# Efficient Spectral Methods for Some Singular Eigenvalue Problems

Article

## Abstract

We propose and analyze some efficient spectral/spectral element methods to solve singular eigenvalue problems related to the Schrödinger operator with an inverse-power potential. For the Schrödinger eigenvalue problem $$-\Delta u +V(x)u=\lambda u$$ with a regular potential $$V(x)=c_1|x|^{-1}$$, we first design an efficient spectral method on a ball of any dimension by adopting the Sobloev-orthogonal basis functions with respect to the Laplacian operator to overwhelm the homogeneous inverse potential and to eliminate the singularity of the eigenfunctions. Then we extend this spectral method to arbitrary polygonal domains by the mortar element method with each corner covered by a circular sector and origin covered by a circular disc. Furthermore, for the Schrödinger eigenvalue problem with a singular potential $$V(x)=c_3|x|^{-3}$$, we devise a novel spectral method by modifying the former Sobloev-orthogonal bases to fit the stronger singularity. As in the case of $$|x|^{-1}$$ potential, this approach can be extended to arbitrary polygonal domains by the mortar element method as well. Finally, for the singular elliptic eigenvalue problem $$-\frac{\partial ^2}{\partial x^2}u-\frac{1}{x^2}\frac{\partial ^2}{\partial y^2}u =\lambda u$$ on rectangles, we propose a novel spectral method by using tensorial bases composed of the $$L^2$$- and $$H^1$$-simultaneously orthogonal functions in the y-direction and the Sobolev-orthogonal functions with respect to the Schrödinger operator with an inverse-square potential in the x-direction. Numerical experiments indicate that all our methods possess exponential orders of convergence, and are superior to the existing polynomial based spectral/spectral element methods and hp-adaptive methods.

## Keywords

Singular equation Inverse-power potential Eigenvalues Spectral/spectral element method Exponential order

## Mathematics Subject Classification

Primary 65N35 65N25 35Q40

## References

1. 1.
Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
2. 2.
Bernardi, C., Maday, Y., Patera, A.T.: Domain Decomposition by the Mortar Element Method. In: Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters. Springer, Dordrecht, pp. 269–286 (1993)Google Scholar
3. 3.
B$$\check{a}$$cut$$\check{a}$$, C., Nistor, V., Zikatanov, L.T.: Improving the rate of convergence of ’high order finite elements’ on polygons and domains with cusps, Numer. Math. 100, 165–184 (2005)Google Scholar
4. 4.
Babu$$\check{s}$$ka, I., Gui, W.: The $$h$$, $$p$$ and $$h$$-$$p$$ versions of the finite element method in 1 dimension. Part II. The error analysis of the $$h$$- and $$h$$-$$p$$ versions. Numer. Math. 49(6), 613–658 (1986)Google Scholar
5. 5.
Babu$$\check{s}$$ka, I.M., Guo, B.: Approximation properties of the $$h$$-$$p$$ version of finite element method. Comput. Methods Appl. Mech. Eng. 133, 319–346 (1996)Google Scholar
6. 6.
Cao, D., Han, P.: Solutions to critical elliptic equations with multi-singular inverse square potentials. J. Differ. Equ. 224, 332–372 (2006)
7. 7.
Cagliero, L., Koornwinder, T.H.: Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials. J. Approx. Theory 193, 20–38 (2015)
8. 8.
Case, K.M.: Singular potentials. Phys. Rev. 80(2), 797–806 (1950)
9. 9.
Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge (2001)
10. 10.
Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, Berlin (2013)
11. 11.
Felli, V., Marchini, E.M., Terracini, S.: On Schrödinger operators with multipolar inverse-square potentials. J. Funct. Anal. 250, 265–316 (2007)
12. 12.
Felli, V., Terracini, S.: Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Commun. Part. Differ. Equ. 31, 469–495 (2006)
13. 13.
Frank, W.M., Land, D.J., Spector, R.M.: Singular potentials. Rev. Mod. Phys. 43(1), 36–98 (1971)
14. 14.
Guo, B., Shen, J., Wang, L.: Optimal spectral-Galerkin methods using generalized Jacobi polynomials. J. Sci. Comput. 27, 305–322 (2006)
15. 15.
Guo, B., Sun, W.: The optimal convergence of the $$h$$-$$p$$ version of the finite element method with quasi-uniform meshes. SIAM J. Numer. Anal. 45, 698–730 (2007)
16. 16.
Gu, W., Wang, C., Liaw, B.Y.: Micro-macroscopic coupled modeling of batteries and fuel cells: part 2. Application to nickel–cadmium and nickel–metal hybrid cells. J. Electrochem. Soc. 145, 3418–3427 (1998)
17. 17.
Li, H.: A-priori analysis and the finite element method for a class of degenerate elliptic equaitons. Math. Comput. 78(266), 713–737 (2009)
18. 18.
Li, H., Shen, J.: Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle. Math. Comput. 79, 1621–1646 (2010)
19. 19.
Li, H., Zhang, Z.: Efficient spectral and spectral element methods for eigenvalue problems of Schrodinger equations with an inverse square potential. SIAM J. Sci. Comput. 39(1), A114–A140 (2017)
20. 20.
Motreanu, D., Rădulescu, V.: Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media. Bound. Value Probl. 2005(2), 708605 (2005)
21. 21.
Pao, C.V.: Eigenvalue problems of a degenerate quasilinear elliptic equation. Rocky Mt. J. Math. 40, 305–311 (2010)
22. 22.
Redheffer, R.M., Straus, E.G.: Degenerate elliptic equations. Pac. J. Math. 7(8), 331–345 (2014)
23. 23.
Shu, C.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40(2), 769–791 (2003)
24. 24.
Szegö, G.: Orthogonal Polynomials, vol. XXIII, 4th edn. American Mathematical Society, Colloquium Publications, Providence (1975)
25. 25.
Shortley, G.H.: The inverse-cube central force field in quantum mechanics. Phys. Rev. 38(1), 120–127 (1931)
26. 26.
Shen, J., Tao, T., Wang, L.: Spectral methods: algorithms, analysis and applications. Springer, Berlin (2011)
27. 27.
Wang, C., Gu, W., Liaw, B.Y.: Micro-macroscopic coupled modeling of batteries and fuel cells: part 1. Model development. J. Electrochem. Soc. 145, 3407–3417 (1998)
28. 28.
Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second order elliptic problems. Math. Comp. 83(289), 2101–2126 (2012)
29. 29.
Ying, L.: Finite element approximations to the discrete spectrum of the Schrödinger operator with the Coulomb potential. SIAM J. Numer. Anal. 42(1), 49–74 (2005)

## Authors and Affiliations

• Suna Ma
• 1
• Huiyuan Li
• 2
• Zhimin Zhang
• 1
• 3
1. 1.Beijing Computational Science Research CenterBeijingChina
2. 2.State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of SoftwareChinese Academy of SciencesBeijingChina
3. 3.Department of MathematicsWayne State UniversityDetroitUSA