Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Reflection principles for functions of Neumann and Dirichlet Laplacians
  on open reflection invariant subsets of $\mathbb{R}^d$   For an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to a
hyperplane and with positive part $\Omega_+$, we consider the Neumann/Dirichlet
Laplacians $-\Delta_{N/D,\Omega}$ and $-\Delta_{N/D,\Omega_+}$. Given a Borel
function $\Phi$ on $[0,\infty)$ we apply the spectral functional calculus and
consider the pairs of operators $\Phi(-\Delta_{N,\Omega})$ and
$\Phi(-\Delta_{N,\Omega_+})$, or $\Phi(-\Delta_{D,\Omega})$ and
$\Phi(-\Delta_{D,\Omega_+})$. We prove relations between the integral kernels
for the operators in these pairs, which in particular cases of
$\Omega_+=\mathbb{R}^{d-1}\times(0,\infty)$ and $\Phi_{t}(u)=\exp(-tu)$, $u
\geq 0$, $t>0$, were known as reflection principles for the Neumann/Dirichlet
heat kernels. These relations are then generalized to the context of symmetry
with respect to a finite number of mutually orthogonal hyperplanes.

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The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
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The categories of this paper are: alg-geom