15 research outputs found
An operator-theoretical treatment of the Maskawa-Nakajima equation in the massless abelian gluon model
The Maskawa-Nakajima equation has attracted considerable interest in
elementary particle physics. From the viewpoint of operator theory, we study
the Maskawa-Nakajima equation in the massless abelian gluon model. We first
show that there is a nonzero solution to the Maskawa-Nakajima equation when the
parameter satisfies . Moreover, we show that the solution
is infinitely differentiable and strictly decreasing. We thus conclude that the
massless abelian gluon model generates the nonzero quark mass spontaneously and
exhibits the spontaneous chiral symmetry breaking when . We next
show that there is a unique solution to the Maskawa-Nakajima equation when
, from which we conclude that each quark remains massless and that
the model realizes the chiral symmetry when .Comment: 10 page
The solution to the BCS gap equation and the second-order phase transition in superconductivity
The existence and the uniqueness of the solution to the BCS gap equation of
superconductivity is established in previous papers, but the temperature
dependence of the solution is not discussed. In this paper, in order to show
how the solution varies with the temperature, we first give another proof of
the existence and the uniqueness of the solution and point out that the unique
solution belongs to a certain set. Here this set depends on the temperature
. We define another certain subset of a Banach space consisting of
continuous functions of both and . Here, stands for the kinetic
energy of an electron minus the chemical potential. Let the solution be
approximated by an element of the subset of the Banach space above. We second
show, under this approximation, that the transition to a superconducting state
is a second-order phase transition.Comment: Journal of Mathematical Analysis and Applications, in pres
The solution to the BCS gap equation for superconductivity and its temperature dependence
From the viewpoint of operator theory, we deal with the temperature
dependence of the solution to the BCS gap equation for superconductivity. When
the potential is a positive constant, the BCS gap equation reduces to the
simple gap equation. We first show that there is a unique nonnegative solution
to the simple gap equation, that it is continuous and strictly decreasing, and
that it is of class with respect to the temperature. We next deal with
the case where the potential is not a constant but a function. When the
potential is not a constant, we give another proof of the existence and
uniqueness of the solution to the BCS gap equation, and show how the solution
varies with the temperature. We finally show that the solution to the BCS gap
equation is indeed continuous with respect to both the temperature and the
energy under a certain condition when the potential is not a constant.Comment: In this paper we can set $\varepsilon=0
Persistence of translational symmetry in the BCS model with radial pair interaction
We consider the two-dimensional BCS functional with a radial pair
interaction. We show that the translational symmetry is not broken in a certain
temperature interval below the critical temperature. In the case of vanishing
angular momentum our results carry over to the three-dimensional case.Comment: 17 pages, 1 figur
A lower bound for the BCS functional with boundary conditions at infinity
We consider a many-body system of fermionic atoms interacting via a local
pair potential and subject to an external potential within the framework of BCS
theory. We measure the free energy of the whole sample with respect to the free
energy of a reference state which allows us to define a BCS functional with
boundary conditions at infinity. Our main result is a lower bound for this
energy functional in terms of expressions that typically appear in
Ginzburg-Landau functionals.Comment: 32 page