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On a formula for sets of constant width in 2d
July 2019, 18(4): 2093-2116.
doi: 10.3934/cpaa.2019094
On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction
| 1. | Department of Mathematics, IME-USP, Rua do Matão 1010, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil |
| 2. | Department of Mathematics, DMA-UEM, Av. Colombo, 5790 Jd. Universitário, CEP 87020-900, Maringá, PR, Brazil |
We study analytically and numerically the existence and orbital stability of the peak-standing-wave solutions for the cubic-quintic nonlinear Schródinger equation with a point interaction determined by the delta of Dirac. We study the cases of attractive-attractive and attractive-repulsive nonlinearities and we recover some results in the literature. Via a perturbation method and continuation argument we determine the Morse index of some specific self-adjoint operators that arise in the stability study. Orbital instability implications from a spectral instability result are established. In the case of an attractive-attractive case and an focusing interaction we give an approach based in the extension theory of symmetric operators for determining the Morse index.
Keywords:
Nonlinear stability,
standing wave solutions,
nonlinear Schródinger equation with point interaction,
analytic perturbation theory,
self-adjoint extensions.
Mathematics Subject Classification:
Primary: 35Q55, 37K40, 37K45; Secondary: 47E05, 35J61.
Citation:
Jaime Angulo Pava, César A. Hernández Melo. On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction. Communications on Pure & Applied Analysis,
2019, 18
(4)
: 2093-2116.
doi: 10.3934/cpaa.2019094
![]()
References:
| [1] |
R. Adami, C. Cacciapuoti, D. Finco and D. Noja,
Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415.
doi: 10.1016/j.jde.2016.01.029. |
| [2] |
R. Adami, C. Cacciapuoti, D. Finco and D. Noja,
Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257 (2014), 3738-3777.
doi: 10.1016/j.jde.2014.07.008. |
| [3] |
R. Adami, D. Noja and N. Visciglia,
Constrained energy minimization and ground states for NLS with point defects, Discr. Cont. Dyn. Syst.- B, 18 (2013), 155-1188.
doi: 10.3934/dcdsb.2013.18.1155. |
| [4] |
G. Agrawal, Nonlinear Fiber Optics, Academic Press, 4th edition, 2007. |
| [5] |
S. Albeverio, Z. Brzezniak and L. Dabrowski,
Fundamental solution of the heat and Schródinger equations with point interaction, J. Funct. Anal., 130 (1995), 220-254.
doi: 10.1006/jfan.1995.1068. |
| [6] |
S. Albeverio, F. Gesztesy, R. Krohn and H. Holden, Solvable Models in Quantum Mechanics, AMS Chelsea publishing, 2004., |
| [7] |
S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators, London Mathematical Society Lecture Note Series 271, Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511758904.
|
| [8] |
J. Angulo,
Instability of cnoidal-peak for the NLS-$\delta$ equation, Math. Nachr., 285 (2012), 1572-1602.
doi: 10.1002/mana.201100209. |
| [9] |
J. Angulo and A. Hernandez,
Stability of standing waves for logarithmic Schródinger equation with attractive delta potential, IUMJ, 67 (2018), 471-494.
doi: 10.1512/iumj.2018.67.7273. |
| [10] |
J. Angulo and N. Goloshchapova,
On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph, Discr. Cont. Dyn. Syst.- A, 38 (2018), 5039-5066.
doi: 10.3934/dcds.2018221. |
| [11] |
J. Angulo and N. Goloshchapova, On the standing waves of the NLS-log equation with point interaction on a star graph, preprint, arXiv: 1803.07194. |
| [12] |
J. Angulo and N. Goloshchapova,
Extension theory approach in the stability of the standing waves for NLS equation with point interactions on a star graph, Advances in Differential Equations, 23 (2018), 793-846.
|
| [13] |
J. Angulo and N. Goloshchapova, Stability of standing waves for NLS-log equation with $\delta$-interaction, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 27.
doi: 10.1007/s00030-017-0451-0. |
| [14] |
J. Angulo, O. Lopes and A. Neves,
Instability of travelling waves for weakly coupled KdV systems, Nonlinear Anal., 69 (2008), 1870-1887.
doi: 10.1016/j.na.2007.07.039. |
| [15] |
J. Angulo and F. Natali,
On the instability of periodic waves for dispersive equations, Differential Integral Equations, 29 (2016), 837-874.
|
| [16] |
J. Angulo and G. Ponce,
The nonlinear Schródinger equation with a periodic $\delta$-interaction, Bull. Braz. Math. Soc., New Series, 44 (2013), 497-551.
doi: 10.1007/s00574-013-0024-8. |
| [17] |
G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala and F. Sanchez, Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun, 219 (2003), 427-433. |
| [18] |
V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattices, Modern Physics Letter - B, 18 (2004), 627-651. |
| [19] |
F. A. Berezin and M. A. Shubin, The Schródinger Equation, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1991.
doi: 10.1007/978-94-011-3154-4. |
| [20] |
V. Caudrelier, M. Mintchev and E. Ragoucy, Solving the quantum non-linear Schródinger equation with $\delta$-type impurity, J. Math. Phys., 46 (2005), 042703-1-24.
doi: 10.1063/1.1842353. |
| [21] |
T. Cazenave, Semilinear Schródinger Equations, American Mathematical Society, AMS. Lecture Notes, v. 10, 2003.
doi: 10.1090/cln/010. |
| [22] |
S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan,
Instability of bound states of a nonlinear Schródinger equation with a Dirac Potential, Phys. D, 237 (2008), 1103-1128.
doi: 10.1016/j.physd.2007.12.004. |
| [23] |
K. Datchev and J. Holmer,
Fast soliton scattering by attractive delta impurities, Comm. PDE., 34 (2009), 1074-1173.
doi: 10.1080/03605300903076831. |
| [24] |
K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in gas of sodium atoms, Phys. Rev. Lett., 74 (1995), 3969-3973. |
| [25] |
E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond and V. Skarka, Robust two-dimensional spatial solitons in liquid carbon disulfide, Phys. Rev. Lett., 110 (2013), 013901. |
| [26] |
E. L. Falcão-Filho, C. B. de Araújo and J. J. Rodrigues Jr., High-order nonlinearities of aqueous colloids containing silver nanoparticles, J. Opt. Soc. Am. - B, 24 (2007), 2948-2956. |
| [27] |
R. Fukuizumi and L. Jeanjean,
Stability of standing waves for a nonlinear Schródinger equation with a repulsive Dirac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136.
doi: 10.3934/dcds.2008.21.121. |
| [28] |
R. Fukuizumi, M. Ohta and T. Ozawa,
Nonlinear Schródinger equation with a point defect, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 837-845.
doi: 10.1016/j.anihpc.2007.03.004. |
| [29] |
B. Gaveau and L. S. Schulman,
Explicit time-dependent Schródinger propagators, J. Physics A: Math. Gen., 19 (1986), 1833-1846.
|
| [30] |
F. Genoud, F. B. Malomed and R. Weishäupl,
Stable NLS solitons in a cubic-quintic medium with a delta-function potential, Nonlinear Anal., 133 (2016), 28-50.
doi: 10.1016/j.na.2015.11.016. |
| [31] |
B. V. Gisin, R. Driben and B. A. Malomed, Bistable guided solitons in the cubic-quintic medium, J. Optics B: Quantum and Semiclassical Optics, 6 (2004), S259–S264. |
| [32] |
R. H. Goodman, J. Holmes and M. Weinstein,
Strong NLS soliton-defect interactions, Phys. D, 192 (2004), 215-248.
doi: 10.1016/j.physd.2004.01.021. |
| [33] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry, Ⅰ, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
| [34] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry, Ⅱ, J. Funct. Anal., 94 (1990), 308-348.
doi: 10.1016/0022-1236(90)90016-E. |
| [35] |
D. Henry, J. Perez and W. Wreszinski,
Stability theory for solitary-wave solutions of scalar field equation, Comm. Math. Phys., 85 (1982), 351-361.
|
| [36] |
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. |
| [37] |
M. Kaminaga and M. Ohta,
Stability of standing waves for nonlinear Schrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J.,, 26 (2009), 39-48.
|
| [38] |
T. Kato, Perturbation Theory for Linear Operators, 2$^{nd}$ edition, Springer-Verlang, New York, 1976., |
| [39] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^{nd}$ edition, Universitext, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2181-2. |
| [40] |
S. Le Coz, Y. Martel and P. Raphael, Minimal mass blow up solutions for a double power nonlinear Schródinger equation, preprint, arXiv: 1406.6002.
doi: 10.4171/RMI/899. |
| [41] |
M. Maeda,
Stability and instability of standing waves for 1-dimensional nonlinear Schródinger equation with multiple-power nonlinearity, Kodai Math. J., 31 (2008), 263-271.
doi: 10.2996/kmj/1214442798. |
| [42] |
C. R. Menyuk, Soliton robustness in optical fibers, J. Opt. Soc. Am. - B, 10 (1993), 1585-1591 |
| [43] |
J. Moloney and A. Newell, Nonlinear Optics, CRC, Taylor & Francis Group, Boca Raton. FL. USA, 2018., |
| [44] |
M. A. Naimark, Linear Differential Operators, F. Ungar Pub. Co., New York, 1967., |
| [45] |
M. Ohta,
Stability and Instability of standing waves for one dimensional nonlinear Schródinger equations with double power nonlinearity, Kodai Math. J., 18 (1995), 68-74.
doi: 10.2996/kmj/1138043354. |
| [46] |
M. Ohta,
Instability of bound states for abstract nonlinear Schródinger equations, J. Funct. Anal., 261 (2011), 90-110.
doi: 10.1016/j.jfa.2011.03.010. |
| [47] |
P. Papagiannis, Y. Kominis and K. Hizanidis, Power-and momentum-dependent soliton dynamics in lattices with longitudinal modulation, Phys. Rev. A, 84 (2011), 013820 |
| [48] |
S. Reed and B. Simon, Methods of Modern Mathematical Physics: Analysis of Operators, Academic Press, vol. Ⅳ, 1978. |
| [49] |
H. Sakaguchi and M. Tamura,
Scattering and trapping of nonlinear Schródinger solitons in external potentials, J. Phys. Soc. Japan, 73 (2004), 503-506.
doi: 10.1143/JPSJ.73.503. |
| [50] |
B. T. Seaman, L. D. Car and M. J. Holland, Effect of a potential step or impurity on the Bose-Einstein condensate mean field, Phys. Rev. A, 71 (2005), 033609. |
show all references
References:
| [1] |
R. Adami, C. Cacciapuoti, D. Finco and D. Noja,
Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415.
doi: 10.1016/j.jde.2016.01.029. |
| [2] |
R. Adami, C. Cacciapuoti, D. Finco and D. Noja,
Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257 (2014), 3738-3777.
doi: 10.1016/j.jde.2014.07.008. |
| [3] |
R. Adami, D. Noja and N. Visciglia,
Constrained energy minimization and ground states for NLS with point defects, Discr. Cont. Dyn. Syst.- B, 18 (2013), 155-1188.
doi: 10.3934/dcdsb.2013.18.1155. |
| [4] |
G. Agrawal, Nonlinear Fiber Optics, Academic Press, 4th edition, 2007. |
| [5] |
S. Albeverio, Z. Brzezniak and L. Dabrowski,
Fundamental solution of the heat and Schródinger equations with point interaction, J. Funct. Anal., 130 (1995), 220-254.
doi: 10.1006/jfan.1995.1068. |
| [6] |
S. Albeverio, F. Gesztesy, R. Krohn and H. Holden, Solvable Models in Quantum Mechanics, AMS Chelsea publishing, 2004., |
| [7] |
S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators, London Mathematical Society Lecture Note Series 271, Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511758904.
|
| [8] |
J. Angulo,
Instability of cnoidal-peak for the NLS-$\delta$ equation, Math. Nachr., 285 (2012), 1572-1602.
doi: 10.1002/mana.201100209. |
| [9] |
J. Angulo and A. Hernandez,
Stability of standing waves for logarithmic Schródinger equation with attractive delta potential, IUMJ, 67 (2018), 471-494.
doi: 10.1512/iumj.2018.67.7273. |
| [10] |
J. Angulo and N. Goloshchapova,
On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph, Discr. Cont. Dyn. Syst.- A, 38 (2018), 5039-5066.
doi: 10.3934/dcds.2018221. |
| [11] |
J. Angulo and N. Goloshchapova, On the standing waves of the NLS-log equation with point interaction on a star graph, preprint, arXiv: 1803.07194. |
| [12] |
J. Angulo and N. Goloshchapova,
Extension theory approach in the stability of the standing waves for NLS equation with point interactions on a star graph, Advances in Differential Equations, 23 (2018), 793-846.
|
| [13] |
J. Angulo and N. Goloshchapova, Stability of standing waves for NLS-log equation with $\delta$-interaction, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 27.
doi: 10.1007/s00030-017-0451-0. |
| [14] |
J. Angulo, O. Lopes and A. Neves,
Instability of travelling waves for weakly coupled KdV systems, Nonlinear Anal., 69 (2008), 1870-1887.
doi: 10.1016/j.na.2007.07.039. |
| [15] |
J. Angulo and F. Natali,
On the instability of periodic waves for dispersive equations, Differential Integral Equations, 29 (2016), 837-874.
|
| [16] |
J. Angulo and G. Ponce,
The nonlinear Schródinger equation with a periodic $\delta$-interaction, Bull. Braz. Math. Soc., New Series, 44 (2013), 497-551.
doi: 10.1007/s00574-013-0024-8. |
| [17] |
G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala and F. Sanchez, Experimental and theoretical study of higher-order nonlinearities in chalcogenide glasses, Opt. Commun, 219 (2003), 427-433. |
| [18] |
V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattices, Modern Physics Letter - B, 18 (2004), 627-651. |
| [19] |
F. A. Berezin and M. A. Shubin, The Schródinger Equation, Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 1991.
doi: 10.1007/978-94-011-3154-4. |
| [20] |
V. Caudrelier, M. Mintchev and E. Ragoucy, Solving the quantum non-linear Schródinger equation with $\delta$-type impurity, J. Math. Phys., 46 (2005), 042703-1-24.
doi: 10.1063/1.1842353. |
| [21] |
T. Cazenave, Semilinear Schródinger Equations, American Mathematical Society, AMS. Lecture Notes, v. 10, 2003.
doi: 10.1090/cln/010. |
| [22] |
S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan,
Instability of bound states of a nonlinear Schródinger equation with a Dirac Potential, Phys. D, 237 (2008), 1103-1128.
doi: 10.1016/j.physd.2007.12.004. |
| [23] |
K. Datchev and J. Holmer,
Fast soliton scattering by attractive delta impurities, Comm. PDE., 34 (2009), 1074-1173.
doi: 10.1080/03605300903076831. |
| [24] |
K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Bose-Einstein condensation in gas of sodium atoms, Phys. Rev. Lett., 74 (1995), 3969-3973. |
| [25] |
E. L. Falcão-Filho, C. B. de Araújo, G. Boudebs, H. Leblond and V. Skarka, Robust two-dimensional spatial solitons in liquid carbon disulfide, Phys. Rev. Lett., 110 (2013), 013901. |
| [26] |
E. L. Falcão-Filho, C. B. de Araújo and J. J. Rodrigues Jr., High-order nonlinearities of aqueous colloids containing silver nanoparticles, J. Opt. Soc. Am. - B, 24 (2007), 2948-2956. |
| [27] |
R. Fukuizumi and L. Jeanjean,
Stability of standing waves for a nonlinear Schródinger equation with a repulsive Dirac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136.
doi: 10.3934/dcds.2008.21.121. |
| [28] |
R. Fukuizumi, M. Ohta and T. Ozawa,
Nonlinear Schródinger equation with a point defect, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 837-845.
doi: 10.1016/j.anihpc.2007.03.004. |
| [29] |
B. Gaveau and L. S. Schulman,
Explicit time-dependent Schródinger propagators, J. Physics A: Math. Gen., 19 (1986), 1833-1846.
|
| [30] |
F. Genoud, F. B. Malomed and R. Weishäupl,
Stable NLS solitons in a cubic-quintic medium with a delta-function potential, Nonlinear Anal., 133 (2016), 28-50.
doi: 10.1016/j.na.2015.11.016. |
| [31] |
B. V. Gisin, R. Driben and B. A. Malomed, Bistable guided solitons in the cubic-quintic medium, J. Optics B: Quantum and Semiclassical Optics, 6 (2004), S259–S264. |
| [32] |
R. H. Goodman, J. Holmes and M. Weinstein,
Strong NLS soliton-defect interactions, Phys. D, 192 (2004), 215-248.
doi: 10.1016/j.physd.2004.01.021. |
| [33] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry, Ⅰ, J. Funct. Anal., 74 (1987), 160-197.
doi: 10.1016/0022-1236(87)90044-9. |
| [34] |
M. Grillakis, J. Shatah and W. Strauss,
Stability theory of solitary waves in the presence of symmetry, Ⅱ, J. Funct. Anal., 94 (1990), 308-348.
doi: 10.1016/0022-1236(90)90016-E. |
| [35] |
D. Henry, J. Perez and W. Wreszinski,
Stability theory for solitary-wave solutions of scalar field equation, Comm. Math. Phys., 85 (1982), 351-361.
|
| [36] |
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. |
| [37] |
M. Kaminaga and M. Ohta,
Stability of standing waves for nonlinear Schrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J.,, 26 (2009), 39-48.
|
| [38] |
T. Kato, Perturbation Theory for Linear Operators, 2$^{nd}$ edition, Springer-Verlang, New York, 1976., |
| [39] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2$^{nd}$ edition, Universitext, Springer, New York, 2015.
doi: 10.1007/978-1-4939-2181-2. |
| [40] |
S. Le Coz, Y. Martel and P. Raphael, Minimal mass blow up solutions for a double power nonlinear Schródinger equation, preprint, arXiv: 1406.6002.
doi: 10.4171/RMI/899. |
| [41] |
M. Maeda,
Stability and instability of standing waves for 1-dimensional nonlinear Schródinger equation with multiple-power nonlinearity, Kodai Math. J., 31 (2008), 263-271.
doi: 10.2996/kmj/1214442798. |
| [42] |
C. R. Menyuk, Soliton robustness in optical fibers, J. Opt. Soc. Am. - B, 10 (1993), 1585-1591 |
| [43] |
J. Moloney and A. Newell, Nonlinear Optics, CRC, Taylor & Francis Group, Boca Raton. FL. USA, 2018., |
| [44] |
M. A. Naimark, Linear Differential Operators, F. Ungar Pub. Co., New York, 1967., |
| [45] |
M. Ohta,
Stability and Instability of standing waves for one dimensional nonlinear Schródinger equations with double power nonlinearity, Kodai Math. J., 18 (1995), 68-74.
doi: 10.2996/kmj/1138043354. |
| [46] |
M. Ohta,
Instability of bound states for abstract nonlinear Schródinger equations, J. Funct. Anal., 261 (2011), 90-110.
doi: 10.1016/j.jfa.2011.03.010. |
| [47] |
P. Papagiannis, Y. Kominis and K. Hizanidis, Power-and momentum-dependent soliton dynamics in lattices with longitudinal modulation, Phys. Rev. A, 84 (2011), 013820 |
| [48] |
S. Reed and B. Simon, Methods of Modern Mathematical Physics: Analysis of Operators, Academic Press, vol. Ⅳ, 1978. |
| [49] |
H. Sakaguchi and M. Tamura,
Scattering and trapping of nonlinear Schródinger solitons in external potentials, J. Phys. Soc. Japan, 73 (2004), 503-506.
doi: 10.1143/JPSJ.73.503. |
| [50] |
B. T. Seaman, L. D. Car and M. J. Holland, Effect of a potential step or impurity on the Bose-Einstein condensate mean field, Phys. Rev. A, 71 (2005), 033609. |
Figure 2.
Function $ \omega\to-||\phi_{\omega, Z}||^2 $ for specific values of $ Z $ . Left picture, $ \omega\in(-6, -1) $ , $ Z = -0.86 $ . Right picture, $ \omega\in(-6, -1) $ , $ Z = -0.9 $
Figure 3.
Function $ (\omega, Z)\to-\partial_\omega||\phi_{\omega, Z}||^2 $ , for $ \omega\in(-50, -2) $ . Left picture for $ Z\in(-0.9, -0.8) $ . Right picture for $ Z\in(-0.8, -0.7) $
Figure 4.
Function $ \omega\to-\partial_\omega||\phi_{\omega, Z}||^2 $ for specific values of $ Z $ . Left picture, $ Z = -0.86602 $ and $ \omega\in(-10, -0.8) $ . Right picture, $ Z = -0.86603 $ and $ \omega\in(-5, -1) $
Figure 5.
The function $ \omega\to-||\phi_{\omega, Z}||^2 $ , for $ \lambda_1 = 2 $ , $ \lambda_2 = -1 $ and $ \omega\in (-\frac34, -\frac14) $ . Left picture, $ Z = -1 $ . Right picture, $ Z = 1 $
Figure 6.
The function $ \omega\to-||\phi_{\omega, Z}||^2 $ , for $ \lambda_1 = 4 $ , $ \lambda_2 = -2 $ and $ \omega\in (-\frac32, -\frac14) $ . Left picture, $ Z = -1 $ . Right picture, $ Z = 1 $
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