Small perturbations of canonical systems
The author considers a singular two-dimensional canonical system \(Jy' = -zHy\) on \([0,\infty)\) such that at \(\infty\) Weyl's limit point case holds. Here, \(H\) 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 \(\text{tr }H = 1\) and the class of Nevanlinna functions. In this note the author shows how the Hamiltonian \(H\) of a canonical system changes if its Titchmarsh-Weyl coefficient or the corresponding spectral measure undergoes certain small perturbations. This generalizes results of \textit{H. Dym} and \textit{N. Kravitsky} [Integral Equations Oper. Theory 1, 270-277 (1978; Zbl 0394.45002)] for so-called vibrating strings, in particular a generalization of a construction principle of \textit{I. M. Gelfand} and \textit{B. M. Levitan} [Am. Math. Soc., Translat., Ser. II 1, 253-304 (1955; Zbl 0066.33604)] can be shown.
singular two-dimensional canonical system
scientific article; zbMATH DE number 1542411
scientific article