August  2013, 6(4): 891-908. doi: 10.3934/dcdss.2013.6.891

Intertwining semiclassical solutions to a Schrödinger-Newton system

1. 

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari

2. 

via Orabona 4, 70125 Bari

3. 

Instituto de Matemticas, Universidad Nacional Autnoma de Mxico

4. 

Circuito Exterior, C.U., 04510 Mxico D.F.

5. 

Dipartimento di Matematica ed Applicazioni

6. 

Universit di Milano-Bicocca

7. 

edificio U5, via R. Cozzi 53, I-20125 Milano

Received  October 2011 Revised  February 2012 Published  December 2012

We study the problem\[\begin{cases}\left( -\varepsilon\mathrm{i}\nabla+A(x)\right) ^{2}u+V(x)u=\varepsilon^{-2}\left( \frac{1}{|x|}\ast|u|^{2}\right) u,\\u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),    \varepsilon\nablau+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}),\end{cases}\]where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magneticpotential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exteriorelectric potential, and $\varepsilon$ is a small positive number. If $A=0$ and$\varepsilon=\hbar$ is Planck's constant this problem is equivalent to theSchr?dinger-Newton equations proposed by Penrose in [23] todescribe his view that quantum state reduction occurs due to somegravitational effect. We assume that $A$ and $V$ are compatible with theaction of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for anygiven homomorphism $\tau:G\rightarrow\mathbb{S}^{1}$ into the unit complexnumbers, we show that there is a combined effect of the symmetries and thepotential $V$ on the number of semiclassical solutions $u:\mathbb{R}^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=\tau(g)u(x)$ for all $g\in G$,$x\in\mathbb{R}^{3}$. We also study the concentration behavior of thesesolutions as $\varepsilon 0.$
Citation: Silvia Cingolani, Mnica Clapp, Simone Secchi. Intertwining semiclassical solutions to a Schrödinger-Newton system. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891
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Z. Angew. Math. Phys. 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8.

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Proc. Roy. Soc. Edinburgh, 140 A (2010), 973-1009. doi: 10.1017/S0308210509000584.

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Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229-248. doi: 10.1007/s00030-009-0051-8.

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Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.

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Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019.

[19]

Nonlinearity, 12 (1999), 201-216. doi: 10.1088/0951-7715/12/2/002.

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Commun. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.

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Comm. Math. Phys., 69 (1979), 19-30.

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Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.

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R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.

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Alfred A. Knopf Inc., New York, 2005.

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Nonlinear Analysis, 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.

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Physics Letters A, 280 (2001), 173-176. doi: 10.1016/S0375-9601(01)00059-7.

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PNLDE 24, Birkhäuser, Boston-Basel-Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

show all references

References:
[1]

J. Fixed Point Theory Appl., 10 (2011), 147-180. doi: 10.1007/s11784-011-0053-0.

[2]

Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.

[3]

Nonlinearity, 22 (2009), 2309-2331. doi: 10.1088/0951-7715/22/9/013.

[4]

Commun. Pure Appl. Anal., 9 (2010), 1263-1281. doi: 10.3934/cpaa.2010.9.1263.

[5]

Z. Angew. Math. Phys. 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8.

[6]

Proc. Roy. Soc. Edinburgh, 140 A (2010), 973-1009. doi: 10.1017/S0308210509000584.

[7]

J. Reine Angew. Math., 418 (1991), 1-29. doi: 10.1007/BF02566437.

[8]

Nonlinear Differ. Equ. Appl. (NoDea), 17 (2010), 229-248. doi: 10.1007/s00030-009-0051-8.

[9]

Walter de Gruyter, Berlin-New York, 1987. doi: 10.1515/9783110858372.312.

[10]

in "Séminaire: Équations aux Dérivées Partielles 2003-2004" Exp. No. XIX, 26 pp., Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2004).

[11]

Comm. Math. Phys., 225 (2002), 223-274. doi: 10.1007/s002200100579.

[12]

Nonlinearity, 16 (2003), 101-122. doi: 10.1088/0951-7715/16/1/307.

[13]

Stud. Appl. Math., 57 (1977), 93-105.

[14]

Graduate Studies in Math. 14, Amer. Math. Soc. 1997.

[15]

Ann. Inst. Henry Poincaré, Analyse Non Linéaire, 1 (1984), 109-145 and 223-283.

[16]

Nonlinear Anal. T.M.A., 4 (1980), 1063-1073. doi: 10.1016/0362-546X(80)90016-4.

[17]

Arch. Rational Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.

[18]

Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733-2742. doi: 10.1088/0264-9381/15/9/019.

[19]

Nonlinearity, 12 (1999), 201-216. doi: 10.1088/0951-7715/12/2/002.

[20]

Commun. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.

[21]

Comm. Math. Phys., 69 (1979), 19-30.

[22]

Gen. Rel. Grav., 28 (1996), 581-600. doi: 10.1007/BF02105068.

[23]

R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.

[24]

Alfred A. Knopf Inc., New York, 2005.

[25]

Nonlinear Analysis, 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.

[26]

Physics Letters A, 280 (2001), 173-176. doi: 10.1016/S0375-9601(01)00059-7.

[27]

J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.

[28]

PNLDE 24, Birkhäuser, Boston-Basel-Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

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