Abstract The notion of quasi boundary triples and their Weyl functions is reviewed and applied to self-adjointness and spectral problems for a class of elliptic, formally symmetric, second order partial differential expressions with variable coefficients on bounded domains.
Boundary triples and associated Weyl functions are a powerful and ef- fficient tool to parameterize the self-adjoint extensions of a symmetric operator and to describe their spectral properties. There are numerous papers applying boundary triple techniques to spectral problems for various types of ordinary differential operators in Hilbert spaces; see, e.g. [Behrndt and Langer, 2010; Behrndt, Malamud and Neidhardt, 2008; Behrndt and Trunk, 2007; Brasche, Malamud and Neidhardt, 2002; Brüning, Geyler and Pankrashkin, 2008; Derkach, Hassi and de Snoo, 2003; Gorbachuk and Gorbachuk, 1991; Derkach and Malamud, 1995; Karabash, Kostenko and Malamud, 2009; Kostenko and Malamud, 2010; Posilicano, 2008] and the references therein.
The abstract notion of boundary triples and Weyl functions is strongly inspired by Sturm-Liouville operators on a half-line and their Titchmarsh -Weyl coefficients. To make this more precise, let us consider the ordinary differential expression l = −D2 + q on the positive half-line ℝ+ = (0, ∞), where D denotes the derivative, and suppose that q is a real-valued L∞-function. The maximal operator associated with l in L2(ℝ+) is defined on the Sobolev space H2(ℝ+) and turns out to be the adjoint of the minimal operator S f = l(f), dom S =, where is the subspace of H2(ℝ+) consisting of functions f that satisfy the boundary conditions f(0) = f′(0) = 0.