# American Institute of Mathematical Sciences

July  2013, 18(5): 1155-1188. doi: 10.3934/dcdsb.2013.18.1155

## Constrained energy minimization and ground states for NLS with point defects

 1 Dipartimento di Scienze Matematiche, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy 2 Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 53, 20125 Milano, Italy 3 Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo, 56100 Pisa, Italy

Received  July 2012 Revised  January 2013 Published  March 2013

We investigate the ground states of the one-dimensional nonlinear Schrödinger equation with a defect located at a fixed point. The nonlinearity is focusing and consists of a subcritical power. The notion of ground state can be defined in several (often non-equivalent) ways. We define a ground state as a minimizer of the energy functional among the functions endowed with the same mass. This is the physically meaningful definition in the main fields of application of NLS. In this context we prove an abstract theorem that revisits the concentration-compactness method and which is suitable to treat NLS with inhomogeneities. Then we apply it to three models, describing three different kinds of defect: delta potential, delta prime interaction, and dipole. In the three cases we explicitly compute ground states and we show their orbital stability.
Citation: Riccardo Adami, Diego Noja, Nicola Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1155-1188. doi: 10.3934/dcdsb.2013.18.1155
##### References:
 [1] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Fast solitons on star graphs, Rev. Math. Phys., 23 (2011), 409-451. doi: 10.1142/S0129055X11004345. [2] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, On the structure of critical energy levels for the cubic focusing NLS on star graphs, J. Phys. A: Math. Theor., 45 (2012), 7pp. 192001. doi: 10.1088/1751-8113/45/19/192001. [3] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Stationary states of NLS on star graphs, EPL, 100 (2012), 10003. [4] R. Adami and D. Noja, Existence of dynamics for a 1-d NLS equation perturbed with a generalized point defect, J. Phys. A Math. Theor., 42 (2009), 495302. doi: 10.1088/1751-8113/42/49/495302. [5] R. Adami and D. Noja, Stability and symmetry breaking bifurcation for the ground states of a NLS equation with a $\delta'$ interaction, Commun. Math. Phys., 318 (2013), 247-289. doi: 10.1007/s00220-012-1597-6. [6] R. Adami, D. Noja and A. Sacchetti, On the mathematical description of the effective behaviour of one-dimensional Bose-Einstein condensates with defects, in "Bose-Einstein Condensates: Theory, Characteristics, and Current Research", Nova Publishing, New York (2010). [7] N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Spaces," Ungar, New York, 1963. [8] S. Albeverio, Z. Brzeźniak and L. Dabrowski, Fundamental solutions of the Heat and Schrödinger Equations with point interaction, J. Func. An., 130 (1995), 220-254. doi: 10.1006/jfan.1995.1068. [9] S. Albeverio, F. Gesztesy, R. HΦgh-Krohn and H. Holden, "Solvable Models in Quantum Mechanics, $2^{nd}$ ed., with an Appendix of P. Exner," AMS, Providence, 2005. [10] S. Albeverio and P. Kurasov, "Singular Perturbations of Differential Operators," Cambridge University Press, 2000. doi: 10.1017/CBO9780511758904. [11] J. Bellazzini and N. Visciglia, On the orbital stability for a class of nonautonmous NLS, Indiana Univ. Math. J., 59 (2010), 1211-1230. doi: 10.1512/iumj.2010.59.3907. [12] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I - Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [13] J. Blank, P. Exner and M. Havlicek, "Hilbert Spaces Operators in Quantum Physics," Springer, New York, 2008. [14] C. Bonanno, M. Ghimenti and M. Squassina, Soliton dynamics of NLS with singular potentials, preprint arXiv:1206.1832 (2012). [15] H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999. [16] D. Cao Xiang and A. B. Malomed, Soliton defect collisions in the nonlinear Schr\"odinger equation, Phys. Lett. A, 206 (1985), 177-182. doi: 10.1016/0375-9601(95)00611-6. [17] T. Cazenave, "Semilinear Schrödinger Equations," 10 Courant Lecture Notes in Mathematics, AMS, Providence, 2003. [18] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. [19] T. Cheon and T. Shigehara, Realizing discontinuous wave functions with renormalized short-range potentials, Phys. Lett. A, 243 (1998), 111-116. [20] K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities, Comm. Part. Diff. Eq., 34 (2009), 1074-1113. doi: 10.1080/03605300903076831. [21] P. Deift and J. Park, Long-Time Asymptotics for solutions of the NLS equation with a delta potential and even initial data, Int. Math. Res. Notices, 24 (2011), 5505-5624. doi: 10.1007/s11005-010-0458-5. [22] P. Exner and P. Grosse, Some properties of the one-dimensional generalized point interactions (a torso), preprint mp-arc 99-390, arXiv:math-ph/9910029, (1999). [23] P. Exner, H. Neidhardt and V. A. Zagrebnov, Potential approximations to a $\delta'$: An inverse Klauder phenomenon with norm-resolvent convergence, Comm. Math. Phys., 224 (2001), 593-612. doi: 10.1007/s002200100567. [24] R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential, Disc. Cont. Dyn. Syst. (A), 21 (2008), 129-144. doi: 10.3934/dcds.2008.21.121. [25] R. Fukuizumi, M. Ohta and T. Ozawa, Nonlinear Schrödinger equation with a point defect, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 25 (2008), 837-845. doi: 10.1016/j.anihpc.2007.03.004. [26] Y. Furuhashi, M. Hirokawa, K. Nakahara and Y. Shikano, Role of a phase factor in the boundary condition of a one-dimensional junction, J. Phys. A Math. Theor., 43 (2010), 354010. doi: 10.1088/1751-8113/43/35/354010. [27] Yu. D. Golovaty and R. O. Hryniv, On norm resolvent convergence of Schrödinger operators with $\delta'$-like potentials, J. Phys. A. Math. Theor., 44 (2011), 049802; Corrigendum J. Phys. A. Math. Theor., 44 (2011), 049802. doi: 10.1088/1751-8113/44/4/049802. [28] R. H. Goodman, P. J. Holmes and M. I. Weinstein, Strong NLS soliton-defect interactions, {Physica D.}, 192 (2004), 215-248. doi: 10.1016/j.physd.2004.01.021. [29] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry - I, J. Func. An., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. [30] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry - II, J. Func. An., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E. [31] J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Phys., 274 (2007), 187-216. doi: 10.1007/s00220-007-0261-z. [32] J. Holmer and M. Zworski, Breathing patterns in nonlinear relaxation, Nonlinearity, 22 (2009), 1259-1301. doi: 10.1088/0951-7715/22/6/002. [33] S. Le Coz, R. Fukuizumi, G. Fibich, Y. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Physica D, 237 (2008), 1103-1128. doi: 10.1016/j.physd.2007.12.004. [34] Y. Linzon, R. Morandotti, V. Aimez, V. Ares and S. Bar-Ad, Nonlinear scattering and trapping by local photonic potentials, Phys. Rev. Lett., 99 (2007), 133901. [35] F. Prinari and N. Visciglia, On a minimization problem involving the critical Sobolev exponent, Advanced Nonlinear Studies, 7 (2007), 551-564. [36] M. Reed and B. Simon, "Methods of Modern Mathematical Physics - Vol I," Academic Press, 1980. [37] M. Weinstein, Modulational stability of ground states of nonlinear Schroedinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034. [38] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-68. doi: 10.1002/cpa.3160390103. [39] D. Witthaut, S. Mossmann and H. J. Korsch, Bound and resonance states of the nonlinear Schrödinger equation in simple model systems, J. Phys. A Math. Gen., 38 (2005), 1777-1702. doi: 10.1088/0305-4470/38/8/013. [40] A. V. Zolotaryuk, P. L. Christiansen and S. V. Iermakova, Scattering properties of point dipole interactions, J. Phys. A Math. Gen., 39 (2006), 9329-9338. doi: 10.1088/0305-4470/39/29/023. [41] V. A. Zolotaryuk, Boundary conditions for the states with resonant tunnelling across the $\delta'$-potential, Phys. Lett. A, 374 (2010), 1636-1641. doi: 10.1016/j.physleta.2010.02.005.

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##### References:
 [1] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Fast solitons on star graphs, Rev. Math. Phys., 23 (2011), 409-451. doi: 10.1142/S0129055X11004345. [2] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, On the structure of critical energy levels for the cubic focusing NLS on star graphs, J. Phys. A: Math. Theor., 45 (2012), 7pp. 192001. doi: 10.1088/1751-8113/45/19/192001. [3] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Stationary states of NLS on star graphs, EPL, 100 (2012), 10003. [4] R. Adami and D. Noja, Existence of dynamics for a 1-d NLS equation perturbed with a generalized point defect, J. Phys. A Math. Theor., 42 (2009), 495302. doi: 10.1088/1751-8113/42/49/495302. [5] R. Adami and D. Noja, Stability and symmetry breaking bifurcation for the ground states of a NLS equation with a $\delta'$ interaction, Commun. Math. Phys., 318 (2013), 247-289. doi: 10.1007/s00220-012-1597-6. [6] R. Adami, D. Noja and A. Sacchetti, On the mathematical description of the effective behaviour of one-dimensional Bose-Einstein condensates with defects, in "Bose-Einstein Condensates: Theory, Characteristics, and Current Research", Nova Publishing, New York (2010). [7] N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Spaces," Ungar, New York, 1963. [8] S. Albeverio, Z. Brzeźniak and L. Dabrowski, Fundamental solutions of the Heat and Schrödinger Equations with point interaction, J. Func. An., 130 (1995), 220-254. doi: 10.1006/jfan.1995.1068. [9] S. Albeverio, F. Gesztesy, R. HΦgh-Krohn and H. Holden, "Solvable Models in Quantum Mechanics, $2^{nd}$ ed., with an Appendix of P. Exner," AMS, Providence, 2005. [10] S. Albeverio and P. Kurasov, "Singular Perturbations of Differential Operators," Cambridge University Press, 2000. doi: 10.1017/CBO9780511758904. [11] J. Bellazzini and N. Visciglia, On the orbital stability for a class of nonautonmous NLS, Indiana Univ. Math. J., 59 (2010), 1211-1230. doi: 10.1512/iumj.2010.59.3907. [12] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I - Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [13] J. Blank, P. Exner and M. Havlicek, "Hilbert Spaces Operators in Quantum Physics," Springer, New York, 2008. [14] C. Bonanno, M. Ghimenti and M. Squassina, Soliton dynamics of NLS with singular potentials, preprint arXiv:1206.1832 (2012). [15] H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.2307/2044999. [16] D. Cao Xiang and A. B. Malomed, Soliton defect collisions in the nonlinear Schr\"odinger equation, Phys. Lett. A, 206 (1985), 177-182. doi: 10.1016/0375-9601(95)00611-6. [17] T. Cazenave, "Semilinear Schrödinger Equations," 10 Courant Lecture Notes in Mathematics, AMS, Providence, 2003. [18] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. [19] T. Cheon and T. Shigehara, Realizing discontinuous wave functions with renormalized short-range potentials, Phys. Lett. A, 243 (1998), 111-116. [20] K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities, Comm. Part. Diff. Eq., 34 (2009), 1074-1113. doi: 10.1080/03605300903076831. [21] P. Deift and J. Park, Long-Time Asymptotics for solutions of the NLS equation with a delta potential and even initial data, Int. Math. Res. Notices, 24 (2011), 5505-5624. doi: 10.1007/s11005-010-0458-5. [22] P. Exner and P. Grosse, Some properties of the one-dimensional generalized point interactions (a torso), preprint mp-arc 99-390, arXiv:math-ph/9910029, (1999). [23] P. Exner, H. Neidhardt and V. A. Zagrebnov, Potential approximations to a $\delta'$: An inverse Klauder phenomenon with norm-resolvent convergence, Comm. Math. Phys., 224 (2001), 593-612. doi: 10.1007/s002200100567. [24] R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential, Disc. Cont. Dyn. Syst. (A), 21 (2008), 129-144. doi: 10.3934/dcds.2008.21.121. [25] R. Fukuizumi, M. Ohta and T. Ozawa, Nonlinear Schrödinger equation with a point defect, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 25 (2008), 837-845. doi: 10.1016/j.anihpc.2007.03.004. [26] Y. Furuhashi, M. Hirokawa, K. Nakahara and Y. Shikano, Role of a phase factor in the boundary condition of a one-dimensional junction, J. Phys. A Math. Theor., 43 (2010), 354010. doi: 10.1088/1751-8113/43/35/354010. [27] Yu. D. Golovaty and R. O. Hryniv, On norm resolvent convergence of Schrödinger operators with $\delta'$-like potentials, J. Phys. A. Math. Theor., 44 (2011), 049802; Corrigendum J. Phys. A. Math. Theor., 44 (2011), 049802. doi: 10.1088/1751-8113/44/4/049802. [28] R. H. Goodman, P. J. Holmes and M. I. Weinstein, Strong NLS soliton-defect interactions, {Physica D.}, 192 (2004), 215-248. doi: 10.1016/j.physd.2004.01.021. [29] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry - I, J. Func. An., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9. [30] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry - II, J. Func. An., 94 (1990), 308-348. doi: 10.1016/0022-1236(90)90016-E. [31] J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Comm. Math. Phys., 274 (2007), 187-216. doi: 10.1007/s00220-007-0261-z. [32] J. Holmer and M. Zworski, Breathing patterns in nonlinear relaxation, Nonlinearity, 22 (2009), 1259-1301. doi: 10.1088/0951-7715/22/6/002. [33] S. Le Coz, R. Fukuizumi, G. Fibich, Y. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Physica D, 237 (2008), 1103-1128. doi: 10.1016/j.physd.2007.12.004. [34] Y. Linzon, R. Morandotti, V. Aimez, V. Ares and S. Bar-Ad, Nonlinear scattering and trapping by local photonic potentials, Phys. Rev. Lett., 99 (2007), 133901. [35] F. Prinari and N. Visciglia, On a minimization problem involving the critical Sobolev exponent, Advanced Nonlinear Studies, 7 (2007), 551-564. [36] M. Reed and B. Simon, "Methods of Modern Mathematical Physics - Vol I," Academic Press, 1980. [37] M. Weinstein, Modulational stability of ground states of nonlinear Schroedinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034. [38] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-68. doi: 10.1002/cpa.3160390103. [39] D. Witthaut, S. Mossmann and H. J. Korsch, Bound and resonance states of the nonlinear Schrödinger equation in simple model systems, J. Phys. A Math. Gen., 38 (2005), 1777-1702. doi: 10.1088/0305-4470/38/8/013. [40] A. V. Zolotaryuk, P. L. Christiansen and S. V. Iermakova, Scattering properties of point dipole interactions, J. Phys. A Math. Gen., 39 (2006), 9329-9338. doi: 10.1088/0305-4470/39/29/023. [41] V. A. Zolotaryuk, Boundary conditions for the states with resonant tunnelling across the $\delta'$-potential, Phys. Lett. A, 374 (2010), 1636-1641. doi: 10.1016/j.physleta.2010.02.005.
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