Efficient Jacobi-Galerkin spectral and second-order time discretization scheme for ground and first excited states of nonlinear fractional Schrödinger equation

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Abstract

In this work, we propose an efficient and accurate numerical scheme for computing both the ground and first excited states of the nonlinear fractional Schrödinger eigenvalue problem. The method is based on a fractional gradient flow with discrete normalization, reformulated via the invariant energy quadratization (IEQ) approach to handle the nonlinear term in a linear way. Temporal discretization is performed using a second-order Crank-Nicolson scheme, while spatial discretization employs a Jacobi-Galerkin spectral method, yielding spectral convergence in space. We rigorously prove the discrete energy-decay property with respect to a modified energy functional. The resulting fully discrete scheme can compute different states simply by varying the initial condition, without altering the algorithmic framework. Numerical experiments confirm its high accuracy, efficiency, and robustness, and demonstrate its capability to capture intricate solution structures across a wide range of fractional orders and nonlinear interaction strengths.

Keywords

fractional Schrödinger / eigenvalue / IEQ / time-discrete scheme / Jacobi-Galerkin / spectral method

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References

[1] H. D. Doebner, G. A. Goldin, P. Nattermann, Gauge transformations in quantum mechanics and the unification of nonlinear schrödinger equations, J. Math. Phys., 40 (1999), 49-63. https://doi.org/10.1063/1.532786

[2] V. Banica, R. Luca, N. Tzvetkov, L. Vega, Blow-up for the 1d cubic nls, Commun. Math. Phys., 405 (2024), 11. https://doi.org/10.1007/s00220-023-04906-3

[3] W. Bao, The nonlinear schrödinger equation and applications in bose-einstein condensation and plasma physics, in Dynamics in Models of Coarsening, Coagulation, Condensation and Quantization, (2007), 141-239. https://doi.org/10.1142/9789812770226_0003

[4] G. Fibich, G. Papanicolaou, Self-focusing in the perturbed and unperturbed nonlinear schrödinger equation in critical dimension, SIAM J. Appl. Math., 60 (1999), 183-240. https://doi.org/10.1137/S0036139997322407

[5] N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62 (2000), 3135-3145. https://doi.org/10.1103/PhysRevE.62.3135

[6] N. Laskin, Fractional schrödinger equation, Phys. Rev. E, 66 (2002), 056108. https://doi.org/10.1103/PhysRevE.66.056108

[7] Q. Yang, I. Turner, F. Liu, M. Ilic, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, Comput. Phys. Commun., 33 (2011), 1159-1180. https://doi.org/10.1137/100800634

[8] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. https://doi.org/10.1016/j.bulsci.2011.12.004

[9] A. Lischke, G. F. Pang, M. Gulian, F. Y. Song, C. Glusa, X. N. Zheng, et al., What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404 (2020), 109009. https://doi.org/10.1016/j.jcp.2019.109009

[10] B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555. https://doi.org/10.2478/s13540-012-0038-8

[11] B. Dyda, A. Kuznetsov, M. Kwasnicki, Eigenvalues of the fractional laplace operator in the unit ball, J. London Math. Soc., 95 (2017), 500-518. https://doi.org/10.1112/jlms.12024

[12] E. M. Harrell, Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators, Commun. Partial Differ. Equations, 32 (2007), 401-413. https://doi.org/10.1080/03605300500532889

[13] M. Kwasnicki, Eigenvalues of the fractional laplace operator in the interval, J. Funct. Anal., 262 (2012), 2379-2402. https://doi.org/10.1016/j.jfa.2011.12.004

[14] P. Li, S. T. Yau, On the schrödinger equation and the eigenvalue problem, Commun. Math. Phys., 88 (1983), 309-318. https://doi.org/10.1007/BF01213210

[15] I. M. Singer, B. Wong, S. S. T. Yau, S. T. Yau, An estimate of the gap of the first two eigenvalues in the schrödinger operator, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (1985), 319-333. Available from: https://www.numdam.org/item/?id=ASNSP_1985_4_12_2_319_0.

[16] S. W. Duo, Y. Z. Zhang, Computing the ground and first excited states of the fractional schrödinger equation in an infinite potential well, Commun. Comput. Phys., 18 (2015), 321-350. https://doi.org/10.4208/cicp.300414.120215a

[17] J. P. Borthagaray, L. M. Del Pezzo, S. Martinez, Finite element approximation for the fractional eigenvalue problem, J. Sci. Comput., 77 (2018), 308-329. https://doi.org/10.1007/s10915-018-0710-1

[18] L. Z. Chen, Z. P. Mao, H. Y. Li, Jacobi-Galerkin spectral method for eigenvalue problems of riesz fractional differential equations, preprint, arXiv:1803.03556.

[19] W. Z. Bao, X. R. Ruan, J. Shen, C. T. Sheng, Fundamental gaps of the fractional schrödinger operator, Commun. Math. Sci., 17 (2019), 447-471. https://doi.org/10.4310/CMS.2019.v17.n2.a7

[20] W. Z. Bao, L. Z. Chen, X. Y. Jiang, 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. https://doi.org/10.1016/j.jcp.2020.109733

[21] S. L. Chang, C. S. Chien, B. W. Jeng, An efficient algorithm for the schrödinger-poisson eigenvalue problem, J. Comput. Appl. Math., 205 (2007), 509-532. https://doi.org/10.1016/j.cam.2006.05.013

[22] Y. Ma, L. Z. Chen, A Jacobi-Galerkin spectral method for computing the ground and first excited states of nonlinear fractional schrödinger equation, East Asian J. Appl. Math., 10 (2020), 274-294. https://doi.org/10.4208/eajam.140319.180719

[23] S. N. Ma, H. Y. Li, Z. M. Zhang, H. Chen, L. C. Chen, Efficient spectral methods for eigenvalue problems of the integral fractional laplacian on a ball of any dimension, J. Comput. Math., 42 (2024), 1032-1062. https://doi.org/10.4208/jcm.2304-m2022-0243

[24] X. Antoine, Q. L. Tang, Y. Zhang, On the ground states and dynamics of space fractional nonlinear schrödinger/gross-pitaevskii equations with rotation term and nonlocal nonlinear interactions, J. Comput. Phys., 325 (2016), 74-97. https://doi.org/10.1016/j.jcp.2016.08.009

[25] Y. X. Gou, P. B. Ming, A deep learning method for computing eigenvalues of the fractional schrödinger operator, J. Syst. Sci. Complexity, 37 (2024), 391-412. https://doi.org/10.1007/s11424-024-3250-9

[26] W. Z. Bao, Q. Du, Computing the ground state solution of bose-einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput., 25 (2004), 1674-1697. https://doi.org/10.1137/S1064827503422956

[27] Z. Xu, X. Yang, H. Zhang, Z. Xie, Efficient and linear schemes for anisotropic cahn-hilliard model using the stabilized-invariant energy quadratization (s-ieq) approach, Comput. Phys. Commun., 238 (2019), 36-49. https://doi.org/10.1016/j.cpc.2018.12.019

[28] X. Yang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys., 327 (2016), 294-316. https://doi.org/10.1016/j.jcp.2016.09.029

[29] X. Yang, X. He, A fully-discrete decoupled finite element method for the conserved allen-cahn type phase-field model of three-phase fluid flow system, Comput. Methods Appl. Mech. Eng., 389 (2022), 114376. https://doi.org/10.1016/j.cma.2021.114376

[30] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[31] X. J. Li, C. J. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108-2131. https://doi.org/10.1137/080718942

[32] J. L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin, 1972.

[33] P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011. https://doi.org/10.1137/1.9781611972030

[34] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Publishing Company, Amsterdam, 1978.

[35] Z. P. Mao, S. Chen, J. Shen, Efficient and accurate spectral method using generalized Jacobi functions for solving riesz fractional differential equations, Appl. Numer. Math., 106 (2016), 165-181. https://doi.org/10.1016/j.apnum.2016.04.002