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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Weyl-Titchmarsh $M$-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators


Authors: Steve Clark and Fritz Gesztesy
Journal: Trans. Amer. Math. Soc. 354 (2002), 3475-3534
MSC (2000): Primary 34B20, 34E05, 34L40; Secondary 34A55
Published electronically: April 30, 2002
MathSciNet review: 1911509
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Abstract: We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on $\mathbb{R} $. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.


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Additional Information

Steve Clark
Affiliation: Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, Missouri 65409
Email: sclark@umr.edu

Fritz Gesztesy
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: fritz@math.missouri.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03025-8
PII: S 0002-9947(02)03025-8
Keywords: Weyl--Titchmarsh matrices, high-energy expansions, uniqueness results, trace formulas, Borg theorems, Dirac operators
Received by editor(s): February 15, 2002
Published electronically: April 30, 2002
Additional Notes: Supported in part by NSF grant INT-9810322.
Dedicated: Dedicated to F. V. Atkinson, one of the pioneers of this subject
Article copyright: © Copyright 2002 by the authors