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

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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 $b$.


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Article Information

TitleBoundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices
SourceMethods Funct. Anal. Topology, Vol. 15 (2009), no. 3, 280-300
MathSciNet MR2567312
CopyrightThe Author(s) 2009 (CC BY-SA)

Authors Information

Vadim Mogilevskii
Department of Calculus, Lugans'k National University, 2 Oboronna, Lugans'k, 91011, Ukraine


Citation Example

Vadim Mogilevskii, Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices, Methods Funct. Anal. Topology 15 (2009), no. 3, 280-300.


BibTex

@article {MFAT498,
    AUTHOR = {Mogilevskii, Vadim},
     TITLE = {Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {15},
      YEAR = {2009},
    NUMBER = {3},
     PAGES = {280-300},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=498},
}


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