The effective-interaction theory has been one of the useful and practical methods for solving nuclear many-body problems based on the shell model. Various approaches have been proposed which are constructed in terms of the so-called $widehat{Q}$ box
and its energy derivatives introduced by Kuo {it et al}. In order to find out a method of calculating them we make decomposition of a full Hilbert space into subspaces (the Krylov subspaces) and transform a Hamiltonian to a block-tridiagonal form. This transformation brings about much simplification of the calculation of the $widehat{Q}$ box. In the previous work a recursion method has been derived for calculating the $widehat{Q}$ box analytically on the basis of such transformation of the Hamiltonian. In the present study, by extending the recursion method for the $widehat{Q}$ box, we derive another recursion relation to calculate the derivatives of the $widehat{Q}$ box of arbitrary order. With the $widehat{Q}$ box and its derivatives thus determined we apply them to the calculation of the $E$-independent effective interaction given in the so-called Lee-Suzuki (LS) method for a system with a degenerate unperturbed energy. We show that the recursion method can also be applied to the generalized LS scheme for a system with non-degenerate unperturbed energies. If the Hilbert space is taken to be sufficiently large, the theory provides an exact way of calculating the $widehat{Q}$ box and its derivatives. This approach enables us to perform recursive calculations for the effective interaction to arbitrary order for both systems with degenerate and non-degenerate unperturbed energies.
We have constructed a noncommutative deformation of the holographic QCD (Sakai-Sugimoto) model and evaluated the mass spectrum of low spin vector mesons at finite temperature. The masses of light vector- and pseudovector-meson in the noncommutative h
olographic QCD model reduces over the whole area in the intermediate-temperature regime compared to the commutative case. However, the space noncommutativity does not change the properties of temperature dependence for the mass spectrum of low spin mesons. The masses of meson also decrease with increasing temperature in noncommutative case.
One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories
in nuclear physics have been formulated by use of the so called $hatQ$ box introduced by Kuo et.al. It has been one of the central problems how to calculate the $hatQ$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $hatQ$ box can be expressed in two ways: One is a form of continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures to derive an exact $hatQ$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $hatQ$ box given in this study corresponds to a non-perturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $hatZ$-box approach based on the $hatQ$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.
We consider the noncommutative deformation of the Sakai--Sugimoto model at finite temperature and finite baryon chemical potential. The space noncommutativity is possible to have an influence on the flavor dynamics of the QCD. The critical temperatur
e and critical value of the chemical potential are modified by the space noncommutativity. The influence of the space noncommutativity on the flavor dynamics of the QCD is caused by the Wess--Zumino term in the effective action of the D8-branes. The intermediate temperature phase, in which the gluons deconfine but the chiral symmetry remains broken, is easy to be realized in some region of the noncommutativity parameter.
