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

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Fritz Gesztesy

  2. Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, 84105, Israel

    Vadim Trachenko

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  1. Fritz Gesztesy
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  2. Vadim Trachenko
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Correspondence to Fritz Gesztesy.

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Based upon work supported by the National Science Foundation under Grant No. DMS-0405526 and the Israel Science Foundation under Grant No. 186/01.

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Gesztesy, F., Trachenko, V. A criterion for Hill operators to be spectral operators of scalar type. J Anal Math 107, 287–353 (2009). https://doi.org/10.1007/s11854-009-0012-5

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