Geodesic
Ground States for NLS on Graphs: a Subtle Interplay of Metric and Topology
R. Adami1
1 Dipartimento di Scienze Matematiche “G.L. Lagrange”, Politecnico di Torino C.so Duca degli Abruzzi 24, 10129 Torino, ITALY
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 2, pp. 20-35.

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We review some recent results on the minimization of the energy associated to the nonlinear Schrödinger Equation on non-compact graphs. Starting from seminal results given by the author together with C. Cacciapuoti, D. Finco, and D. Noja for the star graphs, we illustrate the achiements attained for general graphs and the related methods, developed in collaboration with E. Serra and P. Tilli. We emphasize ideas and examples rather than computations or proofs.
DOI : 10.1051/mmnp/201611202

R. Adami 1

1 Dipartimento di Scienze Matematiche “G.L. Lagrange”, Politecnico di Torino C.so Duca degli Abruzzi 24, 10129 Torino, ITALY 
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R. Adami. Ground States for NLS on Graphs: a Subtle Interplay of Metric and Topology. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 2, pp. 20-35. doi : 10.1051/mmnp/201611202. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611202/

[1] R. Adami, C. Cacciapuoti, D. Finco, D. Noja Rev. Math. Phys 2011 409 451

[2] R. Adami, C. Cacciapuoti, D. Finco, D. Noja J. Phys. A 2012 192001

[3] R. Adami, C. Cacciapuoti, D. Finco, D. Noja Europhys. Lett. 2012 10003

[4] R. Adami, C. Cacciapuoti, D. Finco, D. Noja Constrained energy minimization and orbital stability for the NLS equation on a star graph. Ann. Inst. Poincaré, An Non Lin. 2014 1289 1310

[5] R. Adami, C. Cacciapuoti, D. Finco, D. Noja J. Diff. Eq. 2014 3738 3777

[6] R. Adami, D. Noja J. Phys. A: Math. Theor. 2009 495302

[7] R. Adami, E. Serra, P. Tilli Calc. Var. and PDEs 2015 743 761

[8] R. Adami, E. Serra, P. Tilli. Threshold phenomena and existence results for NLS ground states on graphs. Preprint arXiv:1505.03714 (2015), to appear on J. Func. An.

[9] F. Ali Mehmeti. Nonlinear waves in networks. Akademie Verlag, Berlin, 1994.

[10] B. Bellazzini, M. Mintchev J. Phys. A: Math. Gen. 2006 11101

[11] J. von Below. An existence result for semilinear parabolic network equations with dynamical node conditions. In Pitman Research Notes in Mathematical Series 266, Longman, Harlow Essex, 1992, 274–283.

[12] G. Berkolaiko, P. Kuchment. Introduction to quantum graphs. Mathematical Surveys and Monographs, 186. AMS, Providence, RI, 2013.

[13] J. Blank, P. Exner, M. Havlicek. Hilbert space operators in Quantum Physics, Springer, New York, 2008.

[14] E. Bulgakov, A. Sadreev Phys. Rev. B 2011 155304

[15] C. Cacciapuoti, D. Finco, D. Noja Phys. Rev. E 2015 013206

[16] V. Caudrelier Commun. Math. Phys. 2015 893 917

[17] L. Friedlander Extremal properties of eigenvalues for a metric graph Ann. Inst. Fourier 2005 199 211

[18] S. Gnutzman, U. Smilansky, S. Derevyanko Phys. Rev. A 2001 033831

[19] P.G. Kevrekidis, D.J. Frantzeskakis, G. Theocharis, I.G. Kevrekidis Phys. Lett. A 2003 513 522

[20] P. Kuchment Waves in Random Media 2004 S107 S128

[21] E.H. Lieb, M. Loss, Analysis: Second edition. American Mathematical Society, 2001.

[22] J.L. Marzuola, D.E. Pelinovsky. Ground state on the dumbbell graph. Preprint arXiv:1509.04721 (2015).

[23] Z. Sobirov, D. Matrasulov, K. Sabirov, S. Sawada, K. Nakamura Phys. Rev. E 2010 066602

[24] D. Noja Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci 2014

[25] D. Noja, D. Pelinovsky, G. Shaikhova Nonlinearity 2015 2343 2378

[26] L. Tentarelli J. Math. An. and Appl. 2016 291 304

[27] A. Tokuno, M. Oshikawa, E. Demler Phys. Rev. Lett. 2008 140402

[28] H. Uecker, D. Grieser, Z. Sobirov, D. Babajanov, D. Matrasulov Phys. Rev. E 2015 023209

[29] E.J.G. Vidal, R.P. Lima, M.L. Lyra Phys. Rev. E 2011 061137

[30] N. Viet Hung, M. Trippenbach, B. Malomed Phys. Rev. A 2011 05361

[31] I. Zapata, F. Sols Phys. Rev. Lett. 2009 180405

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