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 & 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.  doi: 10.1002/num.20137.  Google Scholar

[2]

A. Bernoff and P. Sternberg, Onset of superconductivity in decreasing fields for general domains,, J. Math. Phys., 39 (1998), 1272.  doi: 10.1063/1.532379.  Google Scholar

[3]

C. Bolley, Modélisation du champ de retard à la condensation d'un supraconducteur par un problème de bifurcation,, RAIRO Mod\'el. Math. Anal. Num\'er., 26 (1992), 235.   Google Scholar

[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. Poincar\'e Phys. Th\'eor., 58 (1993), 189.   Google Scholar

[5]

V. Bonnaillie, "Analyse mathématique de la supraconductivité dans un domaine à coins; méthodes semi-classiques et numériques,", Thèse de doctorat, (2003).   Google Scholar

[6]

V. Bonnaillie, On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners,, Asymptot. Anal., 41 (2005), 215.   Google Scholar

[7]

V. Bonnaillie-Noël and M. Dauge, Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners,, Ann. Henri Poincar\'e, 7 (2006), 899.  doi: 10.1007/s00023-006-0271-y.  Google Scholar

[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.  doi: 10.1016/j.cma.2006.10.041.  Google Scholar

[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., (2011), 00033.  doi: 10.1007/s00033-011-0163-y.  Google Scholar

[10]

V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners,, Reviews in Mathematical Physics, 19 (2007), 607.  doi: 10.1142/S0129055X07003061.  Google Scholar

[11]

S. J. Chapman, Nucleation of superconductivity in decreasing fields. I,, European J. Appl. Math., 5 (1994), 449.  doi: 10.1017/S095679250000156X.  Google Scholar

[12]

M. Dauge and B. Helffer, Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators,, J. Differential Equations, 104 (1993), 243.  doi: 10.1006/jdeq.1993.1071.  Google Scholar

[13]

P.-G. De Gennes and D. Saint-James, Onset of superconductivity in decreasing fields,, Physics Letters, 7 (1963), 306.  doi: 10.1016/0031-9163(63)90047-7.  Google Scholar

[14]

S. Fournais and B. Helffer, Energy asymptotics for type {II superconductors},, Calc. Var., 24 (2005), 341.  doi: 10.1007/s00526-005-0333-x.  Google Scholar

[15]

S. Fournais and B. Helffer, Accurate eigenvalue estimates for the magnetic Neumann Laplacian,, Annales Inst. Fourier, 56 (2006), 1.  doi: 10.5802/aif.2171.  Google Scholar

[16]

S. Fournais and B. Helffer, On the third critical field in Ginzburg-Landau theory,, Comm. Math. Phys., 266 (2006), 153.  doi: 10.1007/s00220-006-0006-4.  Google Scholar

[17]

S. Fournais and B. Helffer, "Spectral Methods in Surface Superconductivity,", Progress in Nonlinear Differential Equations and their Applications, (2010).   Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag 2001., (2001).   Google Scholar

[19]

E. M. Harrell, Double wells,, Comm. Math. Phys., 75 (1980), 239.  doi: 10.1007/BF01212711.  Google Scholar

[20]

P. Hartmann, "Ordinary Differential Equations,", Wiley, (1964).   Google Scholar

[21]

B. Helffer, "Semi-classical Analysis for the Schrödinger Operator and Applications,", volume 1336 of {\em Lecture Notes in Mathematics}, (1336).   Google Scholar

[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.  doi: 10.1006/jfan.1996.0056.  Google Scholar

[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.  doi: 10.1007/BF02829641.  Google Scholar

[24]

B. Helffer and A. Morame, Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case),, Ann. Sci. \'Ecole Norm. Sup., 37 (2004), 105.  doi: 10.1016/j.ansens.2003.04.003.  Google Scholar

[25]

B. Helffer and X.-B. Pan, Upper critical field and location of surface nucleation of superconductivity,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 20 (2003), 145.  doi: 10.1016/S0294-1449(02)00005-7.  Google Scholar

[26]

T. Kato, On the upper and lower bounds of eigenvalues,, J. Phys. Soc. Japan, 4 (1949), 334.  doi: 10.1143/JPSJ.4.334.  Google Scholar

[27]

K. Lu and X.-B. Pan, Eigenvalue problems of Ginzburg-Landau operator in bounded domains,, J. Math. Phys., 40 (1999), 2647.  doi: 10.1063/1.532721.  Google Scholar

[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.  doi: 10.1016/S0167-2789(98)00246-2.  Google Scholar

[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.  doi: 10.1090/S0002-9947-99-02516-7.  Google Scholar

[30]

N. Raymond, On the semiclassical 3D Neumann Laplacian with variable magnetic field,, Asymptot. Anal., 68 (2010), 1.   Google Scholar

[31]

Y. Sibuya, "Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient,", Noth-Holland 1975., (1975).   Google Scholar

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.  doi: 10.1002/num.20137.  Google Scholar

[2]

A. Bernoff and P. Sternberg, Onset of superconductivity in decreasing fields for general domains,, J. Math. Phys., 39 (1998), 1272.  doi: 10.1063/1.532379.  Google Scholar

[3]

C. Bolley, Modélisation du champ de retard à la condensation d'un supraconducteur par un problème de bifurcation,, RAIRO Mod\'el. Math. Anal. Num\'er., 26 (1992), 235.   Google Scholar

[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. Poincar\'e Phys. Th\'eor., 58 (1993), 189.   Google Scholar

[5]

V. Bonnaillie, "Analyse mathématique de la supraconductivité dans un domaine à coins; méthodes semi-classiques et numériques,", Thèse de doctorat, (2003).   Google Scholar

[6]

V. Bonnaillie, On the fundamental state energy for a Schrödinger operator with magnetic field in domains with corners,, Asymptot. Anal., 41 (2005), 215.   Google Scholar

[7]

V. Bonnaillie-Noël and M. Dauge, Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners,, Ann. Henri Poincar\'e, 7 (2006), 899.  doi: 10.1007/s00023-006-0271-y.  Google Scholar

[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.  doi: 10.1016/j.cma.2006.10.041.  Google Scholar

[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., (2011), 00033.  doi: 10.1007/s00033-011-0163-y.  Google Scholar

[10]

V. Bonnaillie-Noël and S. Fournais, Superconductivity in domains with corners,, Reviews in Mathematical Physics, 19 (2007), 607.  doi: 10.1142/S0129055X07003061.  Google Scholar

[11]

S. J. Chapman, Nucleation of superconductivity in decreasing fields. I,, European J. Appl. Math., 5 (1994), 449.  doi: 10.1017/S095679250000156X.  Google Scholar

[12]

M. Dauge and B. Helffer, Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators,, J. Differential Equations, 104 (1993), 243.  doi: 10.1006/jdeq.1993.1071.  Google Scholar

[13]

P.-G. De Gennes and D. Saint-James, Onset of superconductivity in decreasing fields,, Physics Letters, 7 (1963), 306.  doi: 10.1016/0031-9163(63)90047-7.  Google Scholar

[14]

S. Fournais and B. Helffer, Energy asymptotics for type {II superconductors},, Calc. Var., 24 (2005), 341.  doi: 10.1007/s00526-005-0333-x.  Google Scholar

[15]

S. Fournais and B. Helffer, Accurate eigenvalue estimates for the magnetic Neumann Laplacian,, Annales Inst. Fourier, 56 (2006), 1.  doi: 10.5802/aif.2171.  Google Scholar

[16]

S. Fournais and B. Helffer, On the third critical field in Ginzburg-Landau theory,, Comm. Math. Phys., 266 (2006), 153.  doi: 10.1007/s00220-006-0006-4.  Google Scholar

[17]

S. Fournais and B. Helffer, "Spectral Methods in Surface Superconductivity,", Progress in Nonlinear Differential Equations and their Applications, (2010).   Google Scholar

[18]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Springer-Verlag 2001., (2001).   Google Scholar

[19]

E. M. Harrell, Double wells,, Comm. Math. Phys., 75 (1980), 239.  doi: 10.1007/BF01212711.  Google Scholar

[20]

P. Hartmann, "Ordinary Differential Equations,", Wiley, (1964).   Google Scholar

[21]

B. Helffer, "Semi-classical Analysis for the Schrödinger Operator and Applications,", volume 1336 of {\em Lecture Notes in Mathematics}, (1336).   Google Scholar

[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.  doi: 10.1006/jfan.1996.0056.  Google Scholar

[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.  doi: 10.1007/BF02829641.  Google Scholar

[24]

B. Helffer and A. Morame, Magnetic bottles for the Neumann problem: curvature effects in the case of dimension 3 (general case),, Ann. Sci. \'Ecole Norm. Sup., 37 (2004), 105.  doi: 10.1016/j.ansens.2003.04.003.  Google Scholar

[25]

B. Helffer and X.-B. Pan, Upper critical field and location of surface nucleation of superconductivity,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 20 (2003), 145.  doi: 10.1016/S0294-1449(02)00005-7.  Google Scholar

[26]

T. Kato, On the upper and lower bounds of eigenvalues,, J. Phys. Soc. Japan, 4 (1949), 334.  doi: 10.1143/JPSJ.4.334.  Google Scholar

[27]

K. Lu and X.-B. Pan, Eigenvalue problems of Ginzburg-Landau operator in bounded domains,, J. Math. Phys., 40 (1999), 2647.  doi: 10.1063/1.532721.  Google Scholar

[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.  doi: 10.1016/S0167-2789(98)00246-2.  Google Scholar

[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.  doi: 10.1090/S0002-9947-99-02516-7.  Google Scholar

[30]

N. Raymond, On the semiclassical 3D Neumann Laplacian with variable magnetic field,, Asymptot. Anal., 68 (2010), 1.   Google Scholar

[31]

Y. Sibuya, "Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient,", Noth-Holland 1975., (1975).   Google Scholar

[1]

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 & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040

[2]

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

[3]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[4]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

[5]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[6]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387

[7]

So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

[8]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[9]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[10]

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

[11]

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 & Management Optimization, 2014, 10 (2) : 637-663. doi: 10.3934/jimo.2014.10.637

[12]

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 & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024

[13]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

[14]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795

[15]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

[16]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663

[17]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639

[18]

Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems & Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459

[19]

Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154

[20]

Heung Wing Joseph Lee, Chi Kin Chan, Karho Yau, Kar Hung Wong, Colin Myburgh. Control parametrization and finite element method for controlling multi-species reactive transport in a circular pool. Journal of Industrial & Management Optimization, 2013, 9 (3) : 505-524. doi: 10.3934/jimo.2013.9.505

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]