Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point, II — Its relevance to the Mathieu equation and the Legendre equation

Kamimoto, Shingo; Kawai, Takahiro; Takei, Yoshitsugu

doi:10.1016/j.aim.2014.02.028

Cited by 10 publications

(4 citation statements)

References 5 publications

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“…Using the vectors' identities ( r × B) 2 and the canonical angular momentum operator, L ≡ r × i ∇, we can write and simplify the time-independent Schrödinger equation for a charged particle of charge q in an external uniform magnetic field B to…”

Section: The Uniform Magnetic Field Effectsmentioning

confidence: 99%

“…To solve this problem, there are some approximate approaches that can analytically provide accepted solutions such as the WKB, the variational method (VM) and the perturbation theory. Moreover, many numerical methods, such as the Airy function approach, the asymptotic iteration, the Numerov method (NM) and the finite element method [2][3][4][5][6][7][8][9][10][11] have been suggested as solutions. Finding exact solutions to the Schrödinger equation for potentials that prove useful in the modelling of physical phenomena is a very important challenge for a deep understanding of the structures and interactions in such systems.…”

Section: Introductionmentioning

confidence: 99%

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Etingof’s conjecture for quantized quiver varieties

Bezrukavnikov

Losev

2020

Invent. math.

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We propose the Whittaker function approach as a theoretical method for finding exact solutions of a quantum mechanical system placed in the Kratzer potential. We then show that the effect of an external uniform magnetic field on this system can be satisfactorily determined using variational method. By following the onestep treatment suggested in this study, we increase the reliability and the accuracy of the solutions of Schrödinger equation for a quantum mechanical system placed in potential energy and perturbed by a uniform magnetic field that proves to be useful in modelling physical phenomena. We find that the achieved numerical and analytical results agree very well with those already published and those calculated using the Numerov method.

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Section: The Uniform Magnetic Field Effectsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Etingof’s conjecture for quantized quiver varieties

Bezrukavnikov

Losev

2020

Invent. math.

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“…A very partial list of contributions includes the works by Aoki, Kamimoto, Kawai, Koike, Sasaki, and Takei, [36][37][38][39][40][41]. Parallel to this activity has been the classification of WKB geometry (or Stokes graphs), which includes the works by Aoki, Kawai, Takei, Tanda, [42][43][44][45], as well as a detailed analysis of some WKB-theoretic properties of special classes of equations, which includes the works by Aoki, Kamimoto, Kawai, Koike, and Takei [46][47][48][49][50][51][52][53][54].…”

Section: Introductionmentioning

confidence: 99%

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

Nikolaev

2023

Commun. Math. Phys.

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We prove an existence and uniqueness theorem for exact WKB solutions of general singularly perturbed linear second-order ODEs in the complex domain. These include the one-dimensional time-independent complex Schrödinger equation. Notably, our results are valid both in the case of generic WKB trajectories as well as closed WKB trajectories. We also explain in what sense exact and formal WKB solutions form a basis. As a corollary of the proof, we establish the Borel summability of formal WKB solutions for a large class of problems, and derive an explicit formula for the Borel transform.

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“…If the system consists of two particles or more, the solutions become complicated. Hence, approximation and numerical approaches like WKB and the variational methods, the perturbation theory, the airy function approach, the asymptotic iteration, and the finite element methods [4][5][6][7][8][9][10][11] are considered to solve such problems.…”

Section: Introductionmentioning

confidence: 99%

Analytic solutions for vibrational energy levels of the pseudoharmonic potential

Maiz¹

2020

Phys. Scr.

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The non-relativistic analytic solutions of a quantum mechanical system have been calculated for the case of pseudoharmonic potential by using the Whittaker functions approach. A diatomic quantum system is placed in a Pseudoharmonic potential and perturbed by an external magnetic field. The resulting Schrodinger’s equation has been solved exactly to obtain the analytic expressions of vibrational energy levels and associated wave functions. In this work, we have compared the results of six diatomic molecules with those available in the literature.

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Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point, II — Its relevance to the Mathieu equation and the Legendre equation

Cited by 10 publications

References 5 publications

Etingof’s conjecture for quantized quiver varieties

Etingof’s conjecture for quantized quiver varieties

Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs

Analytic solutions for vibrational energy levels of the pseudoharmonic potential

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