Abstract
The KdV hierarchy is a family of evolutions on a Schrödinger operator that preserves its spectrum. Canonical systems are a generalization of Schrödinger operators, that nevertheless share many features with Schrödinger operators. Since this is a very natural generalization, one would expect that it would also be straightforward to build a hierarchy of isospectral evolutions on canonical systems analogous to the KdV hierarchy. Surprisingly, we show that there are many obstructions to constructing a hierarchy of flows on canonical systems that obeys the standard assumptions of the KdV hierarchy. This suggests that we need a more sophisticated approach to develop such a hierarchy, if it is indeed possible to do so.
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Acknowledgements
The authors would like to deeply thank Christian Remling for valuable discussions. The second author would also like to thank Jessica Liang Yei Shan for proofreading help. The first author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03931764 and NRF-2019R1F1A1061300), the second author was supported by the Fundamental Research Grant Scheme from the Malaysian Ministry of Education (Grant Nos: FRGS/1/2018/STG06/XMU/02/1 and FRGS/1/2020/STG06/XMU/02/1) and two Xiamen University Malaysia Research Funds (Grant Nos: XMUMRF/2018-C1/IMAT/0001 and XMUMRF/2020-C5/IMAT/0011).
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Hur, I., Ong, D.C. Restrictions on the Existence of a Canonical System Flow Hierarchy. Integr. Equ. Oper. Theory 93, 40 (2021). https://doi.org/10.1007/s00020-021-02647-3
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DOI: https://doi.org/10.1007/s00020-021-02647-3