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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

Ichinose, Wataru. "On the Feynman path integral for the magnetic Schrödinger equation with a polynomially growing electromagnetic potential." Reviews in Mathematical Physics 32, no. 01 (August 5, 2019): 2050003. http://dx.doi.org/10.1142/s0129055x20500038.

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Анотація:

The Feynman path integrals for the magnetic Schrödinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For example, we can handle electromagnetic potentials [Formula: see text] such that [Formula: see text] “a polynomial of degree [Formula: see text] in [Formula: see text]” [Formula: see text] and [Formula: see text] are polynomials of degree [Formula: see text] in [Formula: see text]. The Feynman path integrals are defined as [Formula: see text]-valued continuous functions with respect to the time variable.

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2

Lévai, Géza. "Potentials from the Polynomial Solutions of the Confluent Heun Equation." Symmetry 15, no. 2 (February 9, 2023): 461. http://dx.doi.org/10.3390/sym15020461.

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Анотація:

Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function Hc(p,β,γ,δ,σ;z). One of these recovers the generalized Laguerre polynomials LN(α), and another one the rationally extended X1 type Laguerre polynomials L^N(α). The two remaining solutions represent previously unknown polynomials that do not form an orthogonal set and exhibit features characteristic of semi-classical orthogonal polynomials. A standard method of generating exactly solvable potentials in the one-dimensional Schrödinger equation is applied to the CHE, and all known potentials with solutions expressed in terms of the generalized Laguerre polynomials within, or outside the Natanzon confluent potential class, are recovered. It is also found that the potentials generated from the two new polynomial systems necessarily depend on the N quantum number. General considerations on the application of the Heun type differential differential equations within the present framework are also discussed.

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3

QUESNE, C. "HIGHER-ORDER SUSY, EXACTLY SOLVABLE POTENTIALS, AND EXCEPTIONAL ORTHOGONAL POLYNOMIALS." Modern Physics Letters A 26, no. 25 (August 20, 2011): 1843–52. http://dx.doi.org/10.1142/s0217732311036383.

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Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

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4

Natanson, Gregory. "Quantization of rationally deformed Morse potentials by Wronskian transforms of Romanovski-Bessel polynomials." Acta Polytechnica 62, no. 1 (February 28, 2022): 100–117. http://dx.doi.org/10.14311/ap.2022.62.0100.

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Анотація:

The paper advances Odake and Sasaki’s idea to re-write eigenfunctions of rationally deformed Morse potentials in terms of Wronskians of Laguerre polynomials in the reciprocal argument. It is shown that the constructed quasi-rational seed solutions of the Schrödinger equation with the Morse potential are formed by generalized Bessel polynomials with degree-independent indexes. As a new achievement we can point to the construction of the Darboux-Crum net of isospectral rational potentials using Wronskians of generalized Bessel polynomials with no positive zeros. One can extend this isospectral family of solvable rational potentials by including ‘juxtaposed’ pairs of Romanovski-Bessel polynomials into the aforementioned polynomial Wronskians which results in deleting the corresponding pairs of bound energy states.

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5

Tezuka, Hirokazu. "Confinement by polynomial potentials." Zeitschrift für Physik C Particles and Fields 65, no. 1 (March 1995): 101–4. http://dx.doi.org/10.1007/bf01571309.

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6

Lehr, H., and C. A. Chatzidimitriou-Dreismann. "Complex scaling of polynomial potentials." Chemical Physics Letters 201, no. 1-4 (January 1993): 278–83. http://dx.doi.org/10.1016/0009-2614(93)85071-u.

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7

Casahorran, J. "Solitary waves and polynomial potentials." Physics Letters A 153, no. 4-5 (March 1991): 199–203. http://dx.doi.org/10.1016/0375-9601(91)90794-9.

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8

QUESNE, C. "RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM." International Journal of Modern Physics A 26, no. 32 (December 30, 2011): 5337–47. http://dx.doi.org/10.1142/s0217751x11054942.

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Анотація:

A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to kth-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (k-1)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invariance.

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9

Brandon, David, Nasser Saad, and Shi-Hai Dong. "On some polynomial potentials ind-dimensions." Journal of Mathematical Physics 54, no. 8 (August 2013): 082106. http://dx.doi.org/10.1063/1.4817857.

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10

Vigo-Aguiar, M. I., M. E. Sansaturio, and J. M. Ferrándiz. "Integrability of Hamiltonians with polynomial potentials." Journal of Computational and Applied Mathematics 158, no. 1 (September 2003): 213–24. http://dx.doi.org/10.1016/s0377-0427(03)00467-9.

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11

Ikhdair, Sameer M., and Ramazan Sever. "Polynomial Solution of Non-Central Potentials." International Journal of Theoretical Physics 46, no. 10 (May 8, 2007): 2384–95. http://dx.doi.org/10.1007/s10773-007-9356-8.

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12

Fernández, Francisco M. "Bound-state eigenvalues for polynomial potentials." Physical Review A 44, no. 5 (September 1, 1991): 3336–39. http://dx.doi.org/10.1103/physreva.44.3336.

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13

Boumenir, Amin. "The recovery of even polynomial potentials." Applied Mathematics and Computation 215, no. 8 (December 2009): 2914–26. http://dx.doi.org/10.1016/j.amc.2009.09.037.

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14

Turan, Mehmet, Rezan Sevinik Adıgüzel, and Ayşe Doğan Çalışır. "Spectrum of the q-Schrödinger equation by means of the variational method based on the discrete q-Hermite I polynomials." International Journal of Modern Physics A 36, no. 03 (January 30, 2021): 2150020. http://dx.doi.org/10.1142/s0217751x21500202.

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Анотація:

In this work, the [Formula: see text]-Schrödinger equations with symmetric polynomial potentials are considered. The spectrum of the model is obtained for several values of [Formula: see text], and the limiting case as [Formula: see text] is considered. The Rayleigh–Ritz variational method is adopted to the system. The discrete [Formula: see text]-Hermite I polynomials are handled as basis in this method. Furthermore, the following potentials with numerous results are presented as applications: [Formula: see text]-harmonic, purely [Formula: see text]-quartic and [Formula: see text]-quartic oscillators. It is also shown that the obtained results confirm the ones that exist in the literature for the continuous case.

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15

Nanayakkara, Asiri. "Zeros of the wave functions of general polynomial potentials." Canadian Journal of Physics 82, no. 12 (December 1, 2004): 1067–75. http://dx.doi.org/10.1139/p04-060.

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Анотація:

Analytic formulae for the moments of zeros of the wave function of the general polynomial potential V(x) = α MzM + αM1zM1 + αM2zM2 +...+ α1z are derived. Since the coefficients αM, α M1,...α1 can be either real or complex constants, these formulae are valid for both Hermitian and non-Hermitian PT-symmetric (P parity operator and T time reversal operator) systems. These analytic formulae for the moments can be used to obtain polynomials, the roots of which are the zeros of the corresponding wave functions. The formulae derived here contain parameters of the potential αM, αM1, ...α1 explicitly and locations of zeros can be calculated very efficiently with them.PACS Nos.: 3.65.w, 3.65.Ge, 3.65.Sq, 4.25.g

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16

Seko, Atsuto. "Tutorial: Systematic development of polynomial machine learning potentials for elemental and alloy systems." Journal of Applied Physics 133, no. 1 (January 7, 2023): 011101. http://dx.doi.org/10.1063/5.0129045.

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Анотація:

Machine learning potentials (MLPs) developed from extensive datasets constructed from density functional theory calculations have become increasingly appealing to many researchers. This paper presents a framework of polynomial-based MLPs, called polynomial MLPs. The systematic development of accurate and computationally efficient polynomial MLPs for many elemental and binary alloy systems and their predictive powers for various properties are also demonstrated. Consequently, many polynomial MLPs are available in a repository website [A. Seko, Polynomial Machine Learning Potential Repository at Kyoto University, https://sekocha.github.io ]. The repository will help many scientists perform accurate and efficient large-scale atomistic simulations and crystal structure searches.

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17

Stachowiak, Tomasz. "On solvable Dirac equation with polynomial potentials." Journal of Mathematical Physics 52, no. 1 (January 2011): 012301. http://dx.doi.org/10.1063/1.3533946.

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18

Katatbeh, Qutaibeh D., Richard L. Hall, and Nasser Saad. "Eigenvalue bounds for polynomial central potentials inddimensions." Journal of Physics A: Mathematical and Theoretical 40, no. 44 (October 16, 2007): 13431–42. http://dx.doi.org/10.1088/1751-8113/40/44/020.

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19

Falconi, M., E. A. Lacomba, and C. Vidal. "Dynamics of Mechanical Systems with Polynomial Potentials." Journal of Dynamics and Differential Equations 26, no. 3 (March 28, 2014): 707–22. http://dx.doi.org/10.1007/s10884-014-9357-2.

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20

Nanayakkara, A., and V. Bandara. "Approximate energy expressions for confining polynomial potentials." Sri Lankan Journal of Physics 3 (December 1, 2002): 17. http://dx.doi.org/10.4038/sljp.v3i0.183.

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21

Cleary, Paul W. "Integrability and orbits in quartic polynomial potentials." Journal of Mathematical Physics 30, no. 10 (October 1989): 2214–25. http://dx.doi.org/10.1063/1.528546.

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22

Bender, Carl M., and Maria Monou. "New quasi-exactly solvable sextic polynomial potentials." Journal of Physics A: Mathematical and General 38, no. 10 (February 24, 2005): 2179–87. http://dx.doi.org/10.1088/0305-4470/38/10/009.

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23

Shin, Kwang C. "Eigenvalues of -symmetric oscillators with polynomial potentials." Journal of Physics A: Mathematical and General 38, no. 27 (June 22, 2005): 6147–66. http://dx.doi.org/10.1088/0305-4470/38/27/005.

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24

Sánchez, Fabián Toledo, Pedro Pablo Cárdenas Alzate, and Carlos Alberto Abello Muñoz. "ON THE DYNAMICS OF THE HODGKIN-HUXLEY NEURAL MODEL: A STABLE ANALYSIS USING HURWITZ POLYNOMIALS." Journal of Southwest Jiaotong University 57, no. 6 (December 30, 2022): 860–72. http://dx.doi.org/10.35741/issn.0258-2724.57.6.74.

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Анотація:

In this paper, a study of the dynamics of the Hodgkin-Huxley neural model is conducted. This stability study is performed using the stability criteria for obtaining Hurwitz polynomials that provide necessary and/or sufficient conditions to analyze the dynamics of the model by studying the location of the roots of the characteristic polynomial associated with it. The main objective of the article is to analyze the stability of the Hodgkin-Huxley neural model using an analytical technique to establish Hurwitz type polynomials. This article describes and presents an analytical method to analyze the stability of the Hodgkin-Huxley neural model based on obtaining Hurwitz polynomials through stability criteria. This technique allows establishing the asymptotic stability of the neural model through the localization of the Eigen values of the matrix associated with the model. The results of this research allow establishing asymptotic stability conditions of the Hodgkin-Huxley neural model that allow analyzing the dynamics of the model. Through this article, it is possible to analyze the dynamics of the Hodgkin-Huxley neural model and allows drawing inferences regarding the initiation and transmission of action potentials in neurons and cardiac cells using the Hurwitz polynomial technique.

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25

Belhaouane, Mohamed Moez, Mohamed Faiez Ghariani, Hela Belkhiria Ayadi, and Naceur Benhadj Braiek. "Improved Results on Robust Stability Analysis and Stabilization for a Class of Uncertain Nonlinear Systems." Mathematical Problems in Engineering 2010 (2010): 1–24. http://dx.doi.org/10.1155/2010/724563.

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Анотація:

This paper deals with the problems of robust stability analysis and robust stabilization for uncertain nonlinear polynomial systems. The combination of a polynomial system stability criterion with an improved robustness measure of uncertain linear systems has allowed the formulation of a new criterion for robustness bound estimation of the studied uncertain polynomial systems. Indeed, the formulated approach is extended to involve the global stabilization of nonlinear polynomial systems with maximization of the stability robustness bound. The proposed method is helpful to improve the existing techniques used in the analysis and control for uncertain polynomial systems. Simulation examples illustrate the potentials of the proposed approach.

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26

Dziubański, Jacek. "A note on Schrödinger operators with polynomial potentials." Colloquium Mathematicum 78, no. 1 (1998): 149–61. http://dx.doi.org/10.4064/cm-78-1-149-161.

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27

Przybylska, Maria. "Finiteness of integrable n-dimensional homogeneous polynomial potentials." Physics Letters A 369, no. 3 (September 2007): 180–87. http://dx.doi.org/10.1016/j.physleta.2007.04.077.

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28

Maiz, F., Moteb M. Alqahtani, N. Al Sdran, and I. Ghnaim. "Sextic and decatic anharmonic oscillator potentials: Polynomial solutions." Physica B: Condensed Matter 530 (February 2018): 101–5. http://dx.doi.org/10.1016/j.physb.2017.11.010.

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29

Solon, Mikhail P., and J. P. H. Esguerra. "Periods of relativistic oscillators with even polynomial potentials." Physics Letters A 372, no. 44 (October 2008): 6608–12. http://dx.doi.org/10.1016/j.physleta.2008.09.021.

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30

Nanayakkara, Asiri, and Isuru Dasanayake. "Analytic semiclassical energy expansions of general polynomial potentials." Physics Letters A 294, no. 3-4 (February 2002): 158–62. http://dx.doi.org/10.1016/s0375-9601(02)00045-2.

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31

Anda, André, Luca De Vico, Thorsten Hansen, and Darius Abramavičius. "Absorption and Fluorescence Lineshape Theory for Polynomial Potentials." Journal of Chemical Theory and Computation 12, no. 12 (November 4, 2016): 5979–89. http://dx.doi.org/10.1021/acs.jctc.6b00997.

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32

Del Monte, Alessio, Nicola Manini, Luca Guido Molinari *, and Gian Paolo Brivio. "Low-energy unphysical saddle in polynomial molecular potentials." Molecular Physics 103, no. 5 (March 10, 2005): 689–96. http://dx.doi.org/10.1080/00268970412331332114.

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33

Nanayakkara, Asiri. "Asymptotic Energy Expansion for Rational Power Polynomial Potentials." Communications in Theoretical Physics 58, no. 5 (November 2012): 645–48. http://dx.doi.org/10.1088/0253-6102/58/5/05.

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34

Meurice, Y. "Arbitrarily accurate eigenvalues for one-dimensional polynomial potentials." Journal of Physics A: Mathematical and General 35, no. 41 (October 2, 2002): 8831–46. http://dx.doi.org/10.1088/0305-4470/35/41/314.

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35

Wang, Yiqian. "Unboundedness in a Duffing Equation with Polynomial Potentials." Journal of Differential Equations 160, no. 2 (January 2000): 467–79. http://dx.doi.org/10.1006/jdeq.1999.3666.

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36

Alhaidari, A. D. "Exact scattering and bound states solutions for novel hyperbolic potentials with inverse square singularity." Modern Physics Letters A 34, no. 03 (January 30, 2019): 1950020. http://dx.doi.org/10.1142/s0217732319500202.

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Анотація:

We use the Tridiagonal Representation Approach (TRA) to obtain exact scattering and bound states solutions of the Schrödinger equation for short-range inverse-square singular hyperbolic potentials. The solutions are series of square integrable functions written in terms of the Jacobi polynomial with the Wilson polynomial as expansion coefficients. The series is finite for the discrete bound states and infinite but bounded for the continuum scattering states.

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37

BRIHAYE, Y., and A. NININAHAZWE. "DIRAC OSCILLATORS AND QUASI-EXACTLY SOLVABLE OPERATORS." Modern Physics Letters A 20, no. 25 (August 20, 2005): 1875–85. http://dx.doi.org/10.1142/s0217732305018128.

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Анотація:

The Dirac equation is formulated in the background of three types of physically relevant potentials: scalar, vector and "Dirac-oscillator" potentials. Assuming these potentials to be spherically-symmetric and with generic polynomial forms in the radial variable, we construct the corresponding radial Dirac equation. Cases where this linear spectral equation is exactly solvable or quasi-exactly solvable are worked out in details. When available, relations between the radial Dirac operator and some super-algebra are pointed out.

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38

Baran, Á., and T. Vertse. "Matching polynomial tails to the cut-off Woods–Saxon potential." International Journal of Modern Physics E 26, no. 11 (November 2017): 1750078. http://dx.doi.org/10.1142/s0218301317500781.

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Анотація:

Cutting off the tail of the Woods–Saxon (WS) and generalized WS (GWS) potentials changes the distribution of the poles of the [Formula: see text]-matrix considerably. Here, we modify the tail of the cut-off WS (CWS) and cut-off generalized WS (CGWS) potentials by attaching Hermite polynomial tails to them beyond the cut. The tails reach the zero value more or less smoothly at the finite ranges of the potential. Reflections of the resonant wave functions can take place at different distances. The starting points of the pole trajectories have been reproduced not only for the real values and the moduli of the starting points, but also for the imaginary parts.

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39

OGURA, WAICHI. "MAPPING BETWEEN TODA AND KDV FLOWS IN THE HERMITIAN MATRIX MODEL." Modern Physics Letters A 06, no. 09 (March 21, 1991): 811–18. http://dx.doi.org/10.1142/s0217732391000841.

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The scaling operators are studied at finite N. We find new singular potentials for which an orthogonal polynomial identity gives the string equation at the double scaling limit. They are free from the degeneracy between even and odd potentials, and provide the mapping between the sl(∞) Toda and the generalized KdV flows. The degeneracy in formal Virasoro conditions are derived explicitly.

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40

Pilipovic, Stevan, and Nenad Teofanov. "On a symbol class of elliptic pseudodifferential operators." Bulletin: Classe des sciences mathematiques et natturalles 123, no. 27 (2002): 57–68. http://dx.doi.org/10.2298/bmat0227057p.

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Анотація:

We consider a class of symbols with prescribed smoothness and growth conditions and give examples of such symbols. The introduced class contains certain polynomial symbols and symbols with more, than polynomial growth in phase space. The corresponding pseudodifferential operators defined as the Weyl transforms of the symbols are elliptic. As an application, we give a result on isomorphisms between modulation spaces. In particular, we show that the Bessel potentials establish such isomorphisms.

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41

Laederich, Stephane, and Mark Lev. "Invariant curves and time-dependent potentials." Ergodic Theory and Dynamical Systems 11, no. 2 (June 1991): 365–78. http://dx.doi.org/10.1017/s0143385700006192.

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AbstractIn this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time-dependent potentials, namely, for systems of the formwhereis a superquadratic polynomial potential with periodic coefficients. As a limiting case, a proof of the stability of Ulam's problem of a particle bouncing between two periodicially moving walls is given.

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42

Nanayakkara, Asiri, and Thilagarajah Mathanaranjan. "Explicit energy expansion for general odd-degree polynomial potentials." Physica Scripta 88, no. 5 (October 18, 2013): 055004. http://dx.doi.org/10.1088/0031-8949/88/05/055004.

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43

Killingbeck, J. P. "Comment on the asymptotic iteration method for polynomial potentials." Journal of Physics A: Mathematical and Theoretical 40, no. 11 (February 28, 2007): 2819–24. http://dx.doi.org/10.1088/1751-8113/40/11/016.

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44

Kelbert, E., A. Hyder, F. Demir, Z. T. Hlousek, and Z. Papp. "Green's operator for Hamiltonians with Coulomb plus polynomial potentials." Journal of Physics A: Mathematical and Theoretical 40, no. 27 (June 19, 2007): 7721–28. http://dx.doi.org/10.1088/1751-8113/40/27/020.

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45

Mehta, Dhagash, Matthew Niemerg, and Chuang Sun. "Statistics of stationary points of random finite polynomial potentials." Journal of Statistical Mechanics: Theory and Experiment 2015, no. 9 (September 16, 2015): P09012. http://dx.doi.org/10.1088/1742-5468/2015/09/p09012.

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46

de Moura, Alessandro P. S., and Patricio S. Letelier. "Fractal basins in Hénon–Heiles and other polynomial potentials." Physics Letters A 256, no. 5-6 (June 1999): 362–68. http://dx.doi.org/10.1016/s0375-9601(99)00209-1.

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47

Killingbeck, J. P. "Comment on the asymptotic iteration method for polynomial potentials." Journal of Physics A: Mathematical and Theoretical 40, no. 16 (March 30, 2007): 4413. http://dx.doi.org/10.1088/1751-8121/40/16/c01.

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48

Saad, Nasser, Richard L. Hall, and Hakan Ciftci. "Study of a class of non-polynomial oscillator potentials." Journal of Physics A: Mathematical and General 39, no. 24 (May 31, 2006): 7745–56. http://dx.doi.org/10.1088/0305-4470/39/24/011.

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49

Dziubański, Jacek, and Paweł Głowacki. "Sobolev spaces related to Schrödinger operators with polynomial potentials." Mathematische Zeitschrift 262, no. 4 (August 12, 2008): 881–94. http://dx.doi.org/10.1007/s00209-008-0404-8.

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50

Madububa, B. I., N. I. Achuko, JohnPaul Chiagoziem Mbagwu, C. I. Jonas, and J. O. Ozuomba. "First and Second-Order Energy Eigenvalues of One-Dimensional Quantum Harmonic and Anharmonic Oscillator with Linear, Quadratic, Cubic and Polynomial Perturbation Potential." ASM Science Journal 17 (May 18, 2022): 1–10. http://dx.doi.org/10.32802/asmscj.2022.950.

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Анотація:

This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).

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