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 & Continuous Dynamical Systems - S, 2013, 6 (4) : 891-908. doi: 10.3934/dcdss.2013.6.891
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show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

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