Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices - Mogilevskii, V. I. (Methods of Functional Analysis and Topology, 2009)
Let l [y] be a formally selfadjoint differential expression of an even order on the interval [0,b , b≤ ∞, with operator coefficients, acting in a separable Hilbert space H. We introduce the concept of deficiency indices n_b± of the expression l at the point b and show that in the case H=∞ any values of n_b± are possible. Moreover the decomposing selfadjoint boundary conditions exist if and only if n_b+=n_b-. Our considerations of differential operators with arbitrary (possibly unequal) deficiency indices are based on the concept of a decomposing D-boundary triplet. Such an approach enables to describe extensions of the minimal operator directly in terms of operator boundary conditions at the ends of the interval [0,b . In particular we describe in a compact form selfadjoint decomposing boundary conditions. Associated to a D-triplet is an m-function, which can be regarded as a gene alization of the classical characteristic (Titchmarsh-Weyl) function. Our definition enables to describe all m-functions (and, therefore, all spectral functions) directly in terms of boundary conditions at the right end b.