Vol. 55, No. 1, 1974

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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

Let H be the Hilbert space of complex vector-valued functions f : [a,) C2 such that f is Lebesgue measurable on [a,) and af(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

D = {f𝜖H : f,f′are absolutely continuous on compact
subintervals of [ a,∞ ),f has compact support
interior to [ a,∞ ) and c(f)𝜖H }.

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
Received: 22 February 1974
Revised: 8 October 1974
Published: 1 November 1974
Robert Lee Anderson