## Certain Liouville properties of eigenfunctions of elliptic operators

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- by Ari Arapostathis, Anup Biswas and Debdip Ganguly PDF
- Trans. Amer. Math. Soc.
**371**(2019), 4377-4409 Request permission

## Abstract:

We present certain Liouville properties of eigenfunctions of second-order elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of second-order elliptic operators. These extend results of Y. Pinchover to the case of*nonsymmetric*operators of Schrödinger type. In particular, we provide an answer to an open problem posed by Pinchover in [

*Comm. Math. Phys.*272 (2007), pp. 75–84, Problem 5]. In addition, we prove a lower bound on the decay of positive supersolutions of general second-order elliptic operators in any dimension, and discuss its implications to the Landis conjecture.

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## Additional Information

**Ari Arapostathis**- Affiliation: Department of Electrical and Computer Engineering, The University of Texas at Austin, 2501 Speedway, EER 7.824, Austin, Texas 78712
- MR Author ID: 26760
- Email: ari@ece.utexas.edu
**Anup Biswas**- Affiliation: Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India
- MR Author ID: 866719
- Email: anup@iiserpune.ac.in
**Debdip Ganguly**- Affiliation: Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pashan, Pune 411008, India
- MR Author ID: 1060705
- Email: debdip@iiserpune.ac.in
- Received by editor(s): November 16, 2017
- Received by editor(s) in revised form: June 9, 2018, and June 25, 2018
- Published electronically: November 26, 2018
- Additional Notes: The research of the first author was supported in part by the Army Research Office through grant W911NF-17-1-001, in part by the National Science Foundation through grant DMS-1715210, and in part by Office of Naval Research through grant N00014-16-1-2956.

The research of the second author was supported in part by an INSPIRE faculty fellowship (IFA13/MA-32) and DST-SERB grant EMR/2016/004810.

The third author was supported in part at the Technion by a fellowship of the Israel Council for Higher Education and the Israel Science Foundation (Grant No. 970/15) founded by the Israel Academy of Sciences and Humanities. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 4377-4409 - MSC (2010): Primary 35J15; Secondary 35A02, 35B40, 35B60
- DOI: https://doi.org/10.1090/tran/7694
- MathSciNet review: 3917226