Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An operator theoretical proof for the second-order phase transition in
  the BCS-Bogoliubov model of superconductivity   We first show some properties such as smoothness and monotone decreasingness
of the solution to the BCS-Bogoliubov gap equation for superconductivity.
Moreover we give the behavior of the solution with respect to the temperature
near the transition temperature. On the basis of these results, dealing with
the thermodynamic potential, we then show that the transition from a normal
conducting state to a superconducting state is a second-order phase transition
in the BCS-Bogoliubov model of superconductivity from the viewpoint of operator
theory. Here we have no magnetic field and we need to introduce a cutoff
$\varepsilon>0$, which is sufficiently small and fixed (see Remark
\ref{rmk:varepsilon}). Moreover we obtain the exact and explicit expression for
the gap in the specific heat at constant volume at the transition temperature.

10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin