Decoupling of deficiency indices and applications to Schrodinger-type operators with possibly strongly singular potentials

作者：Gesztesy Fritz^*; Mitrea Marius; Nenciu Irina; Teschl Gerald

来源：Advances in Mathematics, 2016, 301: 1022-1061.

DOI：10.1016/j.aim.2016.08.008

• Strongly singular potentials
• Deficiency indices
• Self-adjointness

We investigate closed, symmetric L-2(R-n)-realizations H of Schrodinger-type operators (-Delta + V) C-0(infinity)(R-n\Sigma) whose potential coefficient V has a countable number of well-separated singularities on compact sets Sigma(j), j is an element of J, of n-dimensional Lebesgue measure zero, with J subset of N an index set and Sigma = boolean OR(j is an element of J) Sigma(j) We show that the defect, def (H-j), of H can be computed in terms of the individual defects, def (H-j), of closed, symmetric L-2(R-n)-realizations of (-Delta + V-j) C-0(infinity)(R-n\Sigma) with potential coefficient V-j localized around the singularity Sigma(j), j is an element of J, where V = Sigma(j is an element of J) V-j. In particular, we prove
def (H) = Sigma(j is an element of J) def (H-j),
including the possibility that one, and hence both sides equal oo. We first develop an abstract approach to the question of decoupling of deficiency indices and then apply it to the concrete case of Schrodinger-type operators in L-2(R-n). Moreover, we also show how operator (and form) bounds for V relative to H-0 = -Delta H-2(R-n) can be estimated in terms of the operator (and form) bounds of Vi, j is an element of J, relative to He. Again, we first prove an abstract result and then show its applicability to Schxfidinger-type operators in L-2(R-n). Extensions to second-order (locally uniformly) elliptic differential operators on R-n with a possibly strongly singular potential coefficient are treated as well.

• 出版日期2016-10-1