An asymptotic expansion for energy eigenvalues of anharmonic oscillators

In the present contribution, we derive an asymptotic expansion for the energy eigenvalues of anharmonic oscillators for potentials of the form V(x)=κx^2q+ωx²,q=2,3,…$V\left(x\right)=\kappa x^{2q}+\omega x^2,q=2,3,\dots$ as the energy level n$n$ approaches infinity. The asymptotic expansion is obtained using the WKB theory and series reversion. Furthermore, we construct an algorithm for computing the coefficients of the asymptotic expansion for quartic anharmonic oscillators, leading to an efficient and accurate computation of the energy values for n≥6$n\ge 6$.     

Study of the single-particle energy spectrum of disordered systems

In this paper the recursion method is applied to the study of the energy spectrum of strongly disordered systems. The real spatial atomic configuration of amorphous semiconductors is the foundation the theory is based on. The presence of partial disorder (short-range order) is shown to be the reason for the existence of a mobility edge whose position is determined by the asymptotic behaviour of the recursion coefficients of a weakly disordered substitution structure. An analytical expression for the local density of states (LDS) is found consisting of contributions from a point and a continuous spectrum. Strong distortions of the local atomic structure are shown to be the origin of localized states. The spectral strength is defined as the weight factor a localized level has in the LDS. The concept of local resonance states is introduced into the theory of disordered solids. Resonances put structure on the LDS of the continuum. Near isolated sharp resonances an analytical representation of the LDS is found. Both localized levels, spectral strengths, resonance energies and widths, and LDS's are numerically calculated for simple models. The theory is applied tod-electrons on a fcc-lattice.     

On the energy spectra of single-particle states in nuclei

The single-particle spectra of ¹⁶O, ⁴⁰Ca, ⁴⁸Ca and ⁵⁶Ni have been studied using self- consistent field methods and a quadratically velocity-dependent two-nucleon effective (reaction- matrix) interaction. The self-consistent field equations are derived in some detail for the spin- independent interaction which is used. A one-body spin-orbit interaction of the Thomas form is added.     

The asymptotic behaviour of form factors

Similar to the method of S.A. Anikin and O.I. Zavialov a modified renormalization procedure is applied to the already renormalized current operator. This allows the derivation of an identity for the coefficient functions of this operator and the proof of new renormalization group equations. These equations are applied to the theory and the electromagnetic form factor of quarks in QCD. In the last case the one-loop approximation gives the leading behaviour exp(?clnlnQ²lnQ²) for large euclidean values of the momentum transfer Q² = ?q² at the external electromagnetic vertex. If the leading logarithmic terms (g²ln²Q²)ⁿ are extracted from this result, then the behaviour exp(?g²c′ln²Q²) is obtained. Whereas the first result is modified by contributions from two-loop calculations, the result concerning the leading logarithmic terms does not change.     

Remark on the asymptotic form of the surface response function

We carry a numerical investigation of the surface response function χ(r, r′) for large ∥r−r′∥ separation. We show that in that range previous approximations of χ cannot be valid.     

Behaviour of single-particle states and their eigenvalues near zero energy

For a state in a single-particle potential we derive simple expressions that the eigenvalue behavior and normalization integral properties near threshold. For the square-well, the various parameters in these expressions can be derived analytically. For the more realistic Saxon-Woods well-shape, we have made numerical calculations for s- and p-states and obtained values of these parameters and give rules for their evaluation. We show that those parameters determining the negative energy eigenvalues are also involved in determining the positive energy behavior, and the normalization properties. These studies have direct relevance to recent theories of some types of (d, p) and (n, γ) reactions which critically involve threshold behavior.     

Extraction of single-particle mean kinetic energy from macroscopic thermodynamic data

We have devised a simple and practical method for the extraction of single-particle mean kinetic energy from the Helmholtz free energy, when this quantity is known for a pair of isotopes of the element under investigation. This approach is originated from a property of the mass derivative of the Helmholtz free energy, but makes use of a mass incremental ratio evaluated for the aforementioned isotopes. The transformation from mass incremental ratio to mass derivative is done using a self-consistent method based on two approximations, the latter of which is quite surprising and has been thoroughly tested. Practical calculations on solid and liquid H₂ and D₂, where the center-of-mass mean kinetic energy is known from reliable quantum simulation techniques, are accomplished, obtaining a remarkable agreement, in spite of a slight overestimate for the calculated data set over the simulated one. The last part of the study is devoted to mixed systems, like water and ice, where thermodynamic data are abundant and proton mean kinetic energy has been measured via deep inelastic neutron scattering. Here results are less accurate than earlier, with an overestimate of about 9-10%, and, in addition, no substantial variation going from ice to liquid water. Further investigations on the H₂O/D₂O system are suggested.     

Influence of single-particle shell structure on the energy dissipation in heavy-ion collisions

Shell effects in the mean energy dissipated per transferred nucleon are analysed within a statistical model of random transfers and excitations. Quantitative agreement for all 21 analysed reactions is obtained without fitting any parameter to the data.     

Structure of higher order corrections to the Lipatov asymptotic form

High orders of perturbation theory can be calculated by the Lipatov method, whereby they are determined by saddle-point configurations, or instantons, of the corresponding functional integrals. For most field theories, the Lipatov asymptotic form has the functional form ca ^NΓ(N+b) (N is the order of perturbation theory) and the relative corrections to it are series in powers of 1/N. It is shown that this series diverges factorially and its high-order coefficients can be calculated using a procedure similar to the Lipatov one: the Kth expansion coefficient has the form const[ln(S ₁/S ₀)]^?K Γ(K+(r ₁? r ₀)/2), where S ₀ and S ₁ are the values of the action for the first and second instantons of this particular field theory, and r ₀ and r ₁ are the corresponding number of zeroth-order modes; the instantons satisfy the same equation as in the Lipatov method and are assumed to be renumbered in order of their increasing action. This result is universal and is valid in any field theory for which the Lipatov asymptotic form is as specified above.     

The asymptotic form of three-particle wavefunctions and the cross sections

The asymptotic behavior of three-particle wavefunctions in the coordinate representation is examined as all three particles move towards infinity. We investigate both the wave-function for a particle incident on a bound state, and that for three free incident particles. The results are expressed in terms of cross sections, and they lead to a clean separation of the double-scattering terms from the “true” three-particle scattering amplitude. We also derive the relevant reciprocity theorems from time-reversal invariance.     

The contribution of the central region of the electron energy spectrum to the total double photoionization cross section of helium in the asymptotic nonrelativistic energy region

The double photoionization of helium at high photon energies is considered using a nonrelativistic approach. The central region of the energy spectrum and its contribution to the total process cross section and to the ratio between the double and single ionization cross sections are studied. Interelectronic interaction in the initial state is included exactly, whereas the interaction between the fast outgoing electrons is calculated by perturbation theory. A detailed derivation of the expression for the cross section ratio between double and single ionizations is given. The corresponding results obtained by other authors are analyzed and corrected.     

Constraints on the asymptotic behaviour of total cross sections

We derive constraints on the asymptotic behaviour of total cross sections which follow from dispersion relations and measured real parts of the forward scattering amplitudes. For πN and pp scattering, these constraints are calculated using recent results from FNAL and Serpukhov. The relation to other methods is discussed.     

Study of the single-particle energy spectrum of disordered systems: III. The two-band case

A previous investigation of the energy spectrum of strongly disordered systems with the help of the recursion method is generalized to the two-band case. The real atomic structure of the considered systems is taken into consideration. The presence of short-range order within the system is shown to be the reason for the existence of extended states and of mobility edges. Typical fields of application are doped crystalline and amorphous Silicon as well as Germanium. For the two-band case a number of results is obtained: Mobility edges and their physical interpretation, localized and extended states as well as the structural conditions for their coexistence, analytical representation of the local density of states and of the state vectors, resonances and the general form of the total density of states (TDS) in the complete energy region. The spectral strength as function of the energy is shown to be discontinuous at the mobility edges. The TDS has a square root behaviour on both sides of the mobility edges what leads to a pseudogap of vanishing width at these points. This edge behaviour is shown to be related with the divergency the localization length of a state has as the state energy tends to the mobility edges. Model calculations are carried out.     

Investigation of the phase of the electromagnetic pion form factor in the asymptotic region

The phase of the electromagnetic pion form factor G(t) in the asymptotic region (t > 1 (GeV)²) is investigated by the dispersion sum-rule method. Using experimental data for G(t) at t < 1 (GeV)² we find that the mean weighted value of the asymptotic phase does not show strong deviations from the asymptotic value of the phase coming from an extrapolation of the data from the ?-resonance region with the help of the Gounaris-Sakurai formula. The contribution of the hypothetical ?-like resonances to the dispersion sum rules is roughly estimated. These estimates and the results of the calculation of the dispersion sum rules show that the existence of strong ?-like elastic resonances is forbidden in the 1–1.7 GeV $\text{t}$ range at least.     

On the form of the imaginary part of the optical potential

The imaginary part of the optical potential is derived fromMigdal's theory of nuclear structure. It is shown that the strong enhancement of ImV(r) at the nuclear surface is mainly due to a suppression of the effective nucleon-nucleon force by collective effects in the nuclear interior.