Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 75, pp. 1-6.
Title: An abstract approach to some spectral problems of direct sum
differential operators
Author: Maksim S. Sokolov (National Univ. of Uzbekistan)
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.
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.