# American Institute of Mathematical Sciences

July  2015, 20(5): 1377-1391. doi: 10.3934/dcdsb.2015.20.1377

## A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential

 1 Department of Mathematics, Wayne State University, Detroit, MI 48202, United States 2 Fariborz Maseeh Department of Mathematics & Statistics, Portland State University, Portland, OR 97201, United States

Received  August 2013 Revised  January 2015 Published  May 2015

We develop an a posteriori error estimate of hierarchical type for Dirichlet eigenvalue problems of the form $(-\Delta+(c/r)^2)\psi=\lambda \psi$ on bounded domains $\Omega$, where $r$ is the distance to the origin, which is assumed to be in $\overline\Omega$. This error estimate is proven to be asymptotically identical to the eigenvalue approximation error on a family of geometrically-graded meshes. Numerical experiments demonstrate this asymptotic exactness in practice.
Citation: Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377
##### References:
 [1] T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Methods Appl. Sci., 21 (1998), 519-549. doi: 10.1002/(SICI)1099-1476(199804)21:6<519::AID-MMA962>3.0.CO;2-R. [2] I. Babuška, R. B. Kellogg and J. Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math., 33 (1979), 447-471. doi: 10.1007/BF01399326. [3] I. Babuška and J. Osborn, Eigenvalue problems, In Handbook of numerical analysis, Vol.II, Handb. Numer. Anal., II, pages 641-787. North-Holland, Amsterdam, 1991. [4] C. Băcuţă, V. Nistor and L. T. Zikatanov, Improving the rate of convergence of 'high order finite elements' on polygons and domains with cusps, Numer. Math., 100 (2005), 165-184. doi: 10.1007/s00211-005-0588-3. [5] R. E. Bank, PLTMG: A software package for solving elliptic partial differential equations. Users' Guide 10.0, Technical report, University of California at San Diego, Department of Mathematics, 2007. [6] R. E. Bank, L. Grubišić and J. S. Ovall, A framework for robust eigenvalue and eigenvector error estimation and ritz value convergence enhancement, Applied Numer. Math., 66 (2013), 1-29. doi: 10.1016/j.apnum.2012.11.004. [7] H. Chen, L. He and A. Zhou, Finite element approximations of nonlinear eigenvalue problems in quantum physics, Comput. Methods Appl. Mech. Engrg., 200 (2011), 1846-1865. doi: 10.1016/j.cma.2011.02.008. [8] M. Dauge, Elliptic Boundary Value Problems on Corner Domains, volume 1341 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. [9] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, volume 159 of {Applied Mathematical Sciences}, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5. [10] V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, J. Eur. Math. Soc. (JEMS), 13 (2011), 119-174. doi: 10.4171/JEMS/246. [11] V. Felli, E. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discrete Contin. Dyn. Syst., 21 (2008), 91-119. doi: 10.3934/dcds.2008.21.91. [12] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Østergaard Sørensen, Analytic structure of solutions to multiconfiguration equations, J. Phys. A, 42 (2009), 315208, 11pp. doi: 10.1088/1751-8113/42/31/315208. [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. [14] P. Grisvard, Singularities in Boundary Value Problems, volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1992. [15] E. Hunsicker, H. Li, V. Nistor and U. Ville, Analysis of Schrödinger operators with inverse square potentials I: Regularity results in 3D, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55 (2012), 157-178. [16] V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., 16 (1967), 209-292. [17] V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, volume 52 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. [18] V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, volume 85 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2001. [19] R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide, volume 6 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. doi: 10.1137/1.9780898719628. [20] H. Li, A. Mazzucato and V. Nistor, Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains, Electron. Trans. Numer. Anal., 37 (2010), 41-69. [21] H. Li and V. Nistor, Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM, J. Comput. Appl. Math., 224 (2009), 320-338. doi: 10.1016/j.cam.2008.05.009. [22] H. Li and J. S. Ovall, A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential, Numer. Math, 128 (2014), 707-740. doi: 10.1007/s00211-014-0628-y. [23] S. Moroz and R. Schmidt, Nonrelativistic inverse square potential, scale anomaly, and complex extension, Annals of Physics, 325 (2010), 491-513. doi: 10.1016/j.aop.2009.10.002. [24] A. Naga and Z. Zhang, Function value recovery and its application in eigenvalue problems, SIAM J. Numer. Anal., 50 (2012), 272-286. doi: 10.1137/100797709. [25] G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation, [26] L. N. Trefethen and T. Betcke, Computed eigenmodes of planar regions, In Recent advances in differential equations and mathematical physics, volume 412 of Contemp. Math., pages 297-314. Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/conm/412/07783. [27] N. M. Wigley, Asymptotic expansions at a corner of solutions of mixed boundary value problems, J. Math. Mech., 13 (1964), 549-576. [28] H. Wu and D. W. L. Sprung, Inverse-square potential and the quantum vortex, Phys. Rev. A, 49 (1994), 4305-4311. doi: 10.1103/PhysRevA.49.4305.

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##### References:
 [1] T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges, Math. Methods Appl. Sci., 21 (1998), 519-549. doi: 10.1002/(SICI)1099-1476(199804)21:6<519::AID-MMA962>3.0.CO;2-R. [2] I. Babuška, R. B. Kellogg and J. Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math., 33 (1979), 447-471. doi: 10.1007/BF01399326. [3] I. Babuška and J. Osborn, Eigenvalue problems, In Handbook of numerical analysis, Vol.II, Handb. Numer. Anal., II, pages 641-787. North-Holland, Amsterdam, 1991. [4] C. Băcuţă, V. Nistor and L. T. Zikatanov, Improving the rate of convergence of 'high order finite elements' on polygons and domains with cusps, Numer. Math., 100 (2005), 165-184. doi: 10.1007/s00211-005-0588-3. [5] R. E. Bank, PLTMG: A software package for solving elliptic partial differential equations. Users' Guide 10.0, Technical report, University of California at San Diego, Department of Mathematics, 2007. [6] R. E. Bank, L. Grubišić and J. S. Ovall, A framework for robust eigenvalue and eigenvector error estimation and ritz value convergence enhancement, Applied Numer. Math., 66 (2013), 1-29. doi: 10.1016/j.apnum.2012.11.004. [7] H. Chen, L. He and A. Zhou, Finite element approximations of nonlinear eigenvalue problems in quantum physics, Comput. Methods Appl. Mech. Engrg., 200 (2011), 1846-1865. doi: 10.1016/j.cma.2011.02.008. [8] M. Dauge, Elliptic Boundary Value Problems on Corner Domains, volume 1341 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. [9] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, volume 159 of {Applied Mathematical Sciences}, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5. [10] V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, J. Eur. Math. Soc. (JEMS), 13 (2011), 119-174. doi: 10.4171/JEMS/246. [11] V. Felli, E. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discrete Contin. Dyn. Syst., 21 (2008), 91-119. doi: 10.3934/dcds.2008.21.91. [12] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Østergaard Sørensen, Analytic structure of solutions to multiconfiguration equations, J. Phys. A, 42 (2009), 315208, 11pp. doi: 10.1088/1751-8113/42/31/315208. [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. [14] P. Grisvard, Singularities in Boundary Value Problems, volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1992. [15] E. Hunsicker, H. Li, V. Nistor and U. Ville, Analysis of Schrödinger operators with inverse square potentials I: Regularity results in 3D, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55 (2012), 157-178. [16] V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč., 16 (1967), 209-292. [17] V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities, volume 52 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. [18] V. A. Kozlov, V. G. Maz'ya and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, volume 85 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2001. [19] R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide, volume 6 of Software, Environments, and Tools, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. doi: 10.1137/1.9780898719628. [20] H. Li, A. Mazzucato and V. Nistor, Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains, Electron. Trans. Numer. Anal., 37 (2010), 41-69. [21] H. Li and V. Nistor, Analysis of a modified Schrödinger operator in 2D: Regularity, index, and FEM, J. Comput. Appl. Math., 224 (2009), 320-338. doi: 10.1016/j.cam.2008.05.009. [22] H. Li and J. S. Ovall, A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential, Numer. Math, 128 (2014), 707-740. doi: 10.1007/s00211-014-0628-y. [23] S. Moroz and R. Schmidt, Nonrelativistic inverse square potential, scale anomaly, and complex extension, Annals of Physics, 325 (2010), 491-513. doi: 10.1016/j.aop.2009.10.002. [24] A. Naga and Z. Zhang, Function value recovery and its application in eigenvalue problems, SIAM J. Numer. Anal., 50 (2012), 272-286. doi: 10.1137/100797709. [25] G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation, [26] L. N. Trefethen and T. Betcke, Computed eigenmodes of planar regions, In Recent advances in differential equations and mathematical physics, volume 412 of Contemp. Math., pages 297-314. Amer. Math. Soc., Providence, RI, 2006. doi: 10.1090/conm/412/07783. [27] N. M. Wigley, Asymptotic expansions at a corner of solutions of mixed boundary value problems, J. Math. Mech., 13 (1964), 549-576. [28] H. Wu and D. W. L. Sprung, Inverse-square potential and the quantum vortex, Phys. Rev. A, 49 (1994), 4305-4311. doi: 10.1103/PhysRevA.49.4305.
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