Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator

Álvarez, Gabriel; Casares, Carmen

doi:10.1088/0305-4470/33/29/302

Cited by 56 publications

(71 citation statements)

References 27 publications

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“…Again, these calculations can be performed via matrix diagonalization in the basis of harmonic oscillator wavefunctions. The result of such diagonalization procedure for a coupling parameter g = 0.01 is presented in the last line of the Table 9 and is found in perfect agreement with previous calculations [236].…”

Section: Theorem 13supporting

confidence: 88%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

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Section: Theorem 13supporting

confidence: 88%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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“…A second type of resurgent behavior is less generic, yielding a "low order/low order" form of resurgence, in which the fluctuations about all non-perturbative sectors are explicitly encoded in the perturbative expansion about the vacuum sector. This 'constructive' form of resurgence appears to have first been noticed in formulas for the ionization rate for hydrogenic atoms [16], a result that motivated a systematic study byÁlvarez and Casares in the context of one dimensional oscillators [17][18][19][20], in which such explicit perturbative/non-perturbative (P/NP) relations were found in the cubic and quartic oscillator systems. Later studies found further examples of such P/NP relations in the periodic cosine (Mathieu), supersymmetric double-well, radial anharmonic oscillator, and supersymmetric Mathieu potentials [21][22][23][24][25], and more recently in quasi-exactly soluble models [26].…”

Section: Introductionmentioning

confidence: 91%

“…We stress that this manifestation of resurgence is completely constructive: given a certain number of terms of the expansion of u pert ( , N ), the expression (1.4) generates a similar number of terms in the fluctuations about the one-instanton sector, P inst ( , N ). Furthermore, these relations propagate throughout the entire trans-series, so that perturbation theory encodes the fluctuations about each nonperturbative sector [17][18][19][20][21][22][23][24][25]27].…”

Section: Jhep05(2017)087mentioning

confidence: 99%

“…Concerning the generality of these results, we first note that the common feature of the examples studied in [16][17][18][19][20][21][22][23] is that their classical energy-momentum relation p 2 = 2 u − V (x) defines a genus 1 elliptic curve: a torus. In section 2 we formulate a simple geometric argument which shows that for all potentials corresponding to such a genus 1 elliptic curve, the all-orders ("non-perturbative") dual action a D (u, ) is constructively encoded in the ("perturbative") action a(u, ).…”

Section: Jhep05(2017)087mentioning

confidence: 99%

“…There is a systematic procedure in classical geometry for reduction to standard forms, determining the geometric invariants from the elliptic curve [115,116], summarized in appendix A. The examples in [16][17][18][19][20][21][22][23][24][25]27] are all genus 1 (recall that the Hydrogenic Stark problem separates in parabolic coordinates to anharmonic oscillator problems [117]). The general proof of the perturbative/non-perturbative relation for genus 1 systems proceeds in three simple steps:…”

Section: Jhep05(2017)087mentioning

confidence: 99%

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Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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For a wide variety of quantum potentials, including the textbook 'instanton' examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain N = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological c = 3 Landau-Ginzburg models and 'special geometry'. These systems inherit a natural modular structure corresponding to Ramanujan's theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.

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Exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator

Cited by 56 publications

References 27 publications

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

Quantum geometry of resurgent perturbative/nonperturbative relations

Symbolic Computations

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