Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral analysis and stabilization of the dissipative Schr\"{o}dinger
  operator on the tadpole graph   We consider the damped Schr\"odinger semigroup $e^{-it \frac{d^2}{dx^2}}$ on
the tadpole graph ${\mathcal R}$. We first give a careful spectral analysis and
an appropriate decomposition of the kernel of the resolvent. As a consequence
and by showing that the generalized eigenfunctions form a Riesz basis of some
subspace of $L^2({\mathcal R})$, we prove that the corresponding energy decay
exponentially.

21
The categories of this paper are: adap-org
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th