Borel summability for a nonpolynomial potential

We consider the energy levels of a one-dimensional quantum system in the rational potential 1/2x ²±gx ⁴/(1+gx ²). Their perturbation expansions ing are shown to be Borel summable. The proof is flexible enough to allow simple extensions to other nonpolynomial interactions.Physique Mathématique et Théorique, Equipe de Recherche associée au CNRS     

On a class of exactly soluble statistical mechanical models with nonpolynomial interactions

The approximating Hamiltonian method of N. N. Bogolubov, Jr. is generalized to models with nonpolynomial intensive-observable interactions. The original Hamiltonian is proved to be thermodynamically equivalent to one linear in the intensive-observable trial Hamiltonian. We show that the exact expression for the free energy density in the thermodynamic limit can be obtained from a min-max principle for the system with trial Hamiltonian.On leave of absence from Institute for Nuclear Research and Nuclear Energetics, Bulgarian Academy of Sciences, Sofia, Bulgaria.On leave of absence from Institute for Solid State Physics, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

Strong coupling in the quantum field theory with nonlocal nonpolynomial interaction

In the relativistic quantum field theory the representation for theS-matrix elements is obtained for any coupling constantsg in the case of a one component scalar field (x) with nonlocal nonpolynomial interaction _I ()=gU() when the causal function is bounded in the Euclidean region 0D _c (x _E ² D _c (0)< and the function |U(u)|1 for realu. It is proved that the two point Green function is bounded in the physical region of momenta variablep ².     

Borel summability of the unequal double well

Unlike the =0 case, the perturbation series of the unequal double wellp ²+x ²+2gx ³+g ²(1+)x ⁴ are Borel summable to the eigenvalues for any >0.     

Novel summability methods generalizing the Borel method

The paper deals with moment constant summability methods. A method is constructed which provides an analytic continuation of a function regular at the origin onto its Mittag-Leffler (principal) star. In this sense the method is optimal in contrast to the Borel one. In the next a one parameter family of such methods is introduced. The methods may be useful both in field theory and in statistical physics. Applications to the Nevanlinna theorem, the Rayleigh-Schrö-dinger perturbation theory and the dispersion-like integral are given. The proofs of theorems can be easily adapted to the study of the Mellin transform of some entire functions and a simpler proof of asymptotic properties of the gamma function can be obtained.I am indebted to J. Fuka for stimulating discussions and J. Fischer for continuous interest in my work and valuable comments. Financial support of the Czech Literatury Fund is also gratefully acknowledged.     

Existence and Borel summability of resonances in hydrogen stark effect

Existence of resonances in hydrogen Stark effect is proved. It is also proved that the divergent time-independent perturbation expansions are Borel summable to the resonances, and a simple application of the Borel-Padé method for computing their position and width is indicated.Partially supported by G.N.F.M., C.N.R.Partially supported by I.N.F.N., Sezione di Bologna.     

On the p-adic summability of the anharmonic oscillator

Some remarkable arithmetic properties of the anharmonic oscillator are considered. The concept of the p-adic summability is discussed. For the ground-state energy of the anharmonic oscillator the p-adic summability is shown.     

Construction and Borel summability of planar 4-dimensional euclidean field theory

We use the methods of [1] to show that the planar part of the renormalized perturbation theory for ₄ ⁴ -euclidean field theory is Borel-summable on the asymptotically free side of the theory. The Borel sum can therefore be taken as a rigorous definition of theN limit of a massiveN×N matrix model with a +trg ⁴ interaction, hence with wrong sign ofg. Our construction is relevant for a solution of the ultra-violet problem for planar QCD. We also propose a program for studying the structure of the renormalons singularities within the planar world.     

Generating functionals for harmonic expectation values of paths with fixed end points: Feynman diagrams for nonpolynomial interactions

We introduce a general class of generating functionals for the calculation of quantum-mechanical expectation values of arbitrary functionals of fluctuating paths with fixed end points in configuration or momentum space. The generating functionals are calculated explicitly for the harmonic oscillator with time-dependent frequency, and used to derive a smearing formula for correlation functions of polynomial and nonpolynomial functions of time-dependent positions and momenta. This formula summarizes the effect of quantum fluctuations, and serves to derive generalized Wick rules and Feynman diagrams for perturbation expansions of nonpolynomial interactions.     

Absolute summability of the trace relation for certain Schrödinger operators

A recently established general trace formula for one-dimensional Schrödinger operators is systematically studied in the context of short-range potentials, potentials which approach different spatial asymptotes sufficiently fast, and appropriate impurity (defect) interactions in one-dimensional solids. We prove the absolute summability of the trace formula and establish its connections with scattering quantities, such as reflection coefficients, in each case.This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material     

Borel-Le Roy summability of the high temperature expansion for classical continuous systems

For classical gases with suitable pair interactions such that (r)(lnr ^–1) ^p asr0 (p), the Taylor expansion in of the correlation functions and the pressure are summable at=0 by the Borel-Le Roy method of orderp+1.     

On Edwards' model for polymer chains. III. Borel summability

The basic existence theory for Edwards' model of long polymer chains is completed.     

On the Borel summability of planar perturbation series: Transition to Minkowski space

In recent years 't Hooft and Rivasseau proved the Borel summability of planar asymptotically free massive theories in Euclidean space. The corresponding Borel sums in Minkowski space are shown to exist as linear functionals if the Euclidean counterparts are bounded polynomially in momentum space and fulfill certain analyticity conditions. Both can be verified in massive planar wrong sign ₄ ⁴ using Rivasseau's approach. The functionals alternatively are densely defined and unbounded on anL ^p space or bounded on (the whole of) a Banach space with a more restrictive norm.Supported by a scholarship of the NATO scientific committee via the German academic exchange service (DAAD)After October 1, 1987: Max-Planck-Institut für Physik, Föhringer Ring 6, D-8000 München 40, Federal Republic of Germany     

Local Borel summability of Euclidean 311-1311-1311-1a simple proof via differential flow equations

It is shown how the differential flow equation (or, equivalently, the continous renormalization group) method can be employed to give an astonishingly easy proof of the local Borel summability of the renormalized perturbative Euclidean massive ₄ ⁴ .Supported by NSF grant # DMS-9100383     

The distributional Borel summability and the large coupling Φ4 lattice fields

We complete and correct some proofs of an earlier paper on distributional Borel summability and we add an application which can be useful in the discussion of semiclassical problems.     

Borel summability in the disorder parameter of the averaged Green's function for Gaussian disorder

In this note we prove Borel summability in the disorder parameter of the averaged Green's function <G(E,x,y>) _y of tight binding models $$H_V = - \Delta + V$$ with Gaussian disorder $$d\lambda (V) = (2\pi \gamma )^{ - 1/2} \exp ( - V^2 /2\gamma )dV$$ forγ→0 and fixed large |E|. Using this, we can reconstruct the density of states ?(E)γ from the Borel sums of <G(E,x,x>) _y with ImE↗0 and ImE↘0.     

Derivation of a 1/r^2 potential term for QQ in a potential model

We clarify the difference among potential models so far proposed to explain mass spectra of heavylight mesons via transformations of the vacuum.Applying our idea to QQ quarkonium,we obtain the extra term,1/r ^2 with positive coefficient,other than non-relativistic potential terms expected for quarkonium.     

The classical field limit of nonrelativistic bosons. I. Borel summability for bounded potentials

We study the classical field limit as $\text{h}\text{?}\text{→ 0}$ of nonrelativistic many-boson systems with two-body interaction in the neighborhood of a fixed ?-independent solution of the classical evolution equation (namely, the Hartree equation). The unitary group describing the evolution of the quantum system, after multiplication from both sides by suitable ?-dependent Weyl operators which somehow subtract out the classical motion, has an expansion in power series of $\text{h}\text{?}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, the zeroth-order term of which describes the quantum fluctuations of the system around the classical motion. We prove that for bounded two-body interaction potentials and small time intervals, this series is strongly Borel summable; namely, it yields a power series which is Borel summable in vector norm when applied to fixed ?-independent vectors taken from a suitable dense set.