Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Conservative L-systems and the Liv\v{s}ic function   We study the connection between the Liv\v{s}ic class of functions $s(z)$ that
are the characteristic functions of densely defined symmetric operators $\dot
A$ with deficiency indices $(1, 1)$, the characteristic functions $S(z)$ (the
M\"obius transform of $s(z)$) of a maximal dissipative extension $T$ of $\dot
A$ (determined by the von Neumann parameter $\kappa$ of the extension relative
to an appropriate basis in the deficiency subspaces) and the transfer functions
$W_\Theta(z)$ of a conservative L-system $\Theta$ with the main operator $T$.
It is shown that under a natural hypothesis $S(z)$ and $W_\Theta(z)$ are
reciprocal to each other. In particular, when $\kappa=0$,
$W_\Theta(z)=\frac{1}{S(z)}=-\frac{1}{s(z)}$. It is established that the
impedance function of a conservative L-system with the main operator $T$
coincides with the function from the Donoghue class if and only if the von
Neumann parameter vanishes ($\kappa=0$). Moreover, we introduce the generalized
Donoghue class and obtain the criteria for an impedance function to belong to
this class. All results are illustrated by a number of examples.

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The categories of this paper are: hep-lat
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: physics
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an