A criterion for Hill operators to be spectral operators of scalar type

Fritz Gesztesy, Vadim Trachenko

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

Abstract

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrö dinger 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 Schrödinger 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.

Original languageEnglish
Pages (from-to)287-353
Number of pages67
JournalJournal d'Analyse Mathematique
Volume107
Issue number1
DOIs
StatePublished - 1 Jan 2009

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