Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A bound for the eigenvalue counting function for Krein--von Neumann and
  Friedrichs extensions   For an arbitrary open, nonempty, bounded set $\Omega \subset \mathbb{R}^n$,
$n \in \mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider
the closed, strictly positive, higher-order differential operator $A_{\Omega,
2m} (a,b,q)$ in $L^2(\Omega)$ defined on $W_0^{2m,2}(\Omega)$, associated with
the higher-order differential expression $$ \tau_{2m} (a,b,q) :=
\bigg(\sum_{j,k=1}^{n} (-i \partial_j - b_j) a_{j,k} (-i \partial_k -
b_k)+q\bigg)^m, \quad m \in \mathbb{N}, $$ and its Krein--von Neumann extension
$A_{K, \Omega, 2m} (a,b,q)$ in $L^2(\Omega)$. Denoting by $N(\lambda; A_{K,
\Omega, 2m} (a,b,q))$, $\lambda > 0$, the eigenvalue counting function
corresponding to the strictly positive eigenvalues of $A_{K, \Omega, 2m}
(a,b,q)$, we derive the bound $$ N(\lambda; A_{K, \Omega, 2m} (a,b,q)) \leq C
v_n (2\pi)^{-n} \bigg(1+\frac{2m}{2m+n}\bigg)^{n/(2m)} \lambda^{n/(2m)} , \quad
\lambda > 0, $$ where $C = C(a,b,q,\Omega)>0$ (with $C(I_n,0,0,\Omega) =
|\Omega|$) is connected to the eigenfunction expansion of the self-adjoint
operator $\widetilde A_{2m} (a,b,q)$ in $L^2(\mathbb{R}^n)$ defined on
$W^{2m,2}(\mathbb{R}^n)$, corresponding to $\tau_{2m} (a,b,q)$. Here $v_n :=
\pi^{n/2}/\Gamma((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in
$\mathbb{R}^n$.
  Our method of proof relies on variational considerations exploiting the
fundamental link between the Krein--von Neumann extension and an underlying
abstract buckling problem, and on the distorted Fourier transform defined in
terms of the eigenfunction transform of $\widetilde A_{2} (a,b,q)$ in
$L^2(\mathbb{R}^n)$.
  We also consider the analogous bound for the eigenvalue counting function for
the Friedrichs extension $A_{F,\Omega, 2m} (a,b,q)$ in $L^2(\Omega)$ of
$A_{\Omega, 2m} (a,b,q)$.
  No assumptions on the boundary $\partial \Omega$ of $\Omega$ are made.

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The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio