On the quantum number projection

Analytical derivation of number conserving BCS equation

“…Using the generalized Thouless' theorem [4] The evaluation of general matrix elements with rotated and non-rotated quasi-particles using a sort of general-ized Wick's theorem was discussed in [8] for the simple case of pairs of quasi-particles coupled to total angular momentum zero. In order to give a broad view of the present subject we repeat here the essential steps presented in this paper and the prescriptions presented in [9]. In order to evaluate the matrix elements in (5.6) we need essentially to evaluate matrix elements between rotated and non-rotated quasi-particles states of the type <F I H~... S,, SS~*..../,,*IF>.…”

Recursive Construction of the Bosonic Bogoliubov Vacuum State

“…In order to obtain the complete solution, one needs the quantum state transformations from the base states of the old operator to those of the new one. Ultimately, this is equivalent to obtain the relations between linear canonical transformations and the related unitary operators in Hilbert space, which were extensively studied for bosonic and fermionic operators by many authors [5,6,7].…”

On the solution of the number-projected Hartree-Fock-Bogolyubov equations

“…e ′ in (8) are the single-particle energies of the cranked-deformed potential (1) and v in (12)(13)(14) is the uncoupled antisymmetric matrix-element of the two-body delta-interaction…”

“…The integration in (8-11) over the gauge-angle has been performed using the GaussChebyshev quadrature method [12]. In this method, the integration over the gauge-angle is replaced by a summation.…”

“…In this method, the integration over the gauge-angle is replaced by a summation. It can be shown [12] that the optimal number of mesh-points in the summation which eliminates all the components having undesired particle numbers is given by…”