[en] Four new variational principles are given. They contain generalization of the Rayleigh-Ritz principle. It is first shown that the constrained Hartree-Fock method for a collective coordinate Q must include a constraint on the fluctuation of that coordinate (a Q² term) in order to avoid collapse. Then one shows that the regularisation parameter of a singular potential can be used as a trial parameter for the calculation of binding energies or phase shifts, provided one adds a counterterm to the regularized Hamiltonian. This procedure is valid for the G-matrix. In the case of phase shifts, one uses the method of Pade and also distorted waves

No abstract available

[en] A central potential gV(r) (gεC) is considered. A quantum number for the S-matrix poles belonging to the k-plane sheet images Σ_n'⁽¹⁾ of the Riemann surface R_g⁽¹⁾ is introduced. The problem of labeling is solved in the same manner for both resonant and bound states. The labeling of the resonant poles by the radial quantum number of the corresponding bound state pole is not a legitimate labeling because the radial quantum number would change as the potential strength g is varied and we need a label which is independent of the potential strength g. Moreover, the new class resonant state poles that remain in a neighborhood of some special point (stable points) in the k-plane and do not become bound state poles as the potential strength increases to infinity and could not be labeled by a radial quantum number of the bound state pole. It is obvious that the proposed quantum number is very suitable for labeling in such cases. (Author)

[en] The ensemble N-representability problem for the k-th order reduced density matrix (k-RDM) as well as the problem of reconstruction of the N-particle system density matrices (N-DM) from a given k-RDM are studied. The spatial parts of the k-RDM expansion in terms of spin tensorial operators Θ_λ are represented using particular values (at specially chosen Ξ = Ξ_o) of the Radon transform D_Nλ D_Nλ(Ξ) of the N-DM spatial parts (or their sums) D_Nλ(χ'|χ double-prime) (here, Ξ is a d-plane in the n-space Re double-prime of χ = (χ', χ double-prime), with n = 6N, d = 3(N - k), χ' double-bond ' (r'₁,...,r_N'), χ double-prime double-prime triple-bond (r₁ double-prime,...,r_N double-prime)). In this way, the problem is reduced to investigation of the properties of the functions D_Nλ(Ξ). For a normalizable N - DM, it is proved that D_Nλ(Ξ) are bounded functions. The properties of D_Nλ(Ξ) implied by the N-DM permutational symmetry, Hermiticity, and positive definiteness are found. A formal procedure of reconstruction of all N-DM corresponding to a given k-RDM is proposed. 38 refs

[en] Qualitative physical arguments are presented in support of a two-step calculation of the G-matrix for finite nuclei. In the first step the high-energy excitations are included using orthogonalized plane-wave intermediate states, and in the second step the low-energy excitations are added in, using harmonic-oscillator intermediate states. The double counting associated with the inclusion of two-particle intermediate states between successive G-matrix vertices in diagrams of the effective interaction is discussed. (Auth.)

[en] It is shown that the magnitude of the ensemble average of level-level correlations is proportional to Tr(P/√(1-P)) where P is Satchler's penetration matrix. The implications of this result are discussed. (Auth.)

[en] We show that Wigner semi-circle law holds for Hermitian matrices with dependent entries, provided the deviation of the cumulants from the normalised Gaussian case obeys a simple power law bound in the size of the matrix. To establish this result, we use replicas interpreted as a zero-dimensional quantum field theoretical model whose effective potential obey a renormalisation group equation. (letter)

[en] Let ρ the density matrix of a mixed Gaussian state. Assuming that one of the Robertson–Schrödinger uncertainty inequalities is saturated by ρ, e.g. (Δ^ρX₁)²(Δ^ρP₁)²=Δ^ρ(X₁,P₁)²+ 1/4 ℎ², we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that of ρ and having the same variances and covariance Δ^ρX₁, Δ^ρP₁ and Δ^ρ(X₁, P₁) as ρ. This property can be viewed as an analytic version of Gromov’s non-squeezing theorem in the linear case, which implies that the intersection of a symplectic ball by a single plane of conjugate coordinates determines the radius of this ball. We conclude by giving a short geometric proof of the fact that pure Gaussian states are the only quantum states saturating the Robertson–Schrödinger uncertainty inequalities. (paper)

[en] In this paper we construct a new factorized representation of the R-matrix related to the affine algebra $U_q(\stackrel{^<!--\; ^\; -->}{{sl}_n})$ for symmetric tensor representations with arbitrary weights. Using the 3D approach we obtain explicit formulas for the matrix elements of the R-matrix and give a simple proof that a ‘twisted’ R-matrix is stochastic. We also discuss symmetries of the R-matrix, its degenerations and compare our formulas with other results available in the literature. (paper)

[en] Given a non-empty closed convex subset F of density matrices, we formulate conditions that guarantee the existence of an F-morphism (namely, a completely positive trace-preserving linear map that maps F into itself) between two arbitrarily chosen density matrices. While we allow errors in the transition, the corresponding map is required to be an exact F-morphism. Our findings, though purely geometrical, are formulated in a resource-theoretic language and provide a common framework that comprises various resource theories, including the resource theories of bipartite and multipartite entanglement, coherence, athermality, and asymmetric distinguishability. We show how, when specialized to some situations of physical interest, our general results are able to unify and extend previous analyses. We also study conditions for the existence of maximally resourceful states, defined here as density matrices from which any other one can be obtained by means of a suitable F-morphism. Moreover, we quantitatively characterize the paradigmatic tasks of optimal resource dilution and distillation, as special transitions in which one of the two endpoints is maximally resourceful. (paper)