Abstract
We consider a singular two-dimensional canonical systemJy′=−zHy on [0, ∞) such that at ∞ Weyl's limit point case holds. HereH is a measurable, real and nonnegative definite matrix function, called Hamiltonian. From results of L. de Branges it follows that the correspondence between canonical systems and their Titchmarsh-Weyl coefficients is a bijection between the class of all Hamiltonians with trH=1 and the class of Nevanlinna functions. In this note we show how the HamiltonianH of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of H. Dym and N. Kravitsky for so-called vibrating strings, in particular a generalization of a construction principle of I.M. Gelfand and B.M. Levitan can be shown.
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Winkler, H. Small perturbations of canonical systems. Integr equ oper theory 38, 222–250 (2000). https://doi.org/10.1007/BF01200125
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DOI: https://doi.org/10.1007/BF01200125