Boundary triples and Weyl functions of symmetric operators

Selfadjoint extensions of a closed symmetric operator A in a Hilbert
space with equal deficiency indices were described by in the 30s by
J. von Neumann. Another approach, based on the notion of abstract boundary
triple originates to the works of J.W. Calkin and was developed by M.I. Visik,
G.Grubb, F.S.Rofe-Beketov, M.L.Gorbachuck, A.N.Kochubei and others.
By Calkin's approach all selfadjoint extensions of the symmetric operator A can
be parametrized via "multivalued" selfadjoint operators in an auxiliary Hilbert spaces.
Spectral properties of these extensions can be characterized in terms of the abstract
Weyl function, associated to the boundary triple. In the present talk some recent
developments in the theory of boundary triples will be presented. Applications to
boundary value problems for Laplacian operator in bounded domains with smooth and
rough boundaries will be discussed.