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Boundary-value problems for two-dimensional canonical systems

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Abstract

The two-dimensional canonical systemJy′=−ℓHy where the nonnegative Hamiltonian matrix functionH(x) is trace-normed on (0, ∞) has been studied in a function-theoretic way by L. de Branges in [5]–[8]. We show that the Hamiltonian system induces a closed symmetric relation which can be reduced to a, not necessarily densely defined, symmetric operator by means of Kac' indivisible intervals; of. [33], [34]. The “formal” defect numbers related to the system are the defect numbers of this reduced minimal symmetric operator. By using de Branges' one-to-one correspondence between the class of Nevanlinna functions and such canonical systems we extend our canonical system from (0, ∞) to a trace-normed system onℜ which is in the limit-point case at ±∞. This allows us to study all possible selfadjoint realizations of the original system by means of a boundaryvalue problem for the extended canonical system involving an interface condition at 0.

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Authors and Affiliations

  1. Department of Statistics, University of Helsinki, PL 54, 00014, Helsinki, Finland

    Seppo Hassi

  2. Department of Mathematics, University of Groningen, Postbus 800, 9700 AV, Groningen, Nederland

    Henk de Snoo

  3. Institut für Mathematische Stochastik, Technische Universität Dresden, D-01062, Dresden, Deutschland

    Henrik Winkler

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  1. Seppo Hassi
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  2. Henk de Snoo
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  3. Henrik Winkler
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Hassi, S., de Snoo, H. & Winkler, H. Boundary-value problems for two-dimensional canonical systems. Integr equ oper theory 36, 445–479 (2000). https://doi.org/10.1007/BF01232740

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