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Trace Formulas for a Complex KdV Equation

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Abstract

Faddeev and Zakharov determined the trace formulas for the KdV equation with real initial conditions in 1971. We reprove these results for the KdV equation with complex initial conditions. The Lax operator is a Schrödinger operator with complex-valued potentials on the line. The operator has essential spectrum on the half-line plus eigenvalues (counted with algebraic multiplicity) in the complex plane without the positive half-line. We determine series of trace formulas. Here we have a new term: a singular measure, which is absent for real potentials. Moreover, we estimate of sum of the imaginary part of eigenvalues plus the singular measure in terms of the norm of potentials. The proof is based on classical results about the Hardy spaces.

DOI 10.1134/S106192084010096

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Acknowledgments

EK is grateful to Sergei Kuksin (Paris) for discussions about the KdV equetion, to Ari Laptev (London) for discussions about the Schrödinger operators with complex potentials, and to Alexei Alexandrov (St. Petersburg) for discussions and useful comments about Hardy spaces.

Funding

Our study was supported by the RSF grant No 19-71-30002.

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  1. Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia

    E. Korotyaev

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Correspondence to E. Korotyaev.

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Korotyaev, E. Trace Formulas for a Complex KdV Equation. Russ. J. Math. Phys. 31, 112–131 (2024). https://doi.org/10.1134/S106192084010096

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