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Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators

Published online by Cambridge University Press:  20 November 2018

Calvin D. Ahlbrandt ,
Don B. Hinton  and
Roger T. Lewis
Calvin D. Ahlbrandt
Affiliation:
University of Missouri, Columbia, Missouri
Don B. Hinton
Affiliation:
University of Tennessee, Knoxville, Tennessee
Roger T. Lewis
Affiliation:
University of Alabama in Birmingham, Birmingham, Alabama
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1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l by

where y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfying

with inner product

where . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 1 , 01 February 1981 , pp. 229 - 246
Copyright
Copyright © Canadian Mathematical Society 1981

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