Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Hopf lemma for the Schr\"odinger operator   We prove the Hopf boundary point lemma for solutions of the Dirichlet problem
involving the Schr\"odinger operator $- \Delta + V$ with a nonnegative
potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(\Omega)$. More
precisely, if $u \in W_0^{1, 2}(\Omega) \cap L^2(\Omega; V \mathrm{d}x)$
satisfies $- \Delta u + V u = f$ on $\Omega$ for some nonnegative datum $f \in
L^\infty(\Omega)$, $f \not\equiv 0$, then we show that at every point $a \in
\partial\Omega$ where the classical normal derivative $\partial u(a) / \partial
n$ exists and satisfies the Poisson representation formula, one has $\partial
u(a) / \partial n > 0$ if and only if the boundary value problem $$
\begin{cases} \begin{aligned} - \Delta v + V v &= 0 && \text{in $\Omega$,} \\ v
&= \nu && \text{on $\partial\Omega$,} \end{aligned} \end{cases} $$ involving
the Dirac measure $\nu = \delta_a$ has a solution. More generally, we
characterize the nonnegative finite Borel measures $\nu$ on $\partial\Omega$
for which the boundary value problem above has a solution in terms of the set
where the Hopf lemma fails.

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The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph