Fermi and Coulomb correlations in the 21 S state of the helium isoelectronic sequence

The Fermi and Coulomb holes of the 2¹ S state of the helium isoelectronic sequence are investigated. Several interesting differences between the results obtained and those which might be expected on the basis of the corresponding 2³ S state are pointed out and discussed.     

Intramolecular electronic energy transfer via exchange interaction in bichromophoric molecules

A theory is presented for intramolecular electronic energy transfer in bichromophoric molecules. Expressions are given for the donor moiety fluorescence (phosphorescence) decay and for its fluorescence (phosphorescence) quantum yield in terms of the average distance between the donor and acceptor moieties and the donor—acceptor bridge flexibility. Comparison with available experimental data supports the predictions of the analysis.     

Products of class operators of the symmetric group

A combinatorial derivation of the product of the class of three cycles, [(1)^N?3(3)]_N with an arbitrary class operator of the symmetric group S_N is presented. The form of this result suggests a conjecture concerning the expression of the general class operator product in terms of a relatively small number of reduced class coefficients. The conjecture is applied to the determination of the products of [(1)^N?4(4)]_N, [(1)^N?4(2)²]_N, and [(1)^N?5(5)]_N with arbitrary class operators. General expressions for the reduced class coefficients of the simplest type are obtained.     

The Coulomb hole in the 23S state of the helium isoelectronic sequence

The pair distribution functions evaluated for the 2³S state of the helium isoelectronic sequence from the Hart and Herzberg correlated wave functions and those corresponding to the Hartree-Fock approximation are used to determine the shape of the corresponding Coulomb holes. As a consequence of a discontinuity in the Hartree-Fock solution between He and Li⁺, the Coulomb hole has a different shape for He than for Li⁺ and the other isoelectronic ions.     

A partial recurrence relation for reduced class coefficients of the symmetric group

An expression for the product of a single-cycle class [(1)^{N - P}(p)]_N and an arbitrary class [(1)^l1(2)^l …? (N)^lN]_N of the symmetric group has recently been conjectured. This expression involves a sum over a relatively small number of reduced class sums, depending on p indices. A further conjecture is formulated and demonstrated, according to which reduced class coefficients (RCCS ) involving cycles whose length is expressed by means of a single index can be related to corresponding coefficients in the product of [(1)^{N - P+1}(p - 1)]_N with an arbitrary class sum. Consequently, the problem of evaluating the general class sum product reduces to that of obtaining a relatively small set of fundamental RCCS containing no single-index cycles. The conjectures mentioned can be used to evaluate the product [(1)^{N - p}(p)]_N · [(1)^{N - q}(q)]_N in terms of fundamental RCCS that can all be obtained from the product [(r)]_r · [(r)]_r, where r = min(p, q). For the latter product, we use a result due to Boccara.     

Inequalities for the Eigenvalues of Non-Selfadjoint Jacobi Operators

We prove Lieb-Thirring-type bounds on eigenvalues of non-selfadjoint Jacobi operators, which are nearly as strong as those proven previously for the case of selfadjoint operators by Hundertmark and Simon. We use a method based on determinants of operators and on complex function theory, extending and sharpening earlier work of Borichev, Golinskii and Kupin.     

Symmetrized harmonic oscillatorSU 3 states for multi-cluster systems

An algorithm is presented for the construction of single- and multi-cluster harmonic oscillator wave functions that are coupled into well-defined irreducible representations ofSU ₃ as well as of the symmetric groupS _N . The single-cluster harmonic oscillatorSU ₃ wave functions are constructed recursively, using theSU ₃ coefficients of fractional parentage. To construct multi-cluster wave functions with a well-defined permutational symmetry we diagonalize an appropriate set of single-cycle class operators of the symmetric group involving all the constituent particles. The formalism is applicable to the study of multi-cluster systems in nuclear physics where the wave functions are expressed in terms of harmonic oscillatorSU ₃ states     

Eigenvalue Inequalities in Terms of Schatten Norm Bounds on Differences of Semigroups, and Application to Schrödinger Operators

We develop a new method for obtaining bounds on the negative eigenvalues of self-adjoint operators B in terms of a Schatten norm of the difference of the semigroups generated by A and B, where A is an operator with non-negative spectrum. Our method is based on the application of the Jensen identity of complex function theory to a suitably constructed holomorphic function, whose zeros are in one-to-one correspondence with the negative eigenvalues of B. Applying our abstract results, together with bounds on Schatten norms of semigroup differences obtained by Demuth and Van Casteren, to Schr?dinger operators, we obtain inequalities on moments of the sequence of negative eigenvalues, which are different from the Lieb–Thirring inequalities. Guy Katriel: Partially supported by the Minerva Foundation (Germany). Submitted: September 4, 2007. Accepted: December 11, 2007.