TBA equations and quantization conditions

Abstract It has been recently realized that, in the case of polynomial potentials, the exact WKB method can be reformulated in terms of a system of TBA equations. In this paper we study this method in various examples. We develop a graphical procedure due to Toledo, which provides a fast and simple way to study the wall-crossing behavior of the TBA equations. When complemented with exact quantization conditions, the TBA equations can be used to solve spectral problems exactly in Quantum Mechanics. We compute the quantum corrections to the all-order WKB periods in many examples, as well as the exact spectrum for many potentials. In particular, we show how this method can be used to determine resonances in unbounded potentials.

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Spectral Problems for Quasinormal Modes of Black Holes

Universe ◽

10.3390/universe7120476 ◽

2021 ◽

Vol 7 (12) ◽

pp. 476

Author(s):

Yasuyuki Hatsuda ◽

Masashi Kimura

Keyword(s):

Black Holes ◽

Black Hole ◽

Perturbation Theory ◽

Quantum Mechanics ◽

Quasinormal Modes ◽

Review Article ◽

Wkb Method ◽

Spectral Problems ◽

Points Of View ◽

Numerical Treatments

This is an unconventional review article on spectral problems in black hole perturbation theory. Our purpose is to explain how to apply various known techniques in quantum mechanics to such spectral problems. The article includes analytical/numerical treatments, semiclassical perturbation theory, the (uniform) WKB method and useful mathematical tools: Borel summations, Padé approximants, and so forth. The article is not comprehensive, but rather looks into a few examples from various points of view. The techniques in this article are widely applicable to many other examples.

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Wall-crossing invariants: from quantum mechanics to knots

Journal of Experimental and Theoretical Physics ◽

10.1134/s1063776115030206 ◽

2015 ◽

Vol 120 (3) ◽

pp. 549-577 ◽

Cited By ~ 9

Author(s):

D. Galakhov ◽

A. Mironov ◽

A. Morozov

Keyword(s):

Quantum Mechanics ◽

Wall Crossing

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Hyper-asymptotic expansions and instantons

From Random Walks to Random Matrices ◽

10.1093/oso/9780198787754.003.0024 ◽

2019 ◽

pp. 471-492

Author(s):

Jean Zinn-Justin

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Quantum Mechanics ◽

Asymptotic Expansions ◽

Perturbative Expansion ◽

Quantum Field ◽

Wkb Method ◽

Additional Information ◽

Spectral Equation ◽

Double Well Potential

Chapter 24 examines the topic of hyper–asymptotic expansions and instantons. A number of quantum mechanics and quantum field theory (QFT) examples exhibit degenerate classical minima connected by quantum barrier penetration effects. The analysis of the large order behaviour, based on instanton calculus, shows that the perturbative expansion is not Borel summable, and does not define unique functions. An important issue is then what kind of additional information is required to determine the exact expanded functions. While the QFT examples are complicated, and their study is still at the preliminary stage, in quantum mechanics, in the case of some analytic potentials that have degenerate minima (like the quartic double–well potential), the problem has been completely solved. Some examples are described in Chapter 24. There, the perturbative, complete, hyper–asymptotic expansion exhibits the resurgence property. The perturbative expansion can be related to the calculation of the spectral equation via the complex WKB method.

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The intersection between dual potential and sl(2) algebraic spectral problems

International Journal of Modern Physics A ◽

10.1142/s0217751x20502085 ◽

2020 ◽

Vol 35 (32) ◽

pp. 2050208

Author(s):

William H. Pannell

Keyword(s):

Quantum Mechanics ◽

Harmonic Oscillator ◽

Transformation Formula ◽

Algebraic Construction ◽

Spectral Problems ◽

Partner Potential

The relation between certain Hamiltonians, known as dual, or partner Hamiltonians, under the transformation [Formula: see text] has long been used as a method of simplifying spectral problems in quantum mechanics. This paper seeks to examine this further by expressing such Hamiltonians in terms of the generators of sl(2) algebra, which provides another method of solving spectral problems. It appears that doing so greatly restricts the set of allowable potentials, with the only nontrivial potentials allowed being the Coulomb [Formula: see text] potential and the harmonic oscillator [Formula: see text] potential, for both of which the sl(2) expression is already known. It also appears that, by utilizing both the partner potential transformation and the formalism of the Lie-algebraic construction of quantum mechanics, it may be possible to construct part of a Hamiltonian’s spectrum from the quasi-solvability of its partner Hamiltonian.

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Quantum mechanics on (anti)-de Sitter background II: Ramsauer–Townsend effect and WKB method

Modern Physics Letters A ◽

10.1142/s021773231850150x ◽

2018 ◽

Vol 33 (26) ◽

pp. 1850150 ◽

Cited By ~ 13

Author(s):

Won Sang Chung ◽

Hassan Hassanabadi

Keyword(s):

Quantum Mechanics ◽

Energy Level ◽

Harmonic Oscillator ◽

Wkb Method ◽

De Sitter ◽

Quantum Harmonic Oscillator ◽

One Dimensional ◽

Sitter Background ◽

The One

Based on the one-dimensional quantum mechanics on (anti)-de Sitter background [W. S. Chung and H. Hassanabadi, Mod. Phys. Lett. A 32, 26 (2107)], we discuss the Ramsauer–Townsend effect. We also formulate the WKB method for the quantum mechanics on (anti)-de Sitter background to discuss the energy level of the quantum harmonic oscillator and quantum bouncer.

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WKB method for potentials unbounded from below

Modern Physics Letters B ◽

10.1142/s0217984916500032 ◽

2016 ◽

Vol 30 (03) ◽

pp. 1650003 ◽

Cited By ~ 3

Author(s):

Aleksandar Demić ◽

Vitomir Milanović ◽

Jelena Radovanović ◽

Milenko Musić

Keyword(s):

Quantum Mechanics ◽

Bound States ◽

Wave Functions ◽

Wkb Method ◽

Symmetric Potential ◽

Overlap Integral ◽

Entire Spectrum ◽

One Dimensional ◽

Model Based ◽

Exactly Solvable

Bound states degenerated in energy (and differing in parity) may form in one-dimensional quantum mechanics if the potential is unbounded from below. We focus on symmetric potential and present quasi-exactly solvable (QES) model based on WKB method. The application of this method is limited on slow-changing potentials. We consider the overlap integral of WKB wave functions [Formula: see text] and [Formula: see text] which correspond to energies [Formula: see text] and [Formula: see text], and by setting [Formula: see text], we determine the type of spectrum depending on parameter [Formula: see text] which arises from this method. For finite value [Formula: see text], we show that the entire spectrum will consist of degenerated bound states.

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