A canonical system is a kind of first-order system of ordinary differential equations on an interval of the real line parametrized by complex numbers. It is known that any solution of a canonical system generates an entire function of the Hermite-Biehler class. In this paper, we deal with the inverse problem to recover a canonical system from a given entire function of the Hermite-Biehler class satisfying appropriate conditions. This inverse problem was solved by de Branges in 1960s. However his results are often not enough to investigate a Hamiltonian of recovered canonical system. In this paper, we present an explicit way to recover a Hamiltonian from a given exponential polynomial belonging to the Hermite-Biehler class. After that, we apply it to study distributions of roots of self-reciprocal polynomials.
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This work was supported by KAKENHI (Grant-in-Aid for Young Scientists (B)) No. 21740004 and No. 25800007 and by French-Japanese Projects “Zeta Functions of Several Variables and Applications” in Japan-France Research Cooperative Program supported by JSPS and CNRS.
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Suzuki, M. An inverse problem for a class of canonical systems and its applications to self-reciprocal polynomials. JAMA 136, 273–340 (2018). https://doi.org/10.1007/s11854-018-0061-8