# American Institute of Mathematical Sciences

## Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space

 1 Beijing Computational Science Research Center, Beijing 100193, China 2 Institute of Software, Chinese Academy of Sciences, Beijing 100190, China 3 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

* Corresponding author: Huiyuan Li

Received  January 2018 Revised  March 2018 Published  June 2018

Fund Project: The research of the second author is partially supported by the National Natural Science Foundation of China (NSFC 91130014, 11471312 and 91430216). The research of the third author is supported in part by the U.S. National Science Foundation (DMS-1419040), the National Natural Science Foundation of China (NSFC 11471031 and 91430216), and the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (NSAF U1530401)

In this article, we propose and analyze some novel spectral methods for the Schödinger equation (including the associated eigenvalue problem) with an inverse square potential on an arbitrary whole space $\mathbb{R}^d$ for any dimension $d$. We start from the investigation that the radial component of the eigenfunctions, corresponding to spherical harmonics of degree $n$, of the Schrödinger operator $\displaystyle -Δ u + \frac{c^2}{r^2}u$ can be expressed by Bessel functions of fractional orders $α_n = \sqrt{c^2+(n+d/2-1)^2}$ together with the multiplier $r^{1-\frac{d}{2}}$. This knowledge helps us to construct the Müntz-Hermite functions as the basis functions to fit the singularities of the eigenfunctions. In return, a novel spectral method is then proposed for solving the Schrödinger eigenvalue problem efficiently. Further, a Galerkin spectral approximation using genuine Hermite functions with a distinct Müntz sequence $\{α_n = α+n+d/2-1\}$ is also proposed for the Schrödinger source problem with a singular solution of type $r^{α}$. Optimal error estimates are then established rigorously for both the source and eigenvalue problems. In contrast to classic Hermite spectral methods using tensorial basis functions, our new methods possess an exponential order of convergence for such singular problems while offer a banded structure of the stiffness and mass matrices. Finally, numerical experiments illustrate the efficiency and spectral accuracy of our new methods.

Citation: Suna Ma, Huiyuan Li, Zhimin Zhang. Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018221
##### References:
 [1] J. M. Almira, Müntz type theorems: Ⅰ, Surveys in Approximation Theory, 3 (2007), 152-194. [2] J. I. Babuška and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, Vol. Ⅱ, North-Holland, Amsterdam, 1991,641–787. [3] C. Bǎcutǎ, V. Nistor and L. T. Zikatanov, Improving the rate of convergence of "high order finite elements" on polygons and domains with cusps, Numerische Mathematik, 100 (2005), 165-184. doi: 10.1007/s00211-005-0588-3. [4] D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differ. Equations, 224 (2006), 332-372. doi: 10.1016/j.jde.2005.07.010. [5] K. M. Case, Singular potentials, Physical Rev., 80 (1950), 797-806. doi: 10.1103/PhysRev.80.797. [6] F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spherical and Balls, Springer, New York, 2013. doi: 10.1007/978-1-4614-6660-4. [7] C. F. Dunkl and Y. Xu, Orthogonal Polynomials of several Variables, Vol. 155, Cambridge University Press, 2014. doi: 10.1017/CBO9781107786134. [8] V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316. doi: 10.1016/j.jfa.2006.10.019. [9] V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Commun. Part. Diff. Eq., 31 (2006), 469-495. doi: 10.1080/03605300500394439. [10] W. M. Frank, D. J. Land and R. M. Spector, Singular potentials, Rev. Modern Phys., 43 (1971), 36-98. doi: 10.1103/RevModPhys.43.36. [11] M. Frigo and S. G. Johnson, The design and implementation of FFTW3, P. IEEE, 93 (2005), 216-231. doi: 10.1109/JPROC.2004.840301. [12] B. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp., 68 (1999), 1067-1078. doi: 10.1090/S0025-5718-99-01059-5. [13] B. Guo and W. Sun, The optimal convergence of the $h$-$p$ version of the finite element method with quasi-uniform meshes, SIAM J. Numer. Anal., 45 (2007), 698-730. doi: 10.1137/05063756X. [14] D. M. Healy Jr., D. Rockmore, P. J. Kostelec and S. Moore, FFTs for the 2-Sphere-Improvements and Variations, J. Fourier. Anal. Appl., 9 (2003), 341-385. doi: 10.1007/s00041-003-0018-9. [15] H. Kalf, U. W. Schmincke, J. Walter and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in Spectral Theory and Differential Equations, Vol. 448 of Lect. Notes in Math. Springer, Berlin, 1975,182–226. [16] H. Li and Y. Xu, Spectral approximations on the unit ball, SIAM J. Numer. Anal., 52 (2014), 2647-2675. doi: 10.1137/130940591. [17] H. Li and Z. Zhang, Efficient spectral and spectral element methods for eigenvalue problems of Schrodinger equations with an inverse square potential, SIAM J. Sci. Comput., 39 (2017), A114-A140. doi: 10.1137/16M1069596. [18] H. Li and J. S. Ovall, A posteriori estimation of hierarchical type for the Schrödinger operator with the inverse square potential, Numer. Math., 128 (2014), 707-740. doi: 10.1007/s00211-014-0628-y. [19] H. Li and J. S. Ovall, A posteriori eigenvalue error estimation for a Schrödinger operator with the inverse square potential, Discrete Continuous Dynam. Systems - B, 20 (2017), 1377-1391. doi: 10.3934/dcdsb.2015.20.1377. [20] 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), 1277-1259. doi: 10.1007/s10444-017-9522-3. [21] Z. Mao and J. Shen, Hermite spectral methods for fractional PDEs in unbounded domains, SIAM J. Sci. Comput., 39 (2017), A1928-A1950. doi: 10.1137/16M1097109. [22] T. E. Pérez and M. A. Piñar, On Sobolev Orthogonality for the Generalized Laguerre Polynomials, J. Approx. Theory., 86 (1996), 278-285. doi: 10.1006/jath.1996.0069. [23] G. W. Reddien, Finite-difference approximations to singular Sturm-Liouville eigenvalue problems, Math. Comp., 30 (1976), 278-282. doi: 10.1090/S0025-5718-1976-0403235-1. [24] J. Shen, T. Tao and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, 2011. doi: 10.1007/978-3-540-71041-7. [25] J. Shen and L. Wang, Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5 (2009), 195-241. [26] 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. doi: 10.1137/15M1052391. [27] T. Suzuki, Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains, Mathematica Bohemici, 139 (2014), 231-238. [28] G. Szegö, Orthogonal Polynomials, 4$^{nd}$ edition, Vol. 23, American Mathematical Society, 1975. [29] X. Xiang and Z. Wang, Generalized Hermite approximations and spectral method for partial differential equations in multiple dimensions, J. Sci. Comput., 57 (2013), 229-253. doi: 10.1007/s10915-013-9703-2.

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##### References:
 [1] J. M. Almira, Müntz type theorems: Ⅰ, Surveys in Approximation Theory, 3 (2007), 152-194. [2] J. I. Babuška and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, Vol. Ⅱ, North-Holland, Amsterdam, 1991,641–787. [3] C. Bǎcutǎ, V. Nistor and L. T. Zikatanov, Improving the rate of convergence of "high order finite elements" on polygons and domains with cusps, Numerische Mathematik, 100 (2005), 165-184. doi: 10.1007/s00211-005-0588-3. [4] D. Cao and P. Han, Solutions to critical elliptic equations with multi-singular inverse square potentials, J. Differ. Equations, 224 (2006), 332-372. doi: 10.1016/j.jde.2005.07.010. [5] K. M. Case, Singular potentials, Physical Rev., 80 (1950), 797-806. doi: 10.1103/PhysRev.80.797. [6] F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spherical and Balls, Springer, New York, 2013. doi: 10.1007/978-1-4614-6660-4. [7] C. F. Dunkl and Y. Xu, Orthogonal Polynomials of several Variables, Vol. 155, Cambridge University Press, 2014. doi: 10.1017/CBO9781107786134. [8] V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316. doi: 10.1016/j.jfa.2006.10.019. [9] V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Commun. Part. Diff. Eq., 31 (2006), 469-495. doi: 10.1080/03605300500394439. [10] W. M. Frank, D. J. Land and R. M. Spector, Singular potentials, Rev. Modern Phys., 43 (1971), 36-98. doi: 10.1103/RevModPhys.43.36. [11] M. Frigo and S. G. Johnson, The design and implementation of FFTW3, P. IEEE, 93 (2005), 216-231. doi: 10.1109/JPROC.2004.840301. [12] B. Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp., 68 (1999), 1067-1078. doi: 10.1090/S0025-5718-99-01059-5. [13] B. Guo and W. Sun, The optimal convergence of the $h$-$p$ version of the finite element method with quasi-uniform meshes, SIAM J. Numer. Anal., 45 (2007), 698-730. doi: 10.1137/05063756X. [14] D. M. Healy Jr., D. Rockmore, P. J. Kostelec and S. Moore, FFTs for the 2-Sphere-Improvements and Variations, J. Fourier. Anal. Appl., 9 (2003), 341-385. doi: 10.1007/s00041-003-0018-9. [15] H. Kalf, U. W. Schmincke, J. Walter and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in Spectral Theory and Differential Equations, Vol. 448 of Lect. Notes in Math. Springer, Berlin, 1975,182–226. [16] H. Li and Y. Xu, Spectral approximations on the unit ball, SIAM J. Numer. Anal., 52 (2014), 2647-2675. doi: 10.1137/130940591. [17] H. Li and Z. Zhang, Efficient spectral and spectral element methods for eigenvalue problems of Schrodinger equations with an inverse square potential, SIAM J. Sci. Comput., 39 (2017), A114-A140. doi: 10.1137/16M1069596. [18] H. Li and J. S. Ovall, A posteriori estimation of hierarchical type for the Schrödinger operator with the inverse square potential, Numer. Math., 128 (2014), 707-740. doi: 10.1007/s00211-014-0628-y. [19] H. Li and J. S. Ovall, A posteriori eigenvalue error estimation for a Schrödinger operator with the inverse square potential, Discrete Continuous Dynam. Systems - B, 20 (2017), 1377-1391. doi: 10.3934/dcdsb.2015.20.1377. [20] 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), 1277-1259. doi: 10.1007/s10444-017-9522-3. [21] Z. Mao and J. Shen, Hermite spectral methods for fractional PDEs in unbounded domains, SIAM J. Sci. Comput., 39 (2017), A1928-A1950. doi: 10.1137/16M1097109. [22] T. E. Pérez and M. A. Piñar, On Sobolev Orthogonality for the Generalized Laguerre Polynomials, J. Approx. Theory., 86 (1996), 278-285. doi: 10.1006/jath.1996.0069. [23] G. W. Reddien, Finite-difference approximations to singular Sturm-Liouville eigenvalue problems, Math. Comp., 30 (1976), 278-282. doi: 10.1090/S0025-5718-1976-0403235-1. [24] J. Shen, T. Tao and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, 2011. doi: 10.1007/978-3-540-71041-7. [25] J. Shen and L. Wang, Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5 (2009), 195-241. [26] 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. doi: 10.1137/15M1052391. [27] T. Suzuki, Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains, Mathematica Bohemici, 139 (2014), 231-238. [28] G. Szegö, Orthogonal Polynomials, 4$^{nd}$ edition, Vol. 23, American Mathematical Society, 1975. [29] X. Xiang and Z. Wang, Generalized Hermite approximations and spectral method for partial differential equations in multiple dimensions, J. Sci. Comput., 57 (2013), 229-253. doi: 10.1007/s10915-013-9703-2.
Approximation errors $\mid \lambda_i-\lambda_{i, N}\mid$ ($\circ:\lambda_1, \triangle:\lambda_2, \nabla:\lambda_3, \square:\lambda_4, \star:\lambda_5$) in the semi-logarithm scale versus $N$ in Case 1
Approximation errors $\mid \lambda_i-\lambda_{i, N}\mid$ ($\circ:\lambda_1, \triangle:\lambda_2, \nabla:\lambda_3, \square:\lambda_4, \star:\lambda_5$) in the semi-logarithm scale versus $N$ in Case 2
Eigenfunctions corresponding to the first 5 distinct eigenvalues in two dimensions with $c = 0.5, \mu_1 = 1.5, \mu_2 = 0.6$ for Case 1
Eigenfunctions corresponding to the first 5 distinct eigenvalues in two dimensions with $c = 0.5, \mu_1 = 1.5, \mu_2 = 0.7, \mu_3 = 0.4$ for Case 2
Radial part of eigenfunctions corresponding to the first 4 distinct eigenvalues in three dimensions with $c = 0.5, \mu_1 = 1.5, \mu_2 = 0.6$ for Case 1
Radial part of eigenfunctions corresponding to the first 4 distinct eigenvalues in three dimensions with $c = 0.5, \mu_1 = 1.5, \mu_2 = 0.7, \mu_3 = 0.4$ for Case 2
Errors in Case 1 of Example 2
Errors in Case 2 of Example 2
$L^2$-error in Case 1 (a) of Example 2 with different $\gamma$
The first 5 smallest eigenvalues with $c = 1/2$ for Case 1
 $d$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $2$ 3.823790007724450 4.781244146051844 6.242937223017229 6.922176684690384 7.760880934238864 $3$ 4.144638453493299 5.498683081266146 6.998876870109267 7.243025130459233 8.526418913534627
 $d$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $2$ 3.823790007724450 4.781244146051844 6.242937223017229 6.922176684690384 7.760880934238864 $3$ 4.144638453493299 5.498683081266146 6.998876870109267 7.243025130459233 8.526418913534627
The first 5 smallest eigenvalues with $c = 2/3$ for Case 1
 $d$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $2$ 4.081988897471611 4.911092010985492 6.315179662193871 7.180375574437544 7.810145624178201 $3$ 4.340187787218772 5.592157488184387 7.057518009298278 7.438574464184707 8.568855163872875
 $d$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $2$ 4.081988897471611 4.911092010985492 6.315179662193871 7.180375574437544 7.810145624178201 $3$ 4.340187787218772 5.592157488184387 7.057518009298278 7.438574464184707 8.568855163872875
The first 5 smallest reference eigenvalues with $c = 1/2$ for Case 2
 $d$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $2$ 5.093501192467833 6.812614993417879 9.642977529439129 11.504311062206813 12.808005821670148 $3$ 5.656512628011969 8.172947462792827 11.192562450175135 12.223367733631648 14.481392043922055
 $d$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $2$ 5.093501192467833 6.812614993417879 9.642977529439129 11.504311062206813 12.808005821670148 $3$ 5.656512628011969 8.172947462792827 11.192562450175135 12.223367733631648 14.481392043922055
The first 5 smallest reference eigenvalues with $c = 2/3$ for Case 2
 $d$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $2$ 5.545505126219459 7.054462965653881 9.788676383214872 12.082446880862650 12.914205877275965 $3$ 6.006261275577311 8.354389805223748 11.315024695812147 12.664846034345008 14.575569286102708
 $d$ $\lambda_1$ $\lambda_2$ $\lambda_3$ $\lambda_4$ $\lambda_5$ $2$ 5.545505126219459 7.054462965653881 9.788676383214872 12.082446880862650 12.914205877275965 $3$ 6.006261275577311 8.354389805223748 11.315024695812147 12.664846034345008 14.575569286102708
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