From: Fritz Gesztesy (fritz@math.missouri.edu)
Date: Wed Apr 04 2007 - 15:54:41 CDT
PDE Seminar
221 General Classroom Building
Thursday, April 26
1:00-1:50 p.m.
Mark M. Malamud (Donetsk, Ukraine)
On Completeness of root vectors of boundary value problems for
Sturm-Liouville equation with general boundary conditions.
Abstract: Consider the Sturm-Liouville equation
$$
-y" + qy =\lambda y$ (1)
$$
subject to the boundary conditions
$$
U_i(y)=0, i=1,2, (2)
$$
on a segment $[0,1]$ with a summbable complex-valued potential $q$.
We recall that a system $\{U_i\}_1^2$ of boundary conditions is called
nondegenerate if $\Delta_0 \not = const$, where $\Delta_0$ is the
characteristic determinant of of the unperturbed problem (1)-(2)
with $q=0$.
In this case the determinant $\Delta_0$ has infinitely many zeros,
that is, the problem (1), (2) with $q=0$ has infinitely many eigenvalues.
It is known that a system of root vectors of a nondegenerate boundary
value problem (1), (2) is complete in $L^2[0,1]$ for any summable $q$.
We consider the case of a degenerate problem (1), (2) ($\Delta_0
\not = const$)
assuming that the characteristic determinant $\Delta(\lambda)$ of
the problem (1), (2) with nonzero $q$ differs from a constant.
It turns out that in this case a system of root vectors of the
(perturbed) problem (1), (2) may be complete while the answer depends
on the potential.
We discuss necessary and sufficient conditions for a potential $q$ in
order that the system of roof vectors of the problem (1), (2) is
complete in $L^2[0,1]$.
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