Lieb–Thirring Type Inequalities for Schrödinger Operators with a Complex-Valued Potential

We present a possible generalization of the Lieb–Thirring inequalities for eigenvalues of Schrödinger operators to the case of complex potentials. We ask for a proof or disproof of this generalization. Some weaker results which have been obtained are reviewed.     

On the discrete spectrum of non-selfadjoint operators

We prove quantitative bounds on the eigenvalues of non-selfadjoint unbounded operators obtained from selfadjoint operators by a perturbation that is relatively-Schatten. These bounds are applied to obtain new results on the distribution of eigenvalues of Schrödinger operators with complex potentials.     

Nonuniversality of commonly used correlation-energy density functionals

The correlation energies of the helium isoelectronic sequence and of Hooke's atom isoelectronic sequence have been evaluated using an assortment of local, gradient, and metagradient density functionals. The results are compared with the exact correlation energies, showing that while several of the more recent density functionals reproduce the exact correlation energies of the helium isoelectronic sequence rather closely, none is satisfactory for Hooke's atom isoelectronic sequence. It is argued that the uniformly acceptable results for the helium sequence can be explained through simple scaling arguments that do not hold for Hooke's atom sequence, so that the latter system provides a more sensitive testing ground for approximate density functionals. This state of affairs calls for further effort towards formulating correlation-energy density functionals that would be truly universal at least for spherically symmetric two-fermion systems.     

q-similarity transformations

Expressions for q-similarity transformations with respect to the symmetric q-exponentials are investigated. It is found that a series in multiple q-commutators can be formulated. The values of the qs in the different q-commutators are obtained by solving systems of algebraic equations.     

A study of the adiabatic connection for two-electron systems

Some aspects of the adiabatic connection method are studied for two-particle spherically symmetric systems. Ground-state wave functions that are constrained by means of a set of moments to have the same density as a corresponding fully interacting system are obtained for noninteracting or partially interacting systems. Local one-body potentials that support these constrained wave functions are generated using a simple method. We examine an interacting two-particle system with a parameter-dependent one-body potential, which for a particular value of that parameter exhibits an intersection between the (3)S and the (3)P states, whereas the 2s and 2p eigenvalues of the corresponding Kohn-Sham potentials do not intersect along with the total energies. These results show that there do exist cases where occupying the orbitals from below in energy may not lead to the ground state, and that the inherent assumptions behind the adiabatic connection can sometimes be violated.     

Spurious complex energies for confining potentials in the complex-coordinate method

Complex-valued energies are obtained analytically by application of the complex coordinate method to confining potentials of the form X^m (m > O). The results are used to analyse some properties of the multiple spectra of complex eigen-values recently obtained by Atabek and Lefebvre for the exponential well potential. Since molecular potential surfaces are commonly described by power-series expansions in the coordinates, our results represent possible pitfalls in the calculation of predissociation resonances by the complex-coordinate method.     

Mapping between local potentials and ground state densities

The mapping between the local one-body potential and the ground state density is discussed. The role of degenerate ground states, which form a set of measure zero in potential space, is shown to be amplified by the unitary transformation between degenerate densities, forming a space filling set in density space. A numerical experiment demonstrating these features is carried out. No indication of non-V representable densities is encountered.     

Bonding criteria for diatomic molecular orbitals and inter-relations among them

Bonding criteria for molecular orbitals in diatomic molecules are discussed. An orbital force criterion is shown to have several conceptual and practical advantages, providing a basis for the investigation of inter-relations among many of the commonly employed criteria.It is found that interconsistency among those criteria is guaranteed, within the framework of Koopmans' Theorem, if the orbital energies are monotonic in the range (R _e, ).The application of the orbital force criterion to the second row homonuclear diatomics exhibits reasonable chemical trends concerning the valence-shell orbitals, as well as indications of a slightly antibonding nature of the inner orbitals.Based on a section of a thesis to be submitted by Y.T. to the Senate of the Technion-Israel Institute of Technology, in partial fulfilment of the requirements for the D.Sc. degree     

An exactly soluble model of quadratic magnetostriction

Spin-phonon coupling, quadratic in the lattice displacements, is studied assuming that the spin interactions are of infinite range. The magnetic phase transition changes from second to first order for a sufficiently strong coupling. The ferromagnetic spinodal temperature becomes infinite and the paramagnetic spinodal temperature becomes zero for couplings of appropriate signs and strength. Renormalization of the phonon frequencies, within both the Einstein and the Debye schem schemes, is found.     

The spherical limit of H: A symmetry dilemma

The spherical limit of H is studied as an example of the occurrence of a symmetry dilemma of a kind which is a general characteristic of truncated one centre expansions. At internuclear distances just somewhat larger than the equilibrium distance the best spherical approximation turns out to be centred off the molecular midpoint and is thus a mixture of σ_g and σ_u. Rather detailed insight into the situation is gained by an analysis based on the electrostatic Hellmann-Feynman theorem.