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An abstract approach to some spectral problems of direct sum differential operators. (English) Zbl 1045.47016
Author’s abstract: “In this paper, we study the common spectral properties of abstract self-adjoint direct sum operators, considered in a direct sum Hilbert space. Applications of such operators arise in the modelling of processes of multi-particle quantum mechanics, quantum field theory and, specifically, in multi-interval boundary problems of differential equations. We show that a direct sum operator does not depend in a straightforward manner on the separate operators involved. That is, on having a set of self-adjoint operators giving a direct sum operator, we show how the spectral representation for this operator depends on the spectral representations for the individual operators (the coordinate operators) involved in forming this sum operator. In particular, it is shown that this problem is not immediately solved by taking a direct sum of the spectral properties of the coordinate operators. Primarily, these results are to be applied to operators generated by a multi-interval quasi-differential system studied, in the earlier works of Ashurov, Everitt, Gesztezy, Kirsch, Markus and Zettl. The abstract approach in this paper indicates the need for further development of spectral theory for direct sum differential operators.”

MSC:
47B25 Linear symmetric and selfadjoint operators (unbounded)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47A16 Cyclic vectors, hypercyclic and chaotic operators
34L05 General spectral theory of ordinary differential operators
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