Continuous spectra of a singular symmetric differential operator on a Hilbert space of vector-valued functions

Robert Lee Anderson

Vol. 55 (1974), No. 1, 1–7

DOI: 10.2140/pjm.1974.55.1

Abstract

Let H be the Hilbert space of complex vector-valued functions f : a,∞ → C² such that f is Lebesgue measurable on a,∞ and ∫ _a^∞f^∗(s)f(s)ds < ∞. Consider the formally self adjoint expression c(y) = −y′′ + Py on a,∞, where y is a 2-vector and P is a 2 × 2 symmetric matrix of continuous real valued functions on a,∞. Let D be the linear manifold in H defined by

Then the operator L defined by L(y) = f(y),y𝜖D, is a real symmetric operator on D. Let L₀ be the minimal closed extension of L. For this class of minimal closed symmetric operators this paper determines sufficient conditions for the continuous spectrum of self adjoint extensions to be the entire real axis. Since the domain, D₀, of L₀ is dense in H, self adjoint extensions of L₀ do exist.

Mathematical Subject Classification 2000

Primary: 47E05

Milestones

Received: 22 February 1974

Revised: 8 October 1974

Published: 1 November 1974