Perturbation Theory for Arbitrary Coupling Strength

Nuclear Physics B - Proceedings Supplements, 2006

Adaptive perturbation is a new method for perturbatively computing the eigenvalues and eigenstates of quantum mechanical Hamiltonians that are widely believed not to be solvable by such methods [1]. The novel feature of adaptive perturbation theory is that it decomposes a given Hamiltonian, H, into an unperturbed part and a perturbation in a way which extracts the leading non-perturbative behavior of the problem exactly. In this talk I will introduce the method in the context of the pure anharmonic oscillator and then apply it to the case of tunneling between symmetric minima. After that, I will show how this method can be applied to field theory. In that discussion I will show how one can non-perturbatively extract the structure of mass, wavefunction and coupling constant renormalization.

2005

Adaptive perturbation is a new method for perturbatively computing the eigenvalues and eigenstates of quantum mechanical Hamiltonians that heretofore were not believed to be obtainable by such methods. The novel feature of adaptive perturbation theory is that it decomposes a given Hamiltonian, H, into an unperturbed part and a perturbation in a way which extracts the leading non-perturbative behavior of the problem exactly. This paper introduces the method in the context of the pure anharmonic oscillator and then goes on to apply it to the case of tunneling between both symmetric and asymmetric minima. It concludes with an introduction to the extension of these methods to the discussion of a quantum field theory. A more complete discussion of this issue will be given in the second paper in this series. This paper will show how to use the method of adaptive perturbation theory to non-perturbatively extract the structure of mass, wavefunction and coupling constant renormalization.

2002

The main issue of this work consists in extracting one or several finite values for the sum of series involved in perturbation theories. It is supposed to work for all cases in which two physical parameters are involved, and makes thorough use of dimensional arguments concerning these physical quantities. Weak and strong coupling expansions are considered as the two faces of a formal expression which is the central object of this method. This so-called extension procedure is systematic. We apply it here to the divergent perturbative expansion of the ground state energy of the anharmonic oscillator of quantum mechanics in zero and one dimension, and provide, given a p-order divergent expansion, an analytical expression of its estimated sum. This method which is inspired by variational procedures provides high degree of accuracy from lower orders of perturbation and seems to have remarkable converge properties in a wide class of expansions concerning physical observables.

International Journal of Modern Physics A, 2005

A self-consistent, nonperturbative approximation scheme is proposed which is potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction HI(ϕ) by a suitable "potential" V(ϕ) which satisfies the following two basic requirements, (i) exact solvability (ES): the "effective" Hamiltonian H0 generated by V(ϕ) is exactly solvable i.e., the spectrum of states |n〉 and the eigenvalues En are known and (ii) equality of quantum averages (EQA): 〈n|HI(ϕ)|n〉 = 〈n|V(ϕ)|n〉 for arbitrary n. The leading order (LO) results for |n〉 and En are thus readily obtained and are found to be accurate to within a few percent of the "exact" results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H0 as the unperturbed Hamiltonian and the modified interaction, λH′(ϕ)≡λ(HI(ϕ) - V(ϕ)), as the perturbation wher...

Modern Physics Letters A, 2005

An algebraic nonperturbative approach is proposed for the analytical treatment of Schrödinger equations with a potential that can be expressed in terms of an exactly solvable piece with an additional potential. Avoiding disadvantages of standard approaches, new handy recursion formulas with the same simple form both for ground and excited states have been obtained. As an illustration the procedure, well adapted to the use of computer algebra, is successfully applied to quartic anharmonic oscillators by means of very simple algebraic manipulations. The trend of the exact values of the energies is rather well reproduced for a large range of values of the coupling constant (g = 0.001–10000).

Theoretical Chemistry Accounts, 2016

The partitioning theory provides optical potentials which lead to the resonance energies. The state of the theory in the eighties can be found in the book "Theory of resonances" of Kukulin et al. [3]. The basic principles presented in this book still hold: use of projectors and analytical continuation. Since that time, the theoretical developments have been influenced by the fast development of computers and the theory entered into the quickly expanding field of non-Hermitian quantum mechanics [4]. Jolicard and Austin [5] incorporated optical potentials in computational schemes, and the justification of their method was given by Riss and Meyer [6]. To avoid confusion, we do not use the term optical potential here, but adopt the more suitable expression complex absorbing potential (CAP), as proposed in Ref. [6]. The acronym CAP refers to an energy-independent complex potential added to the Hamiltonian (see review article [7]). For the past two decades, we contributed to the development of approaches combining CAP and perturbation theory [8-12]. In this article, we recognize a resonance as a pole of a Green's function. In Sect. 2, we first recall the definition of the wave function and the energy in the framework of the partitioning technique. Then, a complex absorbing potential written in the form V CAP = −ı ǫ is added to the Hamiltonian. The operator ǫ generalizes the quantity ǫ of collision theory. The parameter in V CAP anticipates that V CAP does not only produce analytic continuation, but that it is also a perturbation operator. The self-energy is expanded in powers of , and the equivalence between this expansion and a Taylor series is demonstrated in Sect. 3 which focuses on our new findings. Finally, the introduction of a convergence operator allows to discuss the convergence properties of these series. Section 4 is devoted to numerical illustrations. First, a discretized N-dimensional Fano model is used to check the accuracy and the convergence properties of the energies. It is shown Abstract We propose a contribution to the theory of quantum resonances that combines complex absorbing potentials (CAP) with standard perturbation theory. We start from resolvents that depend on two variables, the complex energy z and a perturbation parameter. The wave functions and the energies of the resonances are expanded in powers of. It is shown that the zero-order terms correspond to the standard CAP method and that higher-order corrections are significant. The introduction of a convergence operator allows to control the convergence of the perturbation series. Due to the discretization of the continuum, the series are generally asymptotic. Finally, we relate the perturbation series to numerically convenient Taylor series. The theory is illustrated on two model examples.

Physica A: Statistical Mechanics and its Applications, 2000

We discuss the e ects of resonances in driven quantum systems within the context of quantum averaging techniques in the Floquet representation. We consider in particular iterative methods of KAM type and the extensions needed to take into account resonances. The approach consists in separating the coupling terms into resonant and nonresonant components at a given scale of time and intensity. The nonresonant part can be treated with perturbative techniques, which we formulate in terms of KAM-type unitary transformations that are close to the identity. These can be interpreted as averaging procedures with respect to the dynamics deÿned by e ective uncoupled Hamiltonians. The resonant parts are treated with a di erent kind of unitary transfomations that are not close to the identity, and are adapted to the structure of the resonances. They can be interpreted as a renormalization of the uncoupled Hamiltonian, that yields an e ective dressed Hamiltonian, around which a perturbation expansion can be developed. The combination of these two ingredients provides a strongly improved approximation technique.

Physical review, 2016

Schwinger's formalism in quantum field theory can be easily implemented in the case of scalar theories in D dimension with exponential interactions, such as µ D exp(αφ). In particular, we use the relation exp α δ δJ(x) exp(−Z 0 [J]) = exp(−Z 0 [J + α x ]) with J the external source, and α x (y) = αδ(y − x). Such a shift is strictly related to the normal ordering of exp(αφ) and to a scaling relation which follows by renormalizing µ. Next, we derive a new formulation of perturbation theory for the potentials V (φ) = λ n! : φ n :, using the generating functional associated to : exp(αφ) :. The ∆(0)-terms related to the normal ordering are absorbed at once. The functional derivatives with respect to J to compute the generating functional are replaced by ordinary derivatives with respect to auxiliary parameters. We focus on scalar theories, but the method is general and similar investigations extend to other theories.

Annals of Physics - ANN PHYS N Y, 1977

We develop a convenient functional integration method for performing mean-field approximations in quantum field theories. This method is illustrated by applying it to a self-interacting 4" scalar field theory and a PJ, four-Fermion field theory. To solve the 4" theory we introduce an auxiliary field x and rewrite the Lagrangian so that the interaction term has the form ~4~. The vacuum generating functional is then expressed as a path integral over the fields x and 4. Since the x field is introduced to make the action no more than quadratic in #, we do the 4 integral exactly. Then we use Laplace's method to expand the remaining x integral in an asymptotic series about the mean field x0 . We show that there is a simple diagrammatic interpretation of this expansion in terms of the mean-field propagator for the elementary field 4 and the mean-field bound-state propagator for the composite field x. The 4 and x propagators appear in these diagrams with the same topological structure that would have been obtained by expanding in the same manner a ~4% field theory in which x and 4 are both elementary fields. We therefore argue that by renormalizing these theories so that the mean-field propagators are equivalent, the two theories are described by the same renormalized Green's functions containing the same three parameters, pa, m*, and g. The quartic theory is completely specified by the renormalized masses $' and mp of the x and 4 fields. These two masses determine the coupling constant g = g($, ma). The cubic theory depends on pa and me and a third parameter g, , g = g($, m2, g,), where g,, is the bare coupling constant. We indicate that g($, m2, gO) < g($, mz) with equality obtained only in the limit g,, + co. When g, + co the wavefunction renormalization constant for the x field in the cubic theory vanishes, and the cubic theory becomes identical to the quartic theory. Our approach guarantees that all quartic theories have the same * Sloan Foundation Research Fellow.