Recurrence Relations and General Solution of the Exceptional Hermite Equation

Darboux, M.G.: Sur une proposition relative aux équations linéaires. Comptes Rendus Acad. Sci. 94, 1456 (1882)

MATH  Google Scholar 

Crum, M.M.: Associated Sturm–Liouville systems. Q. J. Math. Oxf. Ser. 6(2), 121 (1955)

Article  MathSciNet  MATH  Google Scholar 

Krein, M.G.: On a continual analogue of a Christoffel formula from the theory of orthogonal polynomials. Dokl. Akad. Nauk SSSR 113, 970 (1957)

MathSciNet  MATH  Google Scholar 

Adler, V.E.: On a modification of Crum’s method. Theor. Math. Phys. 101, 1381 (1994)

Article  MathSciNet  MATH  Google Scholar 

Junker, G.: Supersymmetric Methods in Quantum and Statistical Physics. Springer, Berlin (1996)

Book  MATH  Google Scholar 

Bagchi, B.K.: Supersymmetry in Quantum and Classical Physics. Chapman, Boca Raton (2001)

MATH  Google Scholar 

Andrianov, A., Cannata, F., Ioffe, M., Nishnianidze, D.: Systems with higher-order shape invariance: spectral and algebraic properties. Phys. Lett. A 266, 341 (2000)

Article  ADS  MathSciNet  MATH  Google Scholar 

Fernández, D.J., Hussin, V.: Higher-order SUSY, linearized nonlinear Heisenberg algebras and coherent states. J. Phys. A Math. Gen. 32, 3603 (1999)

Article  ADS  MathSciNet  MATH  Google Scholar 

Gomez-Ullate, D., Kamran, N., Milson, R.: An extension of Bochner’s problem: exceptional invariant subspaces. J. Approx. Theory 162, 987 (2010)

Article  MathSciNet  MATH  Google Scholar 

Gomez-Ullate, D., Kamran, N., Milson, R.: An extended class of orthogonal polynomials defined by a Sturm–Liouville problem. J. Math. Anal. Appl. 359, 352 (2009)

Article  MathSciNet  MATH  Google Scholar 

Odake, S., Sasaki, R.: Infinitely many shape invariant potentials and new orthogonal polynomials. Phys. Lett. B 679, 414 (2009)

Article  ADS  MathSciNet  Google Scholar 

Quesne, C.: Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics. SIGMA 5, 084 (2009)

MathSciNet  MATH  Google Scholar 

Fellows, J.M., Smith, R.A.: Factorization solution of a family of quantum nonlinear oscillators. J. Phys. A 42, 335303 (2009)

Article  MathSciNet  MATH  Google Scholar 

Quesne, C.: Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials. Mod. Phys. Lett. A 26, 1843 (2011)

Article  ADS  MathSciNet  MATH  Google Scholar 

Odake, S., Sasaki, R.: Krein–Adler transformations for shape-invariant potentials and pseudo virtual states. J. Phys. A Math. Theor. 46, 245201 (2013)

Article  ADS  MathSciNet  MATH  Google Scholar 

Gomez-Ullate, D., Grandati, Y., Milson, R.: Extended Krein–Adler theorem for the translationally shape invariant potentials. J. Math. Phys. 55, 043510 (2014)

Article  ADS  MathSciNet  MATH  Google Scholar 

Gomez-Ullate, D., Grandati, Y., Milson, R.: Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A Math. Theor. 47, 015203 (2014)

Article  ADS  MathSciNet  MATH  Google Scholar 

Gomez-Ullate, D., Kasman, A., Kuijlaars, A.B.J., Milson, R.: Recurrence relations for exceptional Hermite polynomials. J. Approx. Theor. 204, 1 (2016)

Article  MathSciNet  MATH  Google Scholar 

Gomez-Ullate, D., Grandati, Y., Milson, R.: Spectral Theory of Exceptional Hermite Polynomials, arXiv:2012.02354 [math.CA]

Marquette, I., Quesne, C.: New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials. J. Math. Phys. 54, 042102 (2013)

Article  ADS  MathSciNet  MATH  Google Scholar 

Marquette, I., Quesne, C.: Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators. J. Phys. A Math. Theor. 46, 155201 (2013)

Article  ADS  MathSciNet  MATH  Google Scholar 

Marquette, I., Quesne, C.: New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems. J. Math. Phys. 54, 102102 (2013)

Article  ADS  MathSciNet  MATH  Google Scholar 

Marquette, I., Quesne, C.: Combined state-adding and state-deleting approaches to type III multi-step rationally extended potentials: Applications to ladder operators and superintegrability. J. Math. Phys. 55, 112103 (2014)

Article  ADS  MathSciNet  MATH  Google Scholar 

Cariñena, J.F., Plyushchay, M.S.: ABC of ladder operators for rationally extended quantum harmonic oscillator systems. J. Phys. A 50(27), 275202 (2017)

Article  MathSciNet  MATH  Google Scholar 

Latini, D., Marquette, I., Zhang, Y.-Z.: Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators. Phys. D Nonlinear Phenom. 440, 133464 (2022)

Article  MathSciNet  MATH  Google Scholar 

Gravel, S.: Hamiltonians separable in Cartesian coordinates and third-order integrals of motion. J. Math. Phys. 45, 1003 (2004)

Article  ADS  MathSciNet  MATH  Google Scholar 

Marquette, I.: Superintegrability with third order integrals of motion, cubic algebras, and supersymmetric quantum mechanics. I. Rational function potentials. J. Math. Phys. 50, 012101 (2009)

Sesma, J.: The generalized quantum isotonic oscillator. J. Phys. A Math. Theor. 43, 185303 (2010)

Article  ADS  MathSciNet  MATH  Google Scholar 

Cariñena, J.F., Perelomov, A.M., Ranada, M.F., Santander, M.: A quantum exactly solvable non-linear oscillator related with the isotonic oscillator. J. Phys. A Math. Theor. 41, 085301 (2008)

Cariñena, J.F., Plyushchay, M.S.: Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator. Phys. Rev. D 94(10), 105022 (2016)

Article  ADS  MathSciNet  Google Scholar 

Chalifour, V., Grundland, A.M.: General solution of the exceptional Hermite differential equation and its minimal surface representation. Ann. Henri Poincaré 21, 3341 (2020)

Article  ADS  MathSciNet  MATH  Google Scholar 

Marquette, I., Sajedi, M., Winternitz, P.: Two-dimensional superintegrable systems from operator algebras in one dimension. J. Phys. A Math. Theor. 52(11), 115202 (2019)

Article  ADS  MATH  Google Scholar 

Filipuk, G., Ishkhangan, A., Derezinski, J.: On the derivatives of the Heun functions. J. Contemp. Math. Anal. 55, 200 (2020)

Article  MathSciNet  MATH  Google Scholar 

Marquette, I., Quesne, C.: Connection between quantum systems involving the fourth Painlevé transcendent and k-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57, 052101 (2016)

Article  ADS  MathSciNet  MATH  Google Scholar 

Zelaya, K., Marquette, I., Hussin, V.: Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials. J. Phys. A Math. Theor. 55, 045205 (2022)

Article  ADS  MathSciNet  MATH  Google Scholar 

Derezinski, J., Ishkhanyan, A., Latosinski, A.: From Heun class equations to Painlevé equations. SIGMA 17, 056 (2021)

留言 (0)

沒有登入
gif