Dear Colleagues,

We continue our Colloquium at the Math Department

Amin Boumenir

Department of Mathematics

State University of West Georgia, Carrollton, GA, USA

Friday, November 7, EMCS 211, 1:00pm. (Attn: Friday, not our standard
Th; 1:00, not our standard 2:00pm.)

The Recovery of Analytical Potentials

Abstract. In 1951, I. M. Gelfand and B. M. Levitan have solved the
inverse spectral problem for the Strum-Liouville

$$-y''(x)+q(x)y(x)=\lambda y(x) for x\in [0,\infty)$$

by integral methods. A year later, M.G. Krein solved the inverse problem
for the string operator

$$-y''(x)=\lambda w(x)y(x) for x\in [0,L)$$

using the theory of entire functions and Stieltjes continuous fractions.
In fact he had developed a set of rules, allowing him to recover the
density $w(x)$  explicitly. Twenty years later Debranges, proved the
uniqueness for the string.

Since then, the inverse problem community has wondered whether we can
borrow M.G. Krein's methods to recover $q(x)$. A brief comparison of
both methods from the operator theoretic view point will summarize the
first part of the talk.

In the second part, we introduce a new method which reconstructs
analytical potentials, using the method of coefficient identification.
Among its advantages, is the possibility of working with complex
eigenvalues and boundary conditions depending on the spectral parameter.
We show how to recover an analytic potential from two spectra. The proof
involves solving recurrence relations using combinatorics and Roman
calculus.

Numerical experiments, convergence and open problems will be discussed
at the end.

The presentation will be of special interest for physicists and
engineers. Students (especially math majors) are encouraged to attend.

        Boris P Belinskiy

        Department of Mathematics, Dept. 6956

        University of Tennessee at Chattanooga

        Ph. (423) 425-4748