# 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 & 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
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