# 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 & 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. 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). 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). [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). 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. 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, (2010). [7] N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Spaces,", Ungar, (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. 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, (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. 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. doi: 10.1007/BF00250555. [13] J. Blank, P. Exner and M. Havlicek, "Hilbert Spaces Operators in Quantum Physics,", Springer, (2008). [14] C. Bonanno, M. Ghimenti and M. Squassina, Soliton dynamics of NLS with singular potentials,, preprint arXiv:1206.1832 (2012)., (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. 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. doi: 10.1016/0375-9601(95)00611-6. [17] T. Cazenave, "Semilinear Schrödinger Equations,", 10 Courant Lecture Notes in Mathematics, 10 (2003). [18] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Comm. Math. Phys., 85 (1982), 549. [19] T. Cheon and T. Shigehara, Realizing discontinuous wave functions with renormalized short-range potentials,, Phys. Lett. A, 243 (1998), 111. [20] K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities,, Comm. Part. Diff. Eq., 34 (2009), 1074. 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. 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, (1999), 99. [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. 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. 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, 25 (2008), 837. 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). 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). 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. 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. 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. 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. doi: 10.1007/s00220-007-0261-z. [32] J. Holmer and M. Zworski, Breathing patterns in nonlinear relaxation,, Nonlinearity, 22 (2009), 1259. 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. 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). [35] F. Prinari and N. Visciglia, On a minimization problem involving the critical Sobolev exponent,, Advanced Nonlinear Studies, 7 (2007), 551. [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. doi: 10.1137/0516034. [38] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Comm. Pure Appl. Math., 39 (1986), 51. 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. 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. 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. doi: 10.1016/j.physleta.2010.02.005.

show all references

##### References:
 [1] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Fast solitons on star graphs,, Rev. Math. Phys., 23 (2011), 409. 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). 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). [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). 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. 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, (2010). [7] N. Akhiezer and I. Glazman, "Theory of Linear Operators in Hilbert Spaces,", Ungar, (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. 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, (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. 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. doi: 10.1007/BF00250555. [13] J. Blank, P. Exner and M. Havlicek, "Hilbert Spaces Operators in Quantum Physics,", Springer, (2008). [14] C. Bonanno, M. Ghimenti and M. Squassina, Soliton dynamics of NLS with singular potentials,, preprint arXiv:1206.1832 (2012)., (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. 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. doi: 10.1016/0375-9601(95)00611-6. [17] T. Cazenave, "Semilinear Schrödinger Equations,", 10 Courant Lecture Notes in Mathematics, 10 (2003). [18] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Comm. Math. Phys., 85 (1982), 549. [19] T. Cheon and T. Shigehara, Realizing discontinuous wave functions with renormalized short-range potentials,, Phys. Lett. A, 243 (1998), 111. [20] K. Datchev and J. Holmer, Fast soliton scattering by attractive delta impurities,, Comm. Part. Diff. Eq., 34 (2009), 1074. 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. 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, (1999), 99. [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. 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. 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, 25 (2008), 837. 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). 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). 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. 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. 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. 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. doi: 10.1007/s00220-007-0261-z. [32] J. Holmer and M. Zworski, Breathing patterns in nonlinear relaxation,, Nonlinearity, 22 (2009), 1259. 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. 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). [35] F. Prinari and N. Visciglia, On a minimization problem involving the critical Sobolev exponent,, Advanced Nonlinear Studies, 7 (2007), 551. [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. doi: 10.1137/0516034. [38] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations,, Comm. Pure Appl. Math., 39 (1986), 51. 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. 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. 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. doi: 10.1016/j.physleta.2010.02.005.
 [1] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [2] Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005 [3] Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136 [4] Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009 [5] Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 [6] Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure & Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867 [7] Dongdong Qin, Xianhua Tang, Qingfang Wu. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1261-1280. doi: 10.3934/cpaa.2019061 [8] Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120 [9] Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359 [10] Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060 [11] Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054 [12] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [13] Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091 [14] Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263 [15] Teresa D'Aprile. Some existence and concentration results for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2002, 1 (4) : 457-474. doi: 10.3934/cpaa.2002.1.457 [16] Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117 [17] Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067 [18] D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 [19] Tetsu Mizumachi. Instability of bound states for 2D nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 413-428. doi: 10.3934/dcds.2005.13.413 [20] Xu Zhang. On the concentration of semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5389-5413. doi: 10.3934/dcds.2018238

2018 Impact Factor: 1.008