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

Abstract

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 L 0 in the Hilbert space ℒ m 2 ( J; w ) of all complex, m -dimensional vector-valued functions y on J satisfying with inner product where . All selfadjoint extensions of L 0 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.