## Order problem for canonical systems and a conjecture of Valent

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## Abstract:

We establish a sharp upper estimate for the order of a canonical system in terms of the Hamiltonian. This upper estimate becomes an equality in the case of Krein strings. As an application we prove a conjecture of Valent about the order of a certain class of Jacobi matrices with polynomial coefficients.## References

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## Additional Information

**R. Romanov**- Affiliation: Department of Mathematical Physics and Laboratory of Quantum Networks, Faculty of Physics, St. Petersburg State University, 198504, St. Petersburg, Russia
- Email: morovom@gmail.com
- Received by editor(s): September 22, 2014
- Received by editor(s) in revised form: February 9, 2015
- Published electronically: May 3, 2016
- Additional Notes: This work was supported in part by the Austrian Science Fund (FWF) project I 1536–N25, the Russian Foundation for Basic Research, grants 13-01-91002-ANF and 12-01-00215, and by Project SPbSU 11.38.263.2014
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 1061-1078 - MSC (2010): Primary 34L15, 47B36
- DOI: https://doi.org/10.1090/tran6686
- MathSciNet review: 3572264