that models the dynamics of many electrons coupled to
their self-generated electromagnetic field and interacting with many nuclei. The
central motivating question of our PhD thesis is whether energetic stability has
any influence over dynamic stability. Concerning this question, we study the
quantum mechanical many-body problem of $N \geq 1$ non-relativistic electrons with
spin interacting with their self-generated classical electromagnetic field and $K \geq 0$
static nuclei. We model the dynamics of the electrons and their self-generated
electromagnetic field using the so-called many-body Maxwell-Pauli equations.
The main result presented is the construction time global, finite-energy, weak
solutions to the many-body Maxwell-Pauli equations under the assumption that
the fine structure constant $\alpha$ and the nuclear charges are sufficiently small to
ensure energetic stability of this system. This result represents an initial step
towards understanding the relationship between energetic stability and dynamic
stability. If time permits, we will discuss several open problems that remain.

Committee members: Prof. Michael Loss (Advisor, School of Mathematics,
Georgia Tech), Prof. Brian Kennedy (School of Physics, Georgia Tech), Prof.
Evans Harrell (School of Mathematics, Georgia Tech), Prof. Federico Bonetto
(School of Mathematics, Georgia Tech), Prof. Chongchun Zeng (School of Math-
ematics, Georgia Tech).