Friedrichs extensions of Schroedinger operators with singular potentials

Abstract

The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator -D^2+ Bx^2 + Ax^{-2} + lambda x^{-alpha} (B>0, A >= 0) in L_2(0, infinity) is studied. We look at two different domains of definition for each of these differential operators in L_2(0, infinity), namely C_0^infinity(0, infinity) and D(T_{2,F})\cap D(M_{lambda, alpha}), where the latter is a subspace of the Sobolev space W_{2,2}(0, infinity). Adjoints of these differential operators on C_0^infinity(0,infinity) 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^\infinity(0,\infinity) in D(T_{2,F})\cap D(M_{λ, α}) in L_2(0,\infinity) lead to the other adjoints. Further density properties C_0^infinity(0,infinity) on D(T_{2,F})\cap D(M_{λ, α}) yield the Friedrichs extension of these differential operators with domains of definition D(T_{2,F})\cap D(M_{lambda, alpha}).