scholarly journals On spectral asymptotic of quasi-exactly solvable quartic potential

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Boris Shapiro ◽  
Miloš Tater
Keyword(s):  
Numerical Study ◽  
Commun Math Phys ◽  
Level Crossings ◽  
Quartic Potential ◽  
Asymptotic Shape ◽  
Exactly Solvable ◽  
Math Phys

AbstractMotivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.

Erratum: Commun. Math. Phys. 217, 127-163 (2001)

2002 ◽  
Vol 225 (1) ◽  
pp. 219-221 ◽  
Author(s):  
Not Available Not Available
Keyword(s):  
Commun Math Phys ◽  
Math Phys

scholarly journals Numerical study of the SWKB condition of novel classes of exactly solvable systems

2020 ◽  
pp. 2150025
Author(s):  
Yuta Nasuda ◽  
Nobuyuki Sawado
Keyword(s):  
Orthogonal Polynomial ◽  
Jacobi Polynomials ◽  
Numerical Study ◽  
Mechanical Systems ◽  
The Other ◽  
Quantum Mechanical ◽  
Shape Invariance ◽  
Exactly Solvable ◽  
Solvable Quantum ◽  
Quantum Mechanical Systems

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the exceptional orthogonal polynomial, even the system possesses the shape invariance. In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein–Adler Hermite, Laguerre and Jacobi polynomials. For all of them, one can always remove the [Formula: see text]-dependency from the condition, and it is satisfied with a certain degree of accuracy.

Erratum: Commun. Math. Phys. 191, 249-264 (1998)

1999 ◽  
Vol 207 (2) ◽  
pp. 495-497
Author(s):  
Not Available Not Available
Keyword(s):  
Commun Math Phys ◽  
Math Phys

scholarly journals Quasi-normal modes for Dirac fields in the Kerr–Newman–de Sitter black holes

2018 ◽  
Vol 16 (04) ◽  
pp. 449-524
Author(s):  
Alexei Iantchenko
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Zeeman Effect ◽  
Normal Modes ◽  
Outer Region ◽  
Dirac Field ◽  
Commun Math Phys ◽  
De Sitter ◽  
Dirac Fields ◽  
Math Phys

We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr–Newman–de Sitter black holes. The resonances split in a way similar to the Zeeman effect. The method is based on the extension to Dirac operators of techniques applied by Dyatlov in [Quasi-normal modes and exponential energy decay for the Kerr–de Sitter black hole, Commun. Math. Phys. 306(1) (2011) 119–163; Asymptotic distribution of quasi-normal modes for Kerr–de Sitter black holes, Ann. Henri Poincaré 13(5) (2012) 1101–1166] to the (uncharged) Kerr–de Sitter black holes. We show that the mass of the Dirac field does not have an effect on the two leading terms in the expansions of resonances. We give an expansion of the solution of the evolution equation for the Dirac fields in the outer region of the slowly rotating Kerr–Newman–de Sitter black hole which implies the exponential decay of the local energy. Moreover, using the [Formula: see text]-normal hyperbolicity of the trapped set and applying the techniques from [Asymptotics of linear waves and resonances with applications to black holes, Commun. Math. Phys. 335 (2015) 1445–1485; Resonance projectors and asymptotics for [Formula: see text]-normally hyperbolic trapped sets, J. Amer. Math. Soc. 28 (2015) 311–381], we give location of the resonance free band and the Weyl-type formula for the resonances in the band near the real axis.

An approach to heavy quarkonium spectra

2014 ◽  
Vol 29 (32) ◽  
pp. 1450170 ◽  
Author(s):  
Y. Cançelik ◽  
B. Gönül
Keyword(s):  
Eigenvalue Problem ◽  
Coulomb Potential ◽  
Mass Spectra ◽  
Bound State ◽  
Considerable Effect ◽  
Heavy Quarkonia ◽  
Heavy Quarkonium ◽  
Related Literature ◽  
Exactly Solvable ◽  
Math Phys

An application of the recently introduced method [M. Çapak et al., J. Math. Phys. 52, 102102 (2011)] to the bound-state eigenvalue problem in the elementary quarkonium potential V(r) = -a/r + br + cr2 is described, proved and illustrated for [Formula: see text] and [Formula: see text] systems. The quasi- and conditionally-exactly solvable spin-averaged mass spectra of heavy quarkonia are obtained in compact forms. The comparison of the present predictions with those of other theories in the related literature, together with the available data, has shown the success of the model used in this work and also revealed that the use of different confinings in the perturbed Coulomb potential descriptions has no considerable effect on the mass spectra of such systems.

Singularities in the complex time plane and exactly solvable dynamical systems

1994 ◽  
Vol 72 (3-4) ◽  
pp. 147-151 ◽  
Author(s):  
W.-H. Steeb ◽  
Assia Fatykhova
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Numerical Study ◽  
Complex Time ◽  
Nonlinear Dynamical ◽  
Singularity Structure ◽  
Exactly Solvable ◽  
Solvable Dynamical Systems

A powerful tool in the study of nonlinear dynamical systems is the investigation of the singularity structure in the complex time plane. In most cases the singularity structure can only be found numerically. Here we give two models that can be solved exactly, i.e., we can give the singularities in the complex time plane. The two models play a central role in quantum mechanics. Then we compare them with the numerical study of the nonlinear differential equation.

Erratum: Commun. Math. Phys. 214, 573-592 (2000)

2001 ◽  
Vol 215 (3) ◽  
pp. 707-707
Author(s):  
Not Available Not Available
Keyword(s):  
Commun Math Phys ◽  
Math Phys

Erratum: Commun. Math. Phys. 208, 521-540 (1999)

2001 ◽  
Vol 220 (2) ◽  
pp. 453-454
Author(s):  
Not Available Not Available
Keyword(s):  
Commun Math Phys ◽  
Math Phys

Some Results on Weak KAM Theory for Time-periodic Tonelli Lagrangian Systems

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Kaizhi Wang ◽  
Yong Li
Keyword(s):  
Kam Theory ◽  
Lagrangian Systems ◽  
Specific Class ◽  
Initial Condition ◽  
Commun Math Phys ◽  
Arbitrary Continuous Function ◽  
Weak Kam Theory ◽  
Definition Of ◽  
Time Periodic ◽  
Math Phys

AbstractThis paper contributes several results on weak KAM theory for time-periodic Tonelli Lagrangian systems. Wang and Yan [Commun. Math. Phys. 309 (2012), 663-691] introduced a new kind of Lax-Oleinik type operator associated with any time-periodic Tonelli Lagrangian. Firstly, using the new operator we give an equivalent definition of the backward weak KAM solution. Then we prove a result on the asymptotic behavior of the new operators with an arbitrary continuous function as initial condition, by taking advantage of the definition mentioned above. Finally, for a specific class of time-periodic Tonelli Lagrangians, we discuss the rate of convergence of the new operators.

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