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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm148, 23 pages, doi:10.1093/imrn/rnm148 published on January 3, 2008
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Copyright © The Author 2008. Published by Oxford University Press.

Complex Zeros of Eigenfunctions of 1D Schrödinger Operators

Hamid Hezari

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

Correspondence: Correspondence to be sent to: hhezari{at}math.jhu.edu

In this article we study the semiclassical distribution of the complex zeros of the eigenfunctions of the 1D Schrödinger operators for the class of real polynomial potentials of even degree, with fixed energy level, E. We show that as hn -> 0 the zeros tend to concentrate on the union of some level curves R (S (zm , z)) = cm where Formula is the complex action, and zm is a turning point. We also calculate these curves for some symmetric and nonsymmetric one-well and double-well potentials. The example of the nonsymmetric double-well potential shows that we can obtain different pictures of complex zeros for different subsequences of hn.


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