Friedrichs extensions of Schrödinger operators with singular potentials

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d2/dx2 +Bx2 +Ax−2 +λx−α (B >0, A 0) in L2(0,∞) is studied.We look at two different domains of definition for each of these differential operators in L2(0,∞), namely C∞0 (0,∞) and D(T2,F )∩D(Mλ,α), where the latter is a subspace of the Sobolev spaceW2,2(0,∞). Adjoints of these differential operators on C∞0 (0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C∞0 (0,∞) in D(T2,F ) ∩ D(Mλ,α) in L2(0,∞) lead to the other adjoints. Further density properties C∞0 (0,∞) in D(T2,F )∩ D(Mλ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T2,F ) ∩ D(Mλ,α).  2004 Elsevier Inc. All rights reserved.