A Criterion for Hill Operators to be Spectral Operators of Scalar Type

TL;DR

This paper establishes necessary and sufficient conditions for Hill operators to be spectral operators of scalar type, revealing that smoothness of the potential is not a determining factor, and develops a functional model and eigenfunction expansion.

Contribution

It provides the first complete characterization of Hill operators as spectral operators of scalar type, independent of potential smoothness, and introduces a functional model and eigenfunction expansion.

Findings

Spectral property is independent of potential smoothness.

Established a functional model for spectral Hill operators.

Developed an eigenfunction expansion for these operators.

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.