Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices

Abstract

Let $l [y]$ be a formally selfadjoint differential expression of an even order on the interval $[0,b \rangle$, $b\leq \infty$, with operator coefficients, acting in a separable Hilbert space $H$. We introduce the concept of deficiency indices $n_{b\pm}$ of the expression $l$ at the point $b$ and show that in the case $\dim H=\infty$ any values of $n_{b\pm}$ 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 $[0,b \rangle$. 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 $b$.