Cauchy-type determinants and integrable systems |
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Authors: | Cornelia Schiebold |
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Institution: | Department of Mathematics, Mid Sweden University, S-851 70 Sundsvall, Sweden |
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Abstract: | It is well known that the Sylvester matrix equation AX + XB = C has a unique solution X if and only if 0 ∉ spec(A) + spec(B). The main result of the present article are explicit formulas for the determinant of X in the case that C is one-dimensional. For diagonal matrices A, B, we reobtain a classical result by Cauchy as a special case.The formulas we obtain are a cornerstone in the asymptotic classification of multiple pole solutions to integrable systems like the sine-Gordon equation and the Toda lattice. We will provide a concise introduction to the background from soliton theory, an operator theoretic approach originating from work of Marchenko and Carl, and discuss examples for the application of the main results. |
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Keywords: | 15A15 35Q51 37K40 37K30 |
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