My research interests are in the spectral theory of differential equations
and the associated function inequalities
that arise. Spectral theory of differential equations is an old subject
and is closely connected with problems
in physics and engineering. Its origins date back to the late 1700's
and early 1800's when eigenfunction
expansions were used to study heat flow and vibrating bodies. Much
modern motivation comes from quantum
mechanics. Here the study of observables requires a mathematical structure
of operators in a certain space.
The time evolution of the quantum mechanical system is governed by
differential equations acting in this space.
In these problems the coefficients of the operator are determined by
the physics of the problem, and spectral
theory attempts to answer certain questions from the behavior of the
coefficients. For example, do eigenvalues
exist (these correspond to frequencies of vibration or to bound states
in quantum mechanics), and what bounds
can be given for their location? Another question is the location of
continuous spectrum, when it exists. In
quantum mechanics continuous spectrum corresponds to energy levels
that give rise to free electrons.
I grew up in Savannah, Tennessee which is a small town in West Tennessee.
It sits on the banks of the Tennessee River and is near the battle of Shiloh.
I spent much of my early years fishing on town branch.
In high school I discovered that I liked mathematics (especially geometry),
and came to the University of
Tennessee to study engineering. After getting a B.S. in electrical
engineering, I decided that I would go to
graduate school in mathematics and received a Ph. D. from the University
of Tennessee. John Neuberger was
my advisor. I then spent six years teaching at the University of Georgia
before returning to the University of
Tennessee. I enjoy hiking and camping especially in the mountains.
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Last update: January, 2002