Classical Hamiltonian perturbation theory without secular terms or small denominators

We consider perturbation theory in ? for the classical Hamiltonian H = H₀ + ?H₁, where H₀ gives rise to a known motion and ? is small. First we demonstrate how the usual secular terms and small denominators arise from a straightforward expansion in ? and argue that they are artifacts of the method. Then we present an alternative perturbation theory based on an analysis of the operator (s ? L)^?1, where s is a complex number and L is the Liouville operator corresponding to H. This perturbation series contains neither secular terms nor small denominators. In the case of almost multiply periodic systems we show, to lowest non-trivial order in ?, how our series reproduces the standard results both in the resonant and nonresonant regions — all in one analytic formula. As a final exercise we demonstrate that energy is conserved at order ?ⁿ⁺¹ when the accuracy of the theory is order ?ⁿ.     

Perturbation theory in terms of electron density

When an inhomogeneous electron gas is subjected to a perturbation, its energy and density both change by small amounts. We calculate the changes in the energy explicitly in terms of the density changes within the density-functional theory of many-electron systems. We also derive the equations for the induced densities, and using these show that a density correct up to order n in terms of the perturbation parameter is sufficient to give energy changes up to order (2n + 1). As a corollary, the even-order energy changes E⁽²ⁿ⁾ are variational with respect to the density changes ?⁽ⁿ⁾. The equations for the induced densities also follow from this corollary. The even-order corollary also gives a variational method of calculating the induced densities. The theory is demonstrated by applying it to calculate the polarizability and hyperpolarizability of the hydrogen, helium and neon atoms.     

Higher-order approximations for the particle-particle propagator

A new type of approximations for many-body Green's functions proposed recently is applied to the particle-particle (pp) propagator for anN-particle fermion system. The new approach which is referred to as the algebraic diagrammatic construction (ADC) is based on an exact resummation of the perturbation series for the pp-propagator in terms of a simple algebraic form introducing energy-independent effective interaction matrix elements and transition amplitudes. These effective quantities are represented by perturbation expansions and can be determined consistently through a given ordern of perturbation theory by comparing the algebraic form with the diagrammatic perturbation series of the pp-propagator through ordern. By this procedure one obtaines a systematic set of approximation schemes (ADC(n)) that represent infinite partial summations for the pp-propagator being complete throughnth order of perturbation theory. The explicit ADC equations forn=1 and 2 are presented and discussed. Comparison is made with the particle-particle random phase approximation (RPA). It is demonstrated that the second-order ADC scheme constitutes an essential step beyond the RPA which is consistent only through first order.     

The stark effect in the 6snf Rydberg series of barium

The polarizabilities of the barium 6snf ³ F ₂ levels with principal quantum numbern ranging from 11 to 40 have been measured in a high resolution laser-atomic-beam experiment. In Coulomb approximation the contributions to the polarizabilities of all known even-parity levels connected with the³ F ₂ levels via the electric dipole operator have been calculated. In this way information on the to a large extent unknown 6sng ³ G ₃ Rydberg series is extracted. The results support earlier observations of a perturbation of this series in betweenn = 18 and 19, but no evidence has been found for a perturbation nearn = 14.     

Multipole mixtures in γ transitions from the reaction 154Sm(n, n′γ)

The angular distributions of γ rays from the reaction ¹⁵⁴Sm(n, n′γ) with respect to the axis of the beam of fast reactor neutrons are measured. The known scheme of levels and γ transitions in ¹⁵⁴Sm is supplemented. The multipole-mixture ratio δ is found for many γ transitions.     

Residue-squaring iterative diagonalization method for perturbation problems

We present a new method for the evaluation of the change in eigenvalues due to a perturbation of strength λ. It is a fast converging iterative method which, at thenth step, gives results accurate to order (2 ⁿ⁺¹?1) in λ. Unlike the Rayleigh-Schrödinger perturbation theory in quantum mechanics, which becomes prohibitively cumbersome when carried to higher orders, the present method involves a routine which remains stralghtforward at all stages.     

The absence of cut-off effects for the fixed-point action in one-loop perturbation theory

In order to support the formal renormalization group arguments that the fixed-point action of an asymptotically free model gives cut-off independent physical predictions in one-loop perturbation theory, we calculate the finite-volume mass gap m(L) in the non-linear σ-model. No cut-off effect of the type g⁴ (a/L)ⁿ is seen for any n. The results are compared with those of the standard and tree level improved symanzik actions.     

Absence of pair production in the CPn−1 model

We prove that the CP^n?1 model does not accomodate pair formation up to third order perturbation theory.     

Relativistic calculation of spectra of 2-2 transitions in O- and F-like atomic ions

We have studied theoretically the structure of 1s²2s^n₁2p^n₂ levels in O- and F-like ions with z = 25 ÷ 40. Relativistic perturbation theory is used for the total interelectronic interaction. Allowance is made for Coulomb and relativistic interelectronic interactions. The first order perturbation theory for retardation of the interaction has been calculated fully. The complete calculation has also been made for the nonrelativistic correction in the second order and the Hartree-Fock part of the nonrelativistic correction in the third order. The Hartree-Fock relativistic correction in the second order has been taken into account fully for F-like ions and partially for O-like ions. Corrections to higher orders for z ? 28 have been found empirically and extrapolated to the region z = 29 ÷ 40. For energies of 2-2 transitions, calculation errors for z ? 28 do not exceed 900 cm^-1 and do not increase with z. Typical errors obtained with conventional, more cumbersome calculations are 5000 cm^-1 at z = 30.     

Vacuum instability in scalar field theories

The class of scalar field theories with interaction gφ^2N?1, are studied using the semi-classical approximation. The imaginary part of the vertex functions generated by tunnelling out of the metastable ground state is calculated to first order. Using this result, the leading asymptotic behaviour of the renormalisation group β function for φ³ field theory is obtained in six dimensions. The validity of this result is discussed in view of the extra singularities which appear when the theory is just renormalisable. The structure of the perturbation expansion for n component φ³ theory is also discussed, and cases in which these theories yield perturbation expansions which are Borel summable, are pointed out.     

Radiative corrections to positronium energy levels

A systematic perturbation theory for the computation of positronium energy levels is presented. The proposed method is based on the Bethe-Salpeter equation and a zeroth-order kernel for which exact solutions are known explicitly. The cancellation of ultraviolet divergences arising in the perturbation series is discussed, and new contributions to the ground-state hyperfine splitting of positronium are evaluated up to order α⁶.     

On the statistical mechanics of dense instanton gases

Instanton gases of two-dimensional?P ^n?1 and four-dimensionalSU(n) Yang-Mills theories are considered. The presumable denseness of instanton gases in these models and the corresponding statistics of instantons lead to a theormodynamic limit in which the coupling constant dependence of non-perturbative quantities is modified by a factor proportional to 1/n compared to the case of a dilute gas. As a consequence the largen limit and the infinite volume limit do not appear to commute. We present a naive droplet model for dense instanton gases which exhibits these features. Possible consequences for the large order behaviour of perturbation series are discussed.     

Overtone transitions of diatomic dimers XH<Superscript>−</Superscript>-XH<Superscript>−</Superscript> in ionic crystals

The resonance interaction of molecular defects XH^– in ionic crystals is considered. In the dipole approximation, the frequencies and intensities of vibrational transitions of diatomic dimers ^mXH^?-ⁿXH^? are found for the exact (m=n) and quasi-exact (m≠n) resonances. The interaction-atrix elements are obtained with regard to the mechanical anharmonicity in the second order of the theory of perturbation for the case of a linear dependence of the dipole moment of a diatomic molecule on the vibrational coordinate. The expressions obtained are applied to the calculation of overtones in the absorption spectrum of dimers ^mSH^?-ⁿSH^? in a KCl crystal.     

Calculation of the fine-structure parameters for np 5 n′f configurations of rare-gas atoms

In the single-configuration approximation, fine-structure parameters are calculated semiempirically in the intermediate coupling scheme for the configurations 2_p ⁵ nf(n=4–6) of NeI, 3p ⁵ nf(n=4–7) of ArI, and 4p ⁵4f of KrI. With the fine-structure parameters obtained, the coefficients of expansion of the wave functions in basis functions of the LS-coupling scheme and the gyromagnetic ratios are calculated. To the authors’ knowledge, analogous data are absent in the literature. The correctness of the fine-structure parameters obtained is confirmed by values of the fine-splitting constant, which is well known for other configurations of rare-gas atoms with an almost filled p shell.     

ESR of exchange coupled Cr3+ pairs in the dimer Di-μ-Diphenylphosphinatoacetyl-Acetonatochromium(III) (DPACr)

DPACr is a dimeric phosphinate in which the pairs of Cr³⁺ ions are superexchange coupled via -OPO- bridges. The consequences of site anisotropy and inequivalence are demonstrated in the ESR spectra and can be derived using a perturbation scheme on the appropriate Hamiltonian. First and second order fine structure, multiplet shifts, g-shifts and g-anisotropies are discussed. This dimeric system is the simplest member of a class of poly(chromium phosphinates).     

Optimized perturbation theory applied to jet cross sections ine + e − annihilation

The optimized perturbation theory proposed by Stevenson to deal with coupling constant scheme dependence is applied to the calculation ofπ _tot and jet multiplicities ine ⁺ e ^? annihilation. The results are compared with those of simple perturbation theory and with recent experimental cluster multiplicities.     

Catoptric tadpoles

Fermionic string perturbation theory is known to suffer from an ambiguity in the form of a total derivative in the moduli space. For a class of backgrounds (including R¹⁰, orbifolds and theories with no U(1) factors in gauge group) we show that these ambiguities for the partition function of heterotic string theory at any genus are proportional to massless physical tadpoles in the theory at lower genera and hence vanish in stable vacua. We also find that in R¹⁰ the cosmological constant at a given genus is proportional to the cosmological constant at lower genera. This enables us to give an inductive argument for the vanishing of the cosmological constant in R¹⁰ to all orders in string perturbation theory. We also address the ambiguity and finiteness of n-point functions. Our results indicate that in R¹⁰ the ambiguity can be absorbed by a renormalization of the string coupling constant and the string tension. The expected sources of divergence in the n-point function in arbitrary tachyon-free backgrounds, besides the usual infrared divergences for d ≤ 4, are shown to be proportional to tadpoles of physical massless fields. For type II strings in arbitrary backgrounds, we show by explicit calculations that the ambiguity vanishes at g = 2.     

Exact computation of high-order perturbational eigensolutions and its application to the analysis of a spectral degeneracy in a bistable diffusion process

The complete spectrum (i.e. for any n = 0, 1, …) of a general eigenvalue problem is obtained exactly through the chosen order in a perturbation parameter ? → 0 by means of a special computer program. The algorithms essentially involve exact (i.e. integer valued) manipulations with polynomials in n. The technique is applied to the analysis of a previously reported triplet degeneracy in bistable diffusion, where the pump parameter κ = 1/? → ?∞. Extremely strong degeneracy is found as it is not lifted through order κ^?30.