# American Institute of Mathematical Sciences

November  2012, 11(6): 2221-2237. doi: 10.3934/cpaa.2012.11.2221

## Harmonic oscillators with Neumann condition on the half-line

 1 IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av Robert Schuman, F-35170 Bruz, France

Received  November 2010 Revised  September 2011 Published  April 2012

We consider the spectrum of the family of one-dimensional self-adjoint operators $-{\mathrm{d}}^2/{\mathrm{d}}t^2+(t-\zeta)^2$, $\zeta\in \mathbb{R}$ on the half-line with Neumann boundary condition. It is well known that the first eigenvalue $\mu(\zeta)$ of this family of harmonic oscillators has a unique minimum when $\zeta\in\mathbb{R}$. This paper is devoted to the accurate computations of this minimum $\Theta_{0}$ and $\Phi(0)$ where $\Phi$ is the associated positive normalized eigenfunction. We propose an algorithm based on finite element method to determine this minimum and we give a sharp estimate of the numerical accuracy. We compare these results with a finite element method.
Citation: Virginie Bonnaillie-Noël. Harmonic oscillators with Neumann condition on the half-line. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2221-2237. doi: 10.3934/cpaa.2012.11.2221
##### References:
 [1] F. Alouges and V. Bonnaillie-Noël, Numerical computations of fundamental eigenstates for the Schrödinger operator under constant magnetic field, Numer. Methods Partial Differential Equations, 22 (2006), 1090-1105. doi: 10.1002/num.20137. [2] A. Bernoff and P. Sternberg, Onset of superconductivity in decreasing fields for general domains, J. Math. Phys., 39 (1998), 1272-1284. doi: 10.1063/1.532379. [3] C. Bolley, Modélisation du champ de retard à la condensation d'un supraconducteur par un problème de bifurcation, RAIRO Modél. Math. Anal. Numér., 26 (1992), 235-287. [4] C. Bolley and B. Helffer, An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material, Ann. Inst. H. Poincaré Phys. Théor., 58 (1993), 189-233. [5] V. Bonnaillie, "Analyse mathématique de la supraconductivité dans un domaine à coins; méthodes semi-classiques et numériques," Thèse de doctorat, Université Paris XI - Orsay, 2003. [6] V. Bonnaillie, On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners, Asymptot. Anal., 41 (2005), 215-258. [7] V. Bonnaillie-Noël and M. Dauge, Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners, Ann. Henri Poincaré, 7 (2006), 899-931. doi: 10.1007/s00023-006-0271-y. [8] V. Bonnaillie-Noël, M. Dauge, D. Martin and G. Vial, Computations of the first eigenpairs for the schrödinger operator with magnetic field, Comput. Methods Appl. Mech. Engng., 196 (2007), 3841-3858. doi: 10.1016/j.cma.2006.10.041. [9] V. Bonnaillie-Noël, M. Dauge, N. Popoff and N. Raymond, Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions, Z. Angew. Math. Phys., DOI 10.1007/s00033-011-0163-y (2011). doi: 10.1007/s00033-011-0163-y. [10] V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners, Reviews in Mathematical Physics, 19 (2007), 607-637. doi: 10.1142/S0129055X07003061. [11] S. J. Chapman, Nucleation of superconductivity in decreasing fields. I, European J. Appl. Math., 5 (1994), 449-468. doi: 10.1017/S095679250000156X. [12] M. Dauge and B. Helffer, Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators, J. Differential Equations, 104 (1993), 243-262. doi: 10.1006/jdeq.1993.1071. [13] P.-G. De Gennes and D. Saint-James, Onset of superconductivity in decreasing fields, Physics Letters, 7 (1963), 306-308. doi: 10.1016/0031-9163(63)90047-7. [14] S. Fournais and B. Helffer, Energy asymptotics for type {II superconductors}, Calc. Var., 24 (2005), 341-376. doi: 10.1007/s00526-005-0333-x. [15] S. Fournais and B. Helffer, Accurate eigenvalue estimates for the magnetic Neumann Laplacian, Annales Inst. Fourier, 56 (2006), 1-67. doi: 10.5802/aif.2171. [16] S. Fournais and B. Helffer, On the third critical field in Ginzburg-Landau theory, Comm. Math. Phys., 266 (2006), 153-196. doi: 10.1007/s00220-006-0006-4. [17] S. Fournais and B. Helffer, "Spectral Methods in Surface Superconductivity," Progress in Nonlinear Differential Equations and their Applications, 77. Birkhäuser Boston Inc., Boston, MA 2010. [18] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag 2001. [19] E. M. Harrell, Double wells, Comm. Math. Phys., 75 (1980), 239-261. doi: 10.1007/BF01212711. [20] P. Hartmann, "Ordinary Differential Equations," Wiley, New-York 1964. [21] B. Helffer, "Semi-classical Analysis for the Schrödinger Operator and Applications," volume 1336 of Lecture Notes in Mathematics, [22] B. Helffer and A. Mohamed, Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., 138 (1996), 40-81. doi: 10.1006/jfan.1996.0056. [23] B. Helffer and A. Morame, Magnetic bottles for the Neumann problem: the case of dimension 3, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 71-84. doi: 10.1007/BF02829641. [24] B. Helffer and A. Morame, Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case), Ann. Sci. École Norm. Sup., 37 (2004), 105-170. doi: 10.1016/j.ansens.2003.04.003. [25] B. Helffer and X.-B. Pan, Upper critical field and location of surface nucleation of superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 145-181. doi: 10.1016/S0294-1449(02)00005-7. [26] T. Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan, 4 (1949), 334-339. doi: 10.1143/JPSJ.4.334. [27] K. Lu and X.-B. Pan, Eigenvalue problems of Ginzburg-Landau operator in bounded domains, J. Math. Phys., 40 (1999), 2647-2670. doi: 10.1063/1.532721. [28] K. Lu and X.-B. Pan, Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity, Phys. D, 127 (1999), 73-104. doi: 10.1016/S0167-2789(98)00246-2. [29] K. Lu and X.-B. Pan, Gauge invariant eigenvalue problems in $R^2$ and in $R_+^2$, Trans. Amer. Math. Soc., 352 (2000), 1247-1276. doi: 10.1090/S0002-9947-99-02516-7. [30] N. Raymond, On the semiclassical 3D Neumann Laplacian with variable magnetic field, Asymptot. Anal., 68 (2010), 1-40. [31] Y. Sibuya, "Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient," Noth-Holland 1975.

show all references

##### References:
 [1] F. Alouges and V. Bonnaillie-Noël, Numerical computations of fundamental eigenstates for the Schrödinger operator under constant magnetic field, Numer. Methods Partial Differential Equations, 22 (2006), 1090-1105. doi: 10.1002/num.20137. [2] A. Bernoff and P. Sternberg, Onset of superconductivity in decreasing fields for general domains, J. Math. Phys., 39 (1998), 1272-1284. doi: 10.1063/1.532379. [3] C. Bolley, Modélisation du champ de retard à la condensation d'un supraconducteur par un problème de bifurcation, RAIRO Modél. Math. Anal. Numér., 26 (1992), 235-287. [4] C. Bolley and B. Helffer, An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material, Ann. Inst. H. Poincaré Phys. Théor., 58 (1993), 189-233. [5] V. Bonnaillie, "Analyse mathématique de la supraconductivité dans un domaine à coins; méthodes semi-classiques et numériques," Thèse de doctorat, Université Paris XI - Orsay, 2003. [6] V. Bonnaillie, On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners, Asymptot. Anal., 41 (2005), 215-258. [7] V. Bonnaillie-Noël and M. Dauge, Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners, Ann. Henri Poincaré, 7 (2006), 899-931. doi: 10.1007/s00023-006-0271-y. [8] V. Bonnaillie-Noël, M. Dauge, D. Martin and G. Vial, Computations of the first eigenpairs for the schrödinger operator with magnetic field, Comput. Methods Appl. Mech. Engng., 196 (2007), 3841-3858. doi: 10.1016/j.cma.2006.10.041. [9] V. Bonnaillie-Noël, M. Dauge, N. Popoff and N. Raymond, Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions, Z. Angew. Math. Phys., DOI 10.1007/s00033-011-0163-y (2011). doi: 10.1007/s00033-011-0163-y. [10] V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners, Reviews in Mathematical Physics, 19 (2007), 607-637. doi: 10.1142/S0129055X07003061. [11] S. J. Chapman, Nucleation of superconductivity in decreasing fields. I, European J. Appl. Math., 5 (1994), 449-468. doi: 10.1017/S095679250000156X. [12] M. Dauge and B. Helffer, Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators, J. Differential Equations, 104 (1993), 243-262. doi: 10.1006/jdeq.1993.1071. [13] P.-G. De Gennes and D. Saint-James, Onset of superconductivity in decreasing fields, Physics Letters, 7 (1963), 306-308. doi: 10.1016/0031-9163(63)90047-7. [14] S. Fournais and B. Helffer, Energy asymptotics for type {II superconductors}, Calc. Var., 24 (2005), 341-376. doi: 10.1007/s00526-005-0333-x. [15] S. Fournais and B. Helffer, Accurate eigenvalue estimates for the magnetic Neumann Laplacian, Annales Inst. Fourier, 56 (2006), 1-67. doi: 10.5802/aif.2171. [16] S. Fournais and B. Helffer, On the third critical field in Ginzburg-Landau theory, Comm. Math. Phys., 266 (2006), 153-196. doi: 10.1007/s00220-006-0006-4. [17] S. Fournais and B. Helffer, "Spectral Methods in Surface Superconductivity," Progress in Nonlinear Differential Equations and their Applications, 77. Birkhäuser Boston Inc., Boston, MA 2010. [18] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag 2001. [19] E. M. Harrell, Double wells, Comm. Math. Phys., 75 (1980), 239-261. doi: 10.1007/BF01212711. [20] P. Hartmann, "Ordinary Differential Equations," Wiley, New-York 1964. [21] B. Helffer, "Semi-classical Analysis for the Schrödinger Operator and Applications," volume 1336 of Lecture Notes in Mathematics, [22] B. Helffer and A. Mohamed, Semiclassical analysis for the ground state energy of a Schrödinger operator with magnetic wells, J. Funct. Anal., 138 (1996), 40-81. doi: 10.1006/jfan.1996.0056. [23] B. Helffer and A. Morame, Magnetic bottles for the Neumann problem: the case of dimension 3, Proc. Indian Acad. Sci. Math. Sci., 112 (2002), 71-84. doi: 10.1007/BF02829641. [24] B. Helffer and A. Morame, Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case), Ann. Sci. École Norm. Sup., 37 (2004), 105-170. doi: 10.1016/j.ansens.2003.04.003. [25] B. Helffer and X.-B. Pan, Upper critical field and location of surface nucleation of superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 145-181. doi: 10.1016/S0294-1449(02)00005-7. [26] T. Kato, On the upper and lower bounds of eigenvalues, J. Phys. Soc. Japan, 4 (1949), 334-339. doi: 10.1143/JPSJ.4.334. [27] K. Lu and X.-B. Pan, Eigenvalue problems of Ginzburg-Landau operator in bounded domains, J. Math. Phys., 40 (1999), 2647-2670. doi: 10.1063/1.532721. [28] K. Lu and X.-B. Pan, Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity, Phys. D, 127 (1999), 73-104. doi: 10.1016/S0167-2789(98)00246-2. [29] K. Lu and X.-B. Pan, Gauge invariant eigenvalue problems in $R^2$ and in $R_+^2$, Trans. Amer. Math. Soc., 352 (2000), 1247-1276. doi: 10.1090/S0002-9947-99-02516-7. [30] N. Raymond, On the semiclassical 3D Neumann Laplacian with variable magnetic field, Asymptot. Anal., 68 (2010), 1-40. [31] Y. Sibuya, "Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient," Noth-Holland 1975.
 [1] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [2] Na Peng, Jiayu Han, Jing An. An efficient finite element method and error analysis for fourth order problems in a spherical domain. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022021 [3] Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic and Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040 [4] Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 [5] Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339 [6] Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 [7] Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153 [8] Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665 [9] Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387 [10] So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343 [11] Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052 [12] Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 [13] Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051 [14] Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure and Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297 [15] Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496 [16] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29 (3) : 2375-2389. doi: 10.3934/era.2020120 [17] Junxiang Li, Yan Gao, Tao Dai, Chunming Ye, Qiang Su, Jiazhen Huo. Substitution secant/finite difference method to large sparse minimax problems. Journal of Industrial and Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637 [18] Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, 2021, 29 (5) : 3141-3170. doi: 10.3934/era.2021031 [19] Brittany Froese Hamfeldt, Jacob Lesniewski. A convergent finite difference method for computing minimal Lagrangian graphs. Communications on Pure and Applied Analysis, 2022, 21 (2) : 393-418. doi: 10.3934/cpaa.2021182 [20] Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024

2020 Impact Factor: 1.916