Economical adjunction of square roots to groups

How large must an overgroup of a given group be in order to contain a square root of any element of the initial group? We give an almost exact answer to this question (the obtained estimate is at most twice worse than the best possible) and state several related open questions.Comment: 5 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V2: minor correction

Perturbation determinants for singular perturbations

For proper extensions of a densely defined closed symmetric operator with trace class resolvent difference the perturbation determinant is studied in the framework of boundary triplet approach to extension theory