Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Weakly coupled bound state of 2D Schr\"odinger operator with
  potential-measure   We consider a self-adjoint two-dimensional Schr\"odinger operator
$H_{\alpha\mu}$, which corresponds to the formal differential expression \[
-\Delta - \alpha\mu, \] where $\mu$ is a finite compactly supported positive
Radon measure on ${\mathbb R}^2$ from the generalized Kato class and $\alpha
>0$ is the coupling constant. It was proven earlier that $\sigma_{\rm
ess}(H_{\alpha\mu}) = [0,+\infty)$. We show that for sufficiently small
$\alpha$ the condition $\sharp\sigma_{\rm d}(H_{\alpha\mu}) = 1$ holds and that
the corresponding unique eigenvalue has the asymptotic expansion $$
  \lambda(\alpha) = -(C_\mu + o(1))\exp\Big(-\tfrac{4\pi}{\alpha\mu({\mathbb
R}^2)}\Big), \qquad \alpha\rightarrow 0+, $$ with a certain constant $C_\mu >
0$. We obtain also the formula for the computation of $C_\mu$. The asymptotic
expansion of the corresponding eigenfunction is provided. The statements of
this paper extend Simon's results, see \cite{Si76}, to the case of
potentials-measures. Also for regular potentials our results are partially new.

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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int