Electron. J. Diff. Eqns., Vol. 2003(2003), No. 75, pp. 1-6.

An abstract approach to some spectral problems of direct sum differential operators

Maksim S. Sokolov

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.

Submitted April 15, 2003. Published July 10, 2003.
Math Subject Classifications: 47B25, 47B37, 47A16, 34L05.
Key Words: Direct sum operators, cyclic vector, spectral representation, unitary transformation

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Maksim S. Sokolov
Mechanics and Mathematics Department
National University of Uzbekistan
Uzbekistan, Tashkent 700095
email: sokolovmaksim@hotbox.ru

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