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We denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian (unitary resp.) if A = A* (AA*= identity matrix resp.). An n ×x n hermitian matrix A is said to be definite (semidefinite resp.) if uAu*vAv* ≥ 0 (uAu*vAv* ≧ 0 resp.) for all nonzero u and v in Fn. If A and B are n × n hermitian matrices, then we say that A and B can be diagonalized simultaneously into blocks of size less than or equal to m (abbreviated to d. s. ≧ m) if there exists a nonsingular matrix U with elements in F such that UAU* = diag{A1,…, Ak} and UBU* = diag{B1…, Bk}, where, for each i = 1, …, k, Ai and Bk are of the same size and the size is ≧ m. In particular, if m = 1, then we say A and B can be diagonalized simultaneously (abbreviated to d. s.).

Semi-analytical solutions of the Schrodinger equation for a particle moving in the electrostatic field of two other particles a fixed distance apart, are derived in such a way that the resulting matrix eigenvalue equations contain real symmetric band matrices. Numerical techniques appropriate to the solution of the two simultaneous matrix eigenvalue equations are described; in particular the bisection method is used to determine precisely the significant truncation order of the matrices for a given numerical precision.

Currently, the most satisfactory version of Brueckner theory from the formal point of view is the generalized-time-ordered series of Brandow. In this formalism the Brueckner-Hartree-Fock (BHF) self-energy for normally empty (“particle”) states does not factorize, and for this reason it is not included in the single-particle potential, U . This feature of the theory leads to differences: (i) between single-particle and separation energies, and (ii) between U and the real part of the optical potential. In article I the time-dependent Green function formulation was used to achieve more extensive factorization, with the consequence that U could be defined to include the leading terms of the on-energy-shell BHF potential for “particles”. In the present paper this earlier expansion is written in terms of Brueckner reaction matrices, and the low orders of the resulting series are investigated. For this task, it has been found more convenient to employ the Fourier-transformed formalism of energy-dependent Green functions, G (ω). First, diagram rules for expansions in the bare interaction, V , and an arbitrary U are given in sufficient generality to apply to finite systems, for which U and f (ω) are not diagonal matrices. The binding energy is expressed in terms of closed, propagator-renormalized, ω-dependent, ground state diagrams including over-counting corrections. Then U is chosen, in correspondence with article I and a paper of Klein, in terms of the real part of the proper self-energy. In the framework employed here U may be defined conveniently to be hermitian. A prescription, stated with remarkable simplicity in the G (ω) formalism, is obtained for converting an expansion in V to one in a general transition or reaction matrix. Our U is the quasiparticle separation-energy potential, containing on-energy-shell self-energies of all topologies. For example, when the reaction matrix is that of Brueckner, the first-order (BHF) and secondorder (“rearrangement energy”) insertions are shown to be included in U both for “particles” and holes. Truncation schemes based on a stationary property of the propagator-renormalized series are explored. The simplest is a renormalized BHF approximation similar to that for Brandow's expansion but differing from it by (a) containing the RBHF potential for “particle” states, which implies a smaller energy gap and greater diffuseness at the Fermi surface; and (b) propagator renormalization (“line weighting”) factors which contain single-particle density matrices (“true occupation probabilities”) as in Brandow's expansion, plus “folded” terms. The present formulation allows a close connection between “virtual” and “valence” single-particle states, while staying close to the computationally successful framework of the Brueckner theory and avoiding complex energies.

If H is a Hermitian matrix and W = AH + HA ∗ is positive definite, then A has as many eigenvalues with positive (negative) real part as H has positive (negative) eigenvalues [5]. Theorems of this type are known as inertia theorems. In this note the rank of the controllability matrix of A and W is used to derive a new inertia theorem. As an application, a result in [8] and [4] on a damping problem of the equation M x + (D + G) xdot; + Kx = 0 is extended.

A new method is devised to include the symmetry in perturbation series, in case the unperturbed hamiltonian has a lower (or different in general) symmetry than the whole problem. We define a unitary transformation such that the transformed unperturbed hamiltonian will have the same symmetry as the interacting system. This new unperturbed hamiltonian is hermitian and has indeed the same eigenvalues as the old one. Usual perturbation expansions can be thence utilized. The method is illustrated through some simple examples. The non-uniqueness of the defined unitary transformation is discussed.

For a given $m \times n$ complex matrix A, it is shown that X satisfies $XAX = X$ if and only if it is expressible in the form \[ X = (EAF)^\dag \], where E and F are Hermitian idempotents and the dagger denotes the Moore–Penrose inverse. In particular, a matrix is idempotent if and only if it is the Moore–Penrose inverse of the product of two Hermitian idempotents: (The “if” part of the latter statement was previously shown by Cline.)If L and M are complementary subspaces of the space of n-dimensional complex vectors, and if $P_{L.M} $ denotes the projector on L along M and $P_L $ the orthogonal projector on L, it is shown that \[ P_{L,M} = \left(P_{M^ \bot } P_L \right)^\dag = \left( \left( I - P_M \right) P_L \right)^\dag \], where $M^ \bot $ denotes the orthogonal complement of M. More generally, \[ P_{L,M} = P_L Y\left( I - P_M \right) \], where Y is an arbitrary matrix satisfying $AYA = A$ with $A = ( I - P_M )P_L $.Afriat has previously shown that \[ P_{L,M} = \left( I - P_L P_M \right) ^{ - 1} P_L...

A Lyapunov transformation is a linear transformation on the set H n of hermitian matrices H ϵ C n,n of the form L A(H) = A∗H + HA, where A ϵ C n,n. Given a positive stable A ϵ C n,n, the Stein-Pfeffer Theorem characterizes those K ϵ H n for which K = L B(H), where B is similar to A and H is positive definite. We give a new proof of this result, and extend it in several directions. The proofs involve the idea of a controllability subspace, employed previously in this context by Snyders and Zakai.