Mathematical Analysis of the BCS-Bogoliubov Theory

Superconductivity is one of the historical landmarks in condensed matter physics. Since Onnes found out the fact that the electrical resistivity of mercury drops to zero below the temperature 4.2K in 1911, the zero electrical resistivity is observed in many metals and alloys. Such a phenomenon is called superconductivity, and the magnetic properties of superconductors as well as their electric properties are also astonishing. For example, the magnetic flux is excluded from the interior of a superconductor. This phenomenon was observed first by Meissner in 1933, and is called the Meissner effect. In 1957 Bardeen, Cooper and Schrieffer [1] proposed the highly successful quantum theory called the BCS theory. The superconducting state and the Hamiltonian they dealt with are called the BCS state and the BCS Hamiltonian, respectively. In 1958 Bogoliubov [2] obtained the results similar to those in the BCS theory using the canonical transformation called the Bogoliubov transformation. This theory is called the Bogoliubov theory. The ground state of the BCS Hamiltonian is discussed by several authors. In 1961 Mattis and Lieb [5] studied the wavefunction of the ground state of the BCS Hamiltonian under the condition that in the ground state, all the electrons in the neighborhood of the Fermi surface are paired. See Richardson [7] and von Delft [3] for the ground state of the BCS Hamiltonian without the condition just above. From the viewpoint of C∗-algebra, Gerisch and Rieckers [4] studied a class of BCS-models to show that there is a unique C∗-dynamical system for each BCS-model. In this paper, first, we reformulate the BCS-Bogoliubov theory of superconductivity from the viewpoint of linear algebra. We define the BCS Hamiltonian on C 2M , where M is a positive integer. We discuss selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous symmetry breaking. Beginning with the gap equation, we give the well-known expression for the BCS state and find the existence of an energy gap. We also show that the BCS state has a lower energy than the normal state. Second, we introduce a new superconducting state explicitly and show from the viewpoint of linear algebra that this new state has a lower energy than the BCS state. Third, beginning with our new gap equation, we show from the viewpoint of linear algebra that we arrive at the results similar to those in the BCS-Bogoliubov theory. See Watanabe [8] for more details. Let L, Kmax > 0 be large enough and let us fix them. For n1, n2, n3 ∈ Z, set