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Preface: Applications of mathematical analysis to problems in theoretical physics
Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $
1. | Département de Mathématiques, Université Libre de Bruxelles, CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgium |
2. | Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via Orabona 4, 70125 Bari, Italy |
3. | Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Roberto Cozzi 55, 20125 Milano, Italy |
$ \begin{gather} \begin{cases} -\varepsilon^{2} \Delta\psi+V(x)\psi = E(x) \psi \quad \text{in $ \mathbb{R}^2$}, \\ -\Delta E = |\psi|^{2} \quad \text{in $ \mathbb{R}^2$}, \end{cases} \end{gather}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S{P_\varepsilon }) $ |
$ \varepsilon $ |
$ V $ |
$ (u_\varepsilon, E_\varepsilon) $ |
$ (SP_\varepsilon) $ |
$ \ge \rightarrow 0 $ |
References:
[1] |
A. Ambrosetti, M. Badiale and S. Cingolani,
Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[2] |
A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^n$, Birkhäuser Verlag, 2006. |
[3] |
D. Bonheure, S. Cingolani and J. Van Schaftingen,
The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272 (2017), 5255-5281.
doi: 10.1016/j.jfa.2017.02.026. |
[4] |
P. Choquard and J. Stubbe,
The one-dimensional Schrödinger-Newton equations, Lett. Math. Phys., 81 (2007), 177-184.
doi: 10.1007/s11005-007-0174-y. |
[5] |
P. Choquard, J. Stubbe and M. Vuffray,
Stationary solutions of the Schrödinger-Newton model — an ODE approach, Differ. Integral Equ., 21 (2008), 665-679.
|
[6] |
S. Cingolani, M. Clapp and S. Secchi,
Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 891-908.
doi: 10.3934/dcdss.2013.6.891. |
[7] |
S. Cingolani and L. Jeanjean,
Stationary waves with prescribed $L^2$-norm for the Schrödinger-Poisson system, SIAM J. Math. Anal., 51 (2019), 3533-3568.
doi: 10.1137/19M1243907. |
[8] |
S. Cingolani, S. Secchi and M. Squassina,
Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.
doi: 10.1017/S0308210509000584. |
[9] |
S. Cingolani and K. Tanaka,
Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam., 35 (2019), 1885-1924.
doi: 10.4171/rmi/1105. |
[10] |
S. Cingolani and T. Weth,
On the planar Schrödinger-Poisson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 169-197.
doi: 10.1016/j.anihpc.2014.09.008. |
[11] |
M. Du and T. Weth,
Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492-3515.
doi: 10.1088/1361-6544/aa7eac. |
[12] |
R. Harrison, I. Moroz and K. P. Tod,
A numerical study of the Schrödinger–Newton equation, Nonlinearity, 16 (2003), 101-122.
doi: 10.1088/0951-7715/16/1/307. |
[13] |
E. Lenzmann,
Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.
doi: 10.2140/apde.2009.2.1. |
[14] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.
doi: 10.1002/sapm197757293. |
[15] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[16] |
S. Masaki,
Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space, Comm. Partial Differential Equations, 35 (2010), 2253-2278.
doi: 10.1080/03605301003717142. |
[17] |
V. Moroz and J. Van Schaftingen,
Semi-classical states for the Choquard equations, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.
doi: 10.1007/s00526-014-0709-x. |
[18] |
I. M. Moroz, R. Penrose and P. Tod,
Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.
doi: 10.1088/0264-9381/15/9/019. |
[19] |
R. Penrose,
On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600.
doi: 10.1007/BF02105068. |
[20] |
R. Penrose,
Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.
doi: 10.1098/rsta.1998.0256. |
[21] |
R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. |
[22] |
J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, preprint, arXiv: 0807.4059v1, 2008. |
[23] |
P. Tod and I. M. Moroz,
An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.
doi: 10.1088/0951-7715/12/2/002. |
[24] |
J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger–Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp.
doi: 10.1063/1.3060169. |
show all references
References:
[1] |
A. Ambrosetti, M. Badiale and S. Cingolani,
Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300.
doi: 10.1007/s002050050067. |
[2] |
A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^n$, Birkhäuser Verlag, 2006. |
[3] |
D. Bonheure, S. Cingolani and J. Van Schaftingen,
The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272 (2017), 5255-5281.
doi: 10.1016/j.jfa.2017.02.026. |
[4] |
P. Choquard and J. Stubbe,
The one-dimensional Schrödinger-Newton equations, Lett. Math. Phys., 81 (2007), 177-184.
doi: 10.1007/s11005-007-0174-y. |
[5] |
P. Choquard, J. Stubbe and M. Vuffray,
Stationary solutions of the Schrödinger-Newton model — an ODE approach, Differ. Integral Equ., 21 (2008), 665-679.
|
[6] |
S. Cingolani, M. Clapp and S. Secchi,
Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 891-908.
doi: 10.3934/dcdss.2013.6.891. |
[7] |
S. Cingolani and L. Jeanjean,
Stationary waves with prescribed $L^2$-norm for the Schrödinger-Poisson system, SIAM J. Math. Anal., 51 (2019), 3533-3568.
doi: 10.1137/19M1243907. |
[8] |
S. Cingolani, S. Secchi and M. Squassina,
Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009.
doi: 10.1017/S0308210509000584. |
[9] |
S. Cingolani and K. Tanaka,
Semi-classical states for the nonlinear Choquard equations: Existence, multiplicity and concentration at a potential well, Rev. Mat. Iberoam., 35 (2019), 1885-1924.
doi: 10.4171/rmi/1105. |
[10] |
S. Cingolani and T. Weth,
On the planar Schrödinger-Poisson systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 169-197.
doi: 10.1016/j.anihpc.2014.09.008. |
[11] |
M. Du and T. Weth,
Ground states and high energy solutions of the planar Schrödinger-Poisson system, Nonlinearity, 30 (2017), 3492-3515.
doi: 10.1088/1361-6544/aa7eac. |
[12] |
R. Harrison, I. Moroz and K. P. Tod,
A numerical study of the Schrödinger–Newton equation, Nonlinearity, 16 (2003), 101-122.
doi: 10.1088/0951-7715/16/1/307. |
[13] |
E. Lenzmann,
Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.
doi: 10.2140/apde.2009.2.1. |
[14] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.
doi: 10.1002/sapm197757293. |
[15] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[16] |
S. Masaki,
Local existence and WKB approximation of solutions to Schrödinger-Poisson system in the two-dimensional whole space, Comm. Partial Differential Equations, 35 (2010), 2253-2278.
doi: 10.1080/03605301003717142. |
[17] |
V. Moroz and J. Van Schaftingen,
Semi-classical states for the Choquard equations, Calc. Var. Partial Differential Equations, 52 (2015), 199-235.
doi: 10.1007/s00526-014-0709-x. |
[18] |
I. M. Moroz, R. Penrose and P. Tod,
Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.
doi: 10.1088/0264-9381/15/9/019. |
[19] |
R. Penrose,
On gravity's role in quantum state reduction, Gen. Rel. Grav., 28 (1996), 581-600.
doi: 10.1007/BF02105068. |
[20] |
R. Penrose,
Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.
doi: 10.1098/rsta.1998.0256. |
[21] |
R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. |
[22] |
J. Stubbe, Bound states of two-dimensional Schrödinger-Newton equations, preprint, arXiv: 0807.4059v1, 2008. |
[23] |
P. Tod and I. M. Moroz,
An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.
doi: 10.1088/0951-7715/12/2/002. |
[24] |
J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger–Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp.
doi: 10.1063/1.3060169. |
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