Strongly singular potentials and essential self-adjointness of singular elliptic operators in C0∞(R n⧹{0})

Abstract

The second-order singular elliptic differential operator T 0u : = 1 k {∑ D j(a jlD lu) + qu} with D j : = i∂ j + b j is considered on C 0 ∞(R n ⧹{0}) ⊂ L 2(R n ; k)( n ϵ N). Conditions in addition to the usual ones are imposed on the coefficients which make T 0 bounded from below but still allow rather “strong negative” singularities of q in the origin. The essential self-adjointness of such operators is proved with the help of a criterion due to Walter [22]. It is shown that the fundamental requirement of this criterion, namely, the operator inequality T 0 ⩾ (1 + δ) e 2 σ ( δ > 0, σ a suitable function exhausting R n ⧹{0}) is closely connected with a, in special cases, well-known inequality of Hardy [7]. (A systematic presentation of generalized Hardy's inequalities with or without boundary terms is given which is based on the behavior of ∫ dt t n − 1p(t) [ p smallest eigenvalue of ( jl )] near 0 or ∞.) In the special case a jl = ¦ x ¦ 2μδ jl, q ⩾ β ¦ x ¦ −(2−2μ) (μ < 1) the essential self-adjointness of T 0 is established for β > (1 − μ) 2 − [ (n − 2 + 2μ) 2 ] 2 the constant being the best possible. Setting μ = 0 this improves a result due to Jörgens [9; cf. also his lectures delivered at the Univ. of Colorado, 1970] who obtained β > 1. For the general one-dimensional Sturm-Liouville operator, a new criterion for the limit point case at x − 0 complementing known criteria of Sears and Friedrichs [Dunford-Schwartz, “Linear Operators II,” p. 1605f.] is obtained. (At the same time an erroneous assumption loc. cit. concerning Friedrichs' criterion is corrected.)