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2000-2001
MATHEMATICS COLLOQUIUM SERIES
“Special Solutions to a Differential - Difference Equation Describing Certain Self-similar Potentials.”
ABSTRACT:Let
be a Schroedinger operator that we factorize as
where
and
.
Then the function
satisfies the Riccati equation
.
If
denotes a new Schroedinger operator, obtained from
by permuting the operator factors
,
then the potential
of
is given by
.
The differential-difference equation
(1) 
,
where
and
are some complex constants, describes potentials of the operator
satisfying the self-similarity constraint
,
where
is the translation operator
.
We prove existence of solutions to (1) that are meromorphic in the
complex
-plane.