- 06-289 Fritz Gesztesy and Vadim Tkachenko
- A Criterion for Hill Operators to be Spectral Operators of Scalar Type
Oct 16, 06
(auto. generated ps),
of related papers
Abstract. We derive necessary and sufficient conditions for a Hill operator
(i.e., a one-dimensional periodic Schr\"odinger operator)
$H=-d^2/dx^2+V$ to be a spectral operator of scalar type. The
conditions show the remarkable fact that the property of a Hill
operator being a spectral operator is independent of smoothness (or
even analyticity) properties of the potential $V$. In the course of
our analysis we also establish a functional model for periodic
Schr\"odinger operators that are spectral operators of scalar type
and develop the corresponding eigenfunction expansion.
The problem of deciding which Hill operators are spectral operators
of scalar type appears to have been open for about 40 years.