On the characterization of the Friedrichs extension of ordinary or elliptic differential operators with a strongly singular potential

Abstract

The second-order singular elliptic differential operator T 0u ≔ 1 k {∑ D s(a stD tu) + qu} with D s ≔ i∂ s + b s is considered on C 0 ∞( G) ⊂ L 2( G; k) where G ≔ {x ¦ x ϵ R n, 0 ⩽ 1 < ¦ x ¦ < m ⩽ ∞, n ⩾ 2} . (The general one-dimensional Sturm-Liouville operator is dealt with on an arbitrary interval ( l, m) of the real line.) Conditions in addition to the usual ones are imposed on the coefficients which make T 0 bounded from below (but still allow strong negative singularities of q at ∂G) so that it possesses a Friedrichs extension T F with domain D(T F) = {u ¦ u ϵ D(T 0 ∗), there exists a sequence {u j} ⊂ C 0 ∞(G) such that ∥u j − u ∥ → 0 and (T 0(u j − u j′), u j − u j′) → 0 as j, j′ → ∞} . (1) Unifying and at the same time simplifying and generalizing ideas to be found in the work of Friedrichs [2, 3] and Kato [19] for the special case a st = δ st , b s = 0, k = 1 it is shown that (1) can be characterized by D(T F) = {u ¦ u ϵ H loc 2(G) ∩ L 2(G; k), ∫ G ∑ a st(D su) ( ̄ D tu) dx < ∞, T 0u ϵ L 2(G; k)} . (2) If ∫ dt t n − 1 a(t) (a(t) smallest eigenvalue of ( a st )) converges at l or m the boundary condition lim inf r→l+ m− ∫ |ξ|=1 |u(rξ)| 2dw n=0 (to be imposed on the “distinguished” representatives of the equivalence classes u ϵ H loc 2( G)) has to be added to (2). Equation (2) is derived without recourse to the theory of sesquilinear forms in Hilbert space so that the condition q + 1 2 u ϵ L 2(G) ( q + positive part of q), which has to be assumed from the start when forms are considered (as Friedrichs and Kato do), can entirely be dispensed with. It is shown that it is a consequence of the other conditions to be imposed on u.