Late terms in the asymptotic expansion for the energy levels of a periodic potential

Stone, Michael; Reeve, J S

doi:10.1103/physrevd.18.4746

Cited by 34 publications

(50 citation statements)

References 11 publications

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“…The trans-series obtained from exact quantization conditions are usually based on instanton solutions to the Euclidean EOM. However, there are no real instanton solutions for the cosh(x) potential, and one needs complex instantons [38] coming from classical trajectories along the imaginary axis in the complex x plane [72,73], where we have a periodic potential. One way to find the appropriate trans-series for the modified Mathieu equation is to start with the cos(x) potential (i.e., the Mathieu equation).…”

Section: Comparison With Previous Results In Quantum Mechanicsmentioning

confidence: 99%

Non-perturbative Quantum Mechanics from Non-perturbative Strings

Codesido

Mariño

Schiappa

2018

Ann. Henri Poincaré

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This work develops a new method to calculate non-perturbative corrections in onedimensional Quantum Mechanics, based on trans-series solutions to the refined holomorphic anomaly equations of topological string theory. The method can be applied to traditional spectral problems governed by the Schrödinger equation, where it both reproduces and extends the results of well-established approaches, such as the exact WKB method. It can be also applied to spectral problems based on the quantization of mirror curves, where it leads to new results on the transseries structure of the spectrum. Various examples are discussed, including the modified Mathieu equation, the double-well potential, and the quantum mirror curves of local P 2 and local F 0 . In all these examples, it is verified in detail that the trans-series obtained with this new method correctly predict the large-order behavior of the corresponding perturbative sectors.

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Section: Comparison With Previous Results In Quantum Mechanicsmentioning

confidence: 99%

Non-perturbative Quantum Mechanics from Non-perturbative Strings

Codesido

Mariño

Schiappa

2018

Ann. Henri Poincaré

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“…Following the original publication the vast application of the method was used to analyze the anharmonic oscillator. A notable exception is [23], which attempted to generalize the method to a Mathieu potential, but encountered a numerical instability of the recurrence relations for level numbers ν > 1. Reference [24] applied the method to the supersymmetric version of the double-well potential.…”

Section: Comments On the Literaturementioning

confidence: 99%

Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica ® package

Sulejmanpašić

Ünsal

2018

Computer Physics Communications

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We discuss a general setup which allows the study of the perturbation theory of an arbitrary, locally harmonic 1D quantum mechanical potential as well as its multi-variable (many-body) generalization. The latter may form a prototype for regularized quantum field theory. We first generalize the method of Bender-Wu, and derive exact recursion relations which allow the determination of the perturbative wave-function and energy corrections to an arbitrary order, at least in principle. For 1D systems, we implement these equations in an easy to use MATHEMATICA ® package we call BenderWu. Our package enables quick home-computer computation of high orders of perturbation theory (about 100 orders in 10-30 seconds, and 250 orders in 1-2h) and enables practical study of a large class of problems in Quantum Mechanics. We have two hopes concerning the BenderWu package. One is that due to resurgence, large amount of non-perturbative information, such as nonperturbative energies and wave-functions (e.g. WKB wave functions), can in principle be extracted from the perturbative data. We also hope that the package may be used as a teaching tool, providing an effective bridge between perturbation theory and non-perturbative physics in textbooks. Finally, we show that for the multi-variable case, the recursion relation acquires a geometric character, and has a structure which allows parallelization to computer clusters.

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“…In fact, the Hamiltonian densities v ℓ (x) for the elliptic potential have no pole of order one at x = 0, therefore, they are linear combinations of ∂ k x ℘(x) with k 0. Then the integrated Hamiltonians appearing in (12) are linear combinations of x and ∂ k x ζ(x) with k 0. In the wave function (15) the phase e ±iνx contains the linear term of x, the coefficients of ν −l are linear combinations of ∂ k x ζ(x) with k 0, probably include a x-independent constant term.…”

Section: Lamé Equationmentioning

confidence: 99%

A New Treatment for Some Periodic Schrödinger Operators II: The Wave Function

He¹

2018

Commun. Theor. Phys.

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Following the approach of our previous paper we continue to study the asymptotic solution of periodic Schrödinger operators. Using the eigenvalues obtained earlier the corresponding asymptotic wave functions are derived. This gives further evidence in favor of the monodromy relations for the Floquet exponent proposed in the previous paper. In particular, the large energy asymptotic wave functions are related to the instanton partition function of N=2 supersymmetric gauge theory with surface operator. A relevant number theoretic dessert is appended. (2010): 35P20, 33E10, 34E10. Mathematics Subject Classification

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Late terms in the asymptotic expansion for the energy levels of a periodic potential

Abstract: We obtain a general formula for the late terms in the perturbation expansion of the energy levels of a periodic potential and compare it with the computed values for the first one hundred terms.

Cited by 34 publications

References 11 publications

Non-perturbative Quantum Mechanics from Non-perturbative Strings

Non-perturbative Quantum Mechanics from Non-perturbative Strings

Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica ® package

A New Treatment for Some Periodic Schrödinger Operators II: The Wave Function

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