The different renorminvariant charges in quantum electrodynamics

Together with the usual renorminvariant charge defined by photon propagator the invariant charge defined by Lagrange function of intensive constant electromagnetic field and the bare invariant charge defined by Z₃-factor are considered. The Gell-Mann-Low functions ψ₂(z) and ψ₃(z) of the last two charges were found in α³-approximation and turned out to be different from one another and from Gell-Mann-Low function ψ₁(z) of the first charge. Their negative α³-terms favour the existence of these functions' zeroes necessary for self-consistency of quantum electrodynamics.     

Dispersion relation for the Gell-Mann-Low function,the spectrality condition and the asymptotic behaviour of the invariant charge

It is pointed out that dispersion relation of the Gell-Mann-Low function together with a hypothesis about the spectrality define the asymptotic behaviour of the Gell-Mann-Low function ψ(g). For the g?⁴-model we get ψ(g) ≈ agatg ? 1.     

Gell-Mann-Low function in QED

The Gell-Mann-Low function β(g) in QED (g is the fine structure constant) is reconstructed. At large g, it behaves as β_∞ g ^α with α≈1 and β_∞≈1.     

(Non)-Abelian gauge field theories of the Fermi gas. Neutron-quark stars

It is indicated that the ground state of Fermi systems with (non)-Abelian gauge interactions has a well defined quantum theory devoid of infrared divergences and mass singularities. This is exploited to develop a systematic quantum theory of the quark gas. The equation of state of the quark gas is evaluated up to second order in the Gell-Mann-Low charge α_S(μ). The analysis based on neutron matter models suggests that the matter in the neutron stars can be in the quark phase provided the color interaction is “moderately” strong i.e. α_S (3 GeV) ? 0.3.     

The Fermi Golden Rule and its Form at Thresholds in Odd Dimensions

Let H be a Schrödinger operator on a Hilbert space , such that zero is a nondegenerate threshold eigenvalue of H with eigenfunction Ψ₀. Let W be a bounded selfadjoint operator satisfying 〈 Ψ₀, WΨ₀〈>0. Assume that the resolvent (H?z)^?1 has an asymptotic expansion around z=0 of the form typical for Schrödinger operators on odd-dimensional spaces. Let H(?) =H+?W for ?>0 and small. We show under some additional assumptions that the eigenvalue at zero becomes a resonance for H(?), in the time-dependent sense introduced by A. Orth. No analytic continuation is needed. We show that the imaginary part of the resonance has a dependence on ? of the form ?^2+(ν/2) with the integer ν≥?1 and odd. This shows how the Fermi Golden Rule has to be modified in the case of perturbation of a threshold eigenvalue. We give a number of explicit examples, where we compute the ``location'' of the resonance to leading order in ?. We also give results, in the case where the eigenvalue is embedded in the continuum, sharpening the existing ones.     

Propagating waves and edge vibrations in anisotropic composite cylinders

The entire dispersive spectra of a cylinder with cylindrical anisotropy are determined from three different algebraic eigenvalue problems deducible from the same finite element formulation. The displacement vector v in this version of the finite element method has the form f(r) exp i(εz + nθ + ωt) with the radial dependence f(r) taken as quadratic interpolation polynomials. Therefore, this discretization procedure allows a cylinder with radially inhomogeneous material properties to be modeled. The three different algebraic eigenvalue problems that emerge depend on whether the axial wave number ε or the natural frequency ω is regarded as the eigenvalue parameter and on the real, purely imaginary or complex nature of ε. For ε specified as real, an eigenvalue problem results for the natural frequencies ω_i for waves propagating along the z-direction of a cylinder of infinite extent. When ε is specified to be purely imaginary, then an algebraic eigenvalue problem governing the edge vibrations (end modes) of a semi-infinite cylinder is obtained. The third eigenvalue problem can be obtained by considering ω to be prescribed and regarding ε as the eigenvalue parameter. The algebraic eigenvalue problem that results is quadratic in the eigenvalue parameter and admits solutions for ε which may be real, purely imaginary or complex. Complex ε's correspond to edge vibrations in a cylinder which are exponentially damped trigonometric wave forms. Moreover, for the case ω = 0, the eigenvalue analysis yields ε as the characteristic inverse decay lengths for systems of elastostatic self-equilibrated edge effects in the context of St. Venant's principle. All the eigenvalue problems are solved by efficient techniques based on subspace iteration. Examples of two four-layer angle-ply cylinders are presented to illustrate this comprehensive finite element analysis.     

Quantum electrodynamics with “two-particle” unitarity: Charged spin-0 bosons

The spin-0 propagator in quantum electrodynamics of charged spinless bosons (“pions”) is investigated in detail via the closest approximation to the Schwinger-Dyson equation in the two-particle unitarity approximation proposed initially by Salam by replacing the photon propagator by its zero order result. The high-energy behavior (in spacelike directions) of the (renormalized) spin-0 propagator Δ(p²) is explicitly derived and a finite electromagnetic mass excitation for the “pion” is obtained. The latter follows as a result of the derived asymptotic behavior of Δ^?1(p²) (p² → ∞) that the unrenormalized mass m₀ ≡ 0. This brings us into contact with earlier work of Johnson, Landau and their collaborators. The fact that m₀² ≡ 0 and that Δ^?1(p²)/_p²=?m² = 0 (m ≠ 0) (in Nambu's sense) gives us an eigenvalue equation for the fine structure constant which may be then computed in terms of no other parameters. This eigenvalue equation is expanded in powers of the coupling constant with all the expansion coefficients being finite. The latter is computed to sixth order in the charge.     

Description of uniform many-particle systems in density operator function space

Assuming the ground state wavefunction, ψ₀, of a boson fluid is known, and writing the excited state wavefunctions in the form Fψ₀, a linear eigenvalue equation of the form $\text{H}\text{F = EF}$ is obtained, where E₀ + $\text{E}$ is the excited state energy, E₀ is the ground state energy, and H is a non-hermitian operator which depends in a simple way upon U ≡ ln ψ₀² instead of the potential energy function. An extremum principle is derived in terms of an auxiliary hermitian Hamiltonian operator, H′. The many-body boson plane-wave basis, ?_n(k₁ … k_n) is used to express U in terms of its Fourier components (ordered conveniently in terms of the number of nonzero arguments), making it possible to calculate matrix elements of ovcirc|H and H′ in that basis. A perturbation theory similar to Brillouin-Wigner perturbation theory is developed for the non-hermitian eigenvalue problem. Nonorthogonal perturbation theory is developed for the correlated basis ?_nψ₀. The requirement that these two perturbation theories be consistent produces useful relationships between the components of U and the static structure functions of ψ₀. These relationships are shown to reduce to previous results in the extreme case of low density and weak interactions.     

On the boundedness of the eigenvalue spectrum of phonon collision operator

Anharmonicity in lattice potential leads to boundedness of the eigenvalue Spectrum of the phonon collision operator. Considering the deviation, ψ_q, in the distribution function of the phonons $\text{N}{}_{\mathrm{q}}\text{, N}{}_{\mathrm{q}}\text{=}\text{N}\text{?}{}_{\mathrm{q}}\text{+}\text{N}\text{?}{}_{\mathrm{q\psi q}}\text{(}\text{N}\text{?}\text{+ 1)}$. bar denoting equilibrium value, as an odd function of the phonon wave vector q it has been possible to obtain a lower bound, μ, on the eigenvalue spectrum of the phonon collision operator P. This satisfies the inequelity relation 0?μ?p_i?λ, where p_i are the eigenvalue of P, and λ is an upper bound on it (as given by Benin). The occurence of μ ensures for the possibility of obtaining a sequence of upper bounds on the lattice thermal transport coefficient.     

Lax form of the quantum mechanical eigenvalue problem

The quantum mechanical eigenvalue problem with the hamiltonian H = H₀ + tV is written as a set of dynamical equations for the eigenvalues x_n(t) and the matrix elements V_nm(t) regarding the parameter t as time. By appropriate changes of variables it can be expressed as a pair of matrix equations with the Lax form, hence we are able to write all the possible constants of the motion explicitly. Implications of these constants to the statistical properties of levels are discussed.     

Gell-Mann-Low function in the ϕ4 theory

An algorithm is proposed for the determination of the asymptotics of a sum of a perturbation series from the given values of its coefficients in the strong-coupling limit. When applied to the ?⁴ theory, the algorithm yields the β(g)∝g ^α behavior with α≈1 at large g for the Gell-Mann-Low function.     

Fragmentation of giant multipole resonances in deformed nuclei

The fragmentation of the giant monopole resonance in deformed nuclei is first studied by coupling the monopole oscillation with the quadrupole oscillation by means of the variational procedure for resonance frequencies. It is shown that, for non-axial symmetry, the monopole oscillation couples with both m = 0 and 2 modes of the quadrupole oscillation and the giant monopole resonance is split into three components, whereas for axial symmetry, the fragmentation is given by E₀(1 + 0.86δ² ± 1.25δ³) and E₀(0.74 ± 0.22δ ? 0.21δ² ± 0.57δ³), where E₀ is the g monopole resonance energy for spherical nuclei, δ is the deformation parameter, and the upper and lower signs stand for prolate and oblate deformations, respectively. The initial fragmentation of the giant quadrupole resonance is seen to be little modified by the coupling, except for the m = 0 mode which is split into two components. The variational method is extended to general multipoles for an ellipsoid and the fragmentation of giant multipole resonances in deformed nuclei is investigated for both axial and non-axial symmetries. A brief discussion is also made about the meaning of the energy eigenvalue involved in the model wave equation in terms of multipole sum rules. The giant dipole resonance for the static octupole deformation is shortly considered. The giant E0 and E3 resonances for largely deformed nuclei are finally examined by solving the spheroidal eigenvalue equation and they are compared with the results of the giant dipole and quadrupole resonances.     

Sur une Valeur propre d'un Operateur

The studied eigenvalue is the smallest one of the operator: $$H_\mu = \mu A* + i\lambda A*(A + A*)A$$ in the orthogonal complement of the vacuum whereA* andA are the creation and annihilation operators. Call this eigenvalueE(μ) as attention is focused on the dependence on μ. This eigenvalue exists and is a positive number for positive values of μ. Using the fact that the inverse ofH _μ is a positive operator, it is proved thatE extends to a positive, increasing, analytic function on the whole real line. In particular,E(0)≠0, contrary to what might have been expected from the fact thatA*(A+A*)A is formally self-adjoint.     

Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case

Consider ?Δ + λV with V short range at a value λ₀ where some eigenvalue e(λ) → 0 as λ ↓ λ₀. We analyze two questions: (i) What is the leading order of e(λ), i.e., for what α does e(λ) ～ c(λ ? λ₀)^α? (ii) Is e(λ) analytic at λ = λ₀ and, if not, what is the natural expansion parameter? The results are highly dimension dependent.     

Renormalization and scaling of a two-dimensional Ising system in a transverse field at Tc>0

A quantum generalization of the Niemeijer and van Leeuwen (N-vL) renormalization group transformation is constructed which allows to study the critical properties (at T_c>0) of a two-dimensional Ising system with a transverse field on a triangular lattice. Explicit calculations are performed in a second order cumulant expansion. Only one fixed point is found corresponding to the same Ising-like hamiltonian given by N-vl. The linearized transformation has a zero eigenvalue associated with Γ, the transverse field strength. The critical properties of the system are briefly discussed; in particular we show that the singular behaviour of the transverse susceptibility at Γ = 0, which turns out to be of the same kind as that of the energy density, is explained by the existence of such a zero eigenvalue. Our arguments suggest a natural extension of this result to three dimensions.     

Universal estimate of the gap for the Kac operator in the convex case

The aim of this paper is to prove that ifV is a strictly convex potential with quadratic behavior at ∞, then the quotient μ₂/μ₁ between the largest eigenvalue and the second eigenvalue of the Kac operator defined on L²(? ^m ) by exp ?V(x)/2 · exp Δ_x · exp ?V(x)/2 where Δ_x is the Laplacian on ? ^m satisfies the condition: $${{\mu _2 } \mathord{\left/ {\vphantom {{\mu _2 } {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}} \right. \kern-\nulldelimiterspace} {\mu _1 {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} \mathord{\left/ {\vphantom {{ \leqslant \exp - \cosh ^{ - 1} (\sigma + 1)} {2,}}} \right. \kern-\nulldelimiterspace} {2,}}}}$$ where σ is such that HessV(x)≥σ>0.     

General lower bounds for resonances in one dimension

Lower bounds are derived for the magnitude of the imaginary parts of the resonance eigenvalues of a Schrödinger operator $$ - \frac{{d^2 }}{{dx^2 }} + V(x)$$ on the line, depending only on the support and bounds ofV and on the real part of the resonance eigenvalue. For example, if the resonance eigenvalue is denotedE +i?, then there existC and ?₀ depending only on ‖E‖_∞ andE such that if the support ofV is contained in an interval of length ? > ?₀, then $$\left| \varepsilon \right| > \frac{{m^3 \sqrt E }}{{(m + \sqrt E )^2 }}\exp ( - m\ell )(1 - C\ell ^{ - 1} ),$$ wherem‖V(x)?E? _∞ ^1/2 .     

Influence of the gyromagnetic term in the effect of an oblique magnetic field on the superparamagnetic relaxation time

The effect of a uniform magnetic field applied at an oblique angle ψ to the easy axis of magnetization on the superparamagnetic (longitudinal or Néel) relaxation time is illustrated by considering the simplest possible case where the field is applied normal to the easy axis. This is accomplished by numerically solving the Fokker-Planck equation for the smallest non-vanishing eigenvalue λ₁. It is demonstrated that the asymptotic formula for the Kramers escape rate for general non-axially symmetric potentials evaluated for the particular case ψ = π/2 yields an acceptable asymptotic approximation to the behaviour of λ₁ for sufficiently high-potential barrier heights and a wide range of values of the dimensionless damping factor. The effect of the gyromagnetic term which gives rise to coupling between the longitudinal and transverse modes of motion corresponds to an increase (higher when the dimensionless damping factor a is smaller) of the smallest non-vanishing eigenvalue λ₁ and so to a decrease of the Néel relaxation time τ.     

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

We study the determinant det(I?K _PII) of an integrable Fredholm operator K _PII acting on the interval (?s, s) whose kernel is constructed out of the Ψ-function associated with the Hastings–McLeod solution of the second Painlevé equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the unitary ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann–Hilbert method, we evaluate the large s-asymptotics of det(I?K _PII) .     

Features of multiplicity distributions in a bootstrap model of inclusive spectra

Using a bootstrap model of inclusive spectra we derive an integral equation satisfied by the generating function for multiplicity distributions. The (semi-asymptotic) solution of this equations has the form Ψ(λ, s) = Ψ(λ, s₀)(s/s₀)^b(λ) where s is the usual energy variable and b(λ) satisfies an eigenvalue equation and is completely determined by the leading particle distribution. Closed formulae for the binomial moments and for correlation coefficients are also given, and in addition we discuss some general features of the bootstrap model. As a phenomenological application we discuss the rate of variation with energy of multiplicity moments. Our results are expected to be representative for multiperipheral-like models.