Friedrichs extensions of Schroedinger operators with singular potentials

von Keviczky, Attila B.; Saad, Nasser; Hall, Richard L.

Friedrichs extensions of Schroedinger operators with singular potentials

Authors

von Keviczky, Attila B.

Saad, Nasser

Hall, Richard L.

Description

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}).
16 pages