My main interest is to understand which of the time honored methods in potential recovery for Schrödinger equations work also for complex potentials. At the same time I work on extensions of such methods to discrete problems and problems on trees. I am also interested in practical implementations of inverse problem algorithms.
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Differential Equations in the Complex Domain
Suppose the coefficient q in the differential equation y''+qy=zy
is a meromorphic function of the independent variable. The requirement that,
for every complex number z, every solution of the equation is also
meromorphic is a very strong one. I am interested in understanding the
implications of this requirement. There are connections to algebraic geometry,
to spectral theory and inverse problems, and to the Korteweg-de Vries hierarchy
and integrable Hamiltonian systems. For instance, if q is an elliptic
function the above requirement forces q to be a solution of some
stationary equation in the Korteweg-de Vries (KdV) hierachy. Stationary
KdV equations may be represented by the equation [P,L]=0 where P
and L are a Lax pair of ordinary linear differential expressions which
are of order 2g+1 and 2, respectively and satisfy the algebraic
relation P^2=R(L) where R is a polynomial of degree 2g+1.
Abelian Functions
Closely related to the above topic is my interest in abelian functions,
in particular hyperelliptic functions. Abelian functions are inverses of
abelian integrals, i.e., integrals of rational functions of x and
an algebraic function of x. If this algebraic function is the square
root of a polynomial one is looking (in general) at hyperelliptic integrals
and hyperelliptic functions. If the degree of the polynomial is 3 or 4
one has elliptic integrals and elliptic functions while for degree 1 or
2 the integrals may be solved in terms of inverses of trigonometric functions.
Naturally the study of abelian functions involves the algebraic curve
(or Riemann surface) associated with the algebraic function under consideration
and this relationship leads back to KdV equations and generalizations thereof.
I was born in 1958 and received my doctoral degree (Dr.rer.nat.) in 1987 from the Technische Universität Braunschweig in Germany. I worked as an assistant there from 1987 through 1990. In September 1990 I came to UAB as an Assistant Professor. I became an Associate Professor in 1994 and a Professor in 1999.
