Open Access
Volume 55, Number 5, September-October 2021
Page(s) 2141 - 2168
Published online 13 October 2021
  1. M.S. Agranovich, Sobolev Spaces, their Generalizations and Elliptic Problems in Smooth and Lipschitz domains. Springer Monographs in Mathematics. Springer, Cham (2015). [Google Scholar]
  2. G.E. Andrews, R. Askey and R. Roy, Special Functions. Cambridge (1999). [Google Scholar]
  3. R. Askey, Orthogonal Polynomials and Special Functions. Society for Industrial and Applied Mathematics (1975). [CrossRef] [Google Scholar]
  4. W. Bao, H. Li and J. Shen, A generalized Laguerre-Fourier-Hermite pseudospectral method for computing the dynamics of rotating Bose-Einstein condensates. SIAM J. Sci. Comput. 31 (2009) 3685–3711. [Google Scholar]
  5. W. Bao, X. Ruan, J. Shen and C. Sheng, Fundamental gaps of the fractional Schrödinger operator. Commun. Math. Sci. 17 (2019) 447–471. [Google Scholar]
  6. W. Bao, L. Chen, X. Jiang and Y. Ma, A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator. J. Comput. Phys. 421 (2020) 109733. [Google Scholar]
  7. P. Borwein, T. Erdélyi and J. Zhang, Müntz systems and orthogonal Müntz-Legendre polynomials. Trans. Amer. Math. Soc. 342 (1994) 523–542. [Google Scholar]
  8. D. Burnett, The distribution of molecular velocities and the mean motion in a non-uniform gas. Proc. London Math. Soc. 40 (1936) 382–435. [Google Scholar]
  9. Z. Cai, Y. Fan and Y. Wang, Burnett spectral method for the spatially homogeneous Boltzmann equation. Comput. & Fluids 200 (2020) 104456. [Google Scholar]
  10. T. Chihara, Generalized Hermite polynomials. Ph.D. thesis, Purdue University (1955). [Google Scholar]
  11. T. Chihara, An Introduction to Orthogonal Polynomials, New York (1978). [Google Scholar]
  12. F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer-Verlag (2013). [Google Scholar]
  13. E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. [CrossRef] [MathSciNet] [Google Scholar]
  14. C.F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables, 2nd edition, vol. 155, Cambridge University Press, Cambridge (2014). [Google Scholar]
  15. Z.J. Duoandikoetxea, Fourier Analysis, vol 29, American Mathematical Society (2001). [Google Scholar]
  16. T. Gerald, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, 2nd edition. Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators. Vol 157. American Mathematical Society, Providence, RI (2014). [Google Scholar]
  17. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 8th edition. Elsevier/Academic Press, Amsterdam (2015). [Google Scholar]
  18. G.A. Hagedorn, Semiclassical quantum mechanics. I. The h→0 limit for coherent states. Comm. Math. Phys. 71 (1980) 77–93. [Google Scholar]
  19. D. Hou and C. Xu, A fractional spectral method with applications to some singular problems. Adv. Comput. Math. 43 (2017) 911–944. [Google Scholar]
  20. Z. Hu and Z. Cai, Burnett spectral method for high-speed rarefied gas flows. SIAM J. Sci. Comput. 42 (2020) 1193–1226. [Google Scholar]
  21. L. Klebanov, Approximation of PDEs with Underlying Continuity Equations . Ph.D. thesis. Technische Universität, München (2015). [Google Scholar]
  22. N. Laskin, Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268 (2000) 298–305. [CrossRef] [MathSciNet] [Google Scholar]
  23. C. Lasser and S. Troppmann, Hagedorn wavepackets in time-frequency and phase space. J. Fourier Anal. Appl. 20 (2014) 679–714. [Google Scholar]
  24. C. Lasser and C. Lubich, Computing quantum dynamics in the semiclassical regime. Acta Numer. 29 (2020) 229–401. [Google Scholar]
  25. N.N. Lebedev, Special Functions and their Applications, edited by Richard A Silverman. Prentice-Hall Inc, Englewood Cliffs, N.J. (1965). [Google Scholar]
  26. F. Liu, Z. Wang and H. Li, A fully diagonalized spectral method using generalized Laguerre functions on the half line. Adv. Comput. Math. 43 (2017) 1227–1259. [Google Scholar]
  27. C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. European Mathematical Society, Zürich (2008). [Google Scholar]
  28. S. Ma, H. Li and Z. Zhang, Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete Contin. Dyn. Syst. Ser. B 24 (2019) 1589–1615. [Google Scholar]
  29. Z. Mao and J. Shen, Hermite spectral methods for fractional PDEs in unbounded domains. SIAM J. Sci. Comput. 39 (2017) A1928–A1950. [Google Scholar]
  30. M. Masjed-Jamei and W. Koepf, Two classes of special functions using Fourier transforms of generalized ultraspherical and generalized Hermite polynomials.a Proc. Amer. Math. Soc. 140 (2012) 2053–2063. [Google Scholar]
  31. C.H. Müntz. Über den Approximationssatz von Weierstrass, in H.A. Schwarz’s Festschrift, Berlin (1914) 303–312. [Google Scholar]
  32. V.I. Osherov and V.G. Ushakov, Analytical solutions of the Schrödinger equation for a hydrogen atom in a uniform electric field. Phys. Rev. A. 95 (2017) 023419. [Google Scholar]
  33. L. Pauling and E.B. Wilson, Introduction to Quantum Mechanics with Applications to Chemistry. McGraw-Hill (1935). [Google Scholar]
  34. L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation. Oxford University Press, Oxford (2003). [Google Scholar]
  35. M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus. Nonselfadjoint operators and related topics.s Oper. Theory Adv. Appl. 73 (1994) 369–396. [Google Scholar]
  36. M. Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys. 192 (1998) 519–542. [Google Scholar]
  37. T.S. Shao, T.C. Chen and R.M. Frank, Tables of zeros and Gaussian weights of certain associated Laguerre polynomials and the related generalized Hermite polynomials. Math. Comp. 18 (1964) 598–616. [Google Scholar]
  38. J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications. Springer (2011). [Google Scholar]
  39. J. Shen and Y. Wang, Müntz-Galerkin methods and applications to mixed Dirichlet-Neumann boundary value problems. SIAM J. Sci. Comput. 38 (2016) A2357–A2381. [Google Scholar]
  40. C. Sheng, J. Shen, T. Tang, L.-L. Wang and H. Yuan, Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains. SIAM J. Numer. Anal. 58 (2020) 2435–2464. [Google Scholar]
  41. E. Schrödinger, Quantisierung als Eigenwertproblem. Annalen der Physik 384 (1926) 361–377. [Google Scholar]
  42. G. Strang and G. Fix, An Analysis of the Finite Element Method. Prentice-Hall Series in Automatic Computation . Prentice-Hall Inc., Englewood Cliffs, N.J (1973). [Google Scholar]
  43. G. Szegö, Orthogonal Polynomials. American Mathematical Society, Providence (1939). [Google Scholar]
  44. T. Tang, The Hermite spectral method for Gaussian-type functions. SIAM J. Sci. Comput. 14 (1993) 594–606. [Google Scholar]
  45. T. Tang, H. Yuan and T. Zhou, Hermite spectral collocation methods for fractional PDEs in unbounded domains. Commun. Comput. Phys. 24 (2018) 1143–1168. [Google Scholar]
  46. A. Yurova, Generalized Anisotropic Hermite Functions and their Applications . Ph.D. thesis. Technische Universität, München (2020). [Google Scholar]
  47. Y. Zhang, X. Liu, M. Belić, et al., Propagation dynamics of a light beam in a fractional Schrödinger equation. Phys. Rev. Lett. 115 (2015) 180403. [PubMed] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you