An introduction to the two-dimensional Schrodinger equation with nonlinear¨ point interactions
https://doi.org/10.17586/2220-8054-2018-9-2-187-195
Abstract
We present an introduction to the nonlinear Schrodinger equation (NLSE) with concentrated nonlinearities in¨ R2. Precisely, taking a cue from the linear problem, we sketch the main challenges and the typical difficulties that arise in the two-dimensional case, and mention some recent results obtained by the authors on local and global well-posedness.
About the Authors
R. Carlone
Universita “Federico II” di Napoli, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
Italy
Italy
via Cinthia, I-80126, Napoli
M. Correggi
“Sapienza” Universita di Roma, Dipartimento di Matematica P.le Aldo Moro
Italy
Italy
P.le Aldo Moro, 5, 00185, Roma
L. Tentarelli
“Sapienza” Universita di Roma, Dipartimento di Matematica
Italy
Italy
P.le Aldo Moro, 5, 00185, Roma
References
1. Cacciapuoti C., Carlone R., Figari R. A solvable model of a tracking chamber. Rep. Math. Phys., 2007, 59(3), P. 337–349.
2. Carlone R., Figari R., Negulescu C. A model of a quantum particle in a quantum environment: a numerical study. Commun. Comput. Phys., 2015, 18(1), P. 247–262.
3. Carlone R., Figari R., Negulescu C. The quantum beating and its numerical simulation. J. Math. Anal. Appl., 2017, 450(2), P. 1294–1316.
4. Figari R., Teta A. From quantum to classical world: emergence of trajectories in a quantum system. Math. Mech. Complex Syst., 2016, 4(3–4), P. 235–254.
5. Cacciapuoti C., Figari R., Posilicano A. Effective equation for a system of mechanical oscillators in an acoustic field. Asymptot. Anal., 2015, 91(3–4), P. 253–264.
6. Cacciapuoti C., Fermi D., Posilicano A. Relative-zeta and Casimir energy for a semitransparent hyperplane selecting transverse modes. Advances in quantum mechanics, P. 71–97, Springer INdAM Ser., 18, Springer, Cham, 2017.
7. Cacciapuoti C., Carlone R., Figari R. Perturbations of eigenvalues embedded at threshold: two-dimensional solvable models. J. Math. Phys., 2011, 52(8), 083515, 12 pp.
8. Albeverio S., Gesztesy F., Høegh-Krohn R., Holden H. Solvable Models in Quantum Mechanics. Texts and Monographs in Physics, Springer-Verlag, New York, 1988.
9. Cacciapuoti C., Carlone R., Noja D., Posilicano A. The one-dimensional Dirac equation with concentrated nonlinearity. SIAM J. Math. Anal., 2017, 49(3), P. 2246–2268.
10. Carlone R., Malamud M., Posilicano A. On the spectral theory of Gesztesy-Seba realizations of 1-D Dirac operators with point interactionsˇ on a discrete set. J. Differential Equations, 2013, 254(9), P. 3835–3902.
11. Adami R., Cacciapuoti C., Finco D., Noja D. Fast solitons on star graphs. Rev. Math. Phys., 2011, 23(4), P. 409–451.
12. Adami R., Cacciapuoti C., Finco D., Noja D. On the structure of critical energy levels for the cubic focusing NLS on star graphs. J. Phys. A, 2012, 45(19), 192001, 7 pp.
13. Adami R., Cacciapuoti C., Finco D., Noja D. Constrained energy minimization and orbital stability for the NLS equation on a star graph. Ann. Inst. H. Poincare Anal. Non Lin´ eaire´ , 2014, 31(6), P. 1289–1310.
14. Adami R., Cacciapuoti C., Finco D., Noja D. Variational properties and orbital stability of standing waves for NLS equation on a star graph. J. Differential Equations, 2014, 257(10), P. 3738–3777.
15. Adami R., Serra E., Tilli P. Lack of ground state for NLSE on bridge-type graphs. Mathematical technology of networks, 1–11. Springer Proc. Math. Stat., 128. Springer, Cham, 2015.
16. Adami R., Serra E., Tilli P. NLS ground states on graphs. Calc. Var. Partial Differential Equations, 2015, 54(1), P. 743–761.
17. Adami R., Serra E., Tilli P. Threshold phenomena and existence results for NLS ground states on graphs. J. Funct. Anal., 2016, 271(1), P. 201–223.
18. Adami R., Serra E., Tilli P. Negative energy ground states for the L2–critical NLSE on metric graphs. Comm. Math. Phys., 2017, 352(1), P. 387–406.
19. Berkolaiko G., Kuchment P. Introduction to quantum graphs, Mathematical Surveys and Monographs 186, AMS, Providence, RI, 2013.
20. Cacciapuoti C., Finco D., Noja D. Topology–induced bifurcations for the nonlinear Schrodinger equation on the tadpole graph.¨ Phys. Rev. E (3), 2015, 91(1), article number 013206, 8 pp.
21. Gnutzmann S., Smilansky U., Derevyanko S. Stationary scattering from a nonlinear network. Phys. Rev. A, 2011, 83(3), article number 033831, 6 pp.
22. Noja D. Nonlinear Schrodinger equation on graphs: recent results and open problems.¨ Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2014, 372(2007), 20130002, 20 pp.
23. Noja D., Pelinovsky D., Shaikhova G. Bifurcations and stability of standing waves in the nonlinear Schrodinger equation on the tadpole¨ graph. Nonlinearity, 2015, 28(7), P. 2343–2378.
24. Serra E., Tentarelli L. Bound states of the NLS equation on metric graphs with localized nonlinearities. J. Differential Equations, 2016, 260(7), P. 5627–5644.
25. Serra E., Tentarelli L. On the lack of bound states for certain NLS equations on metric graphs. Nonlinear Anal., 2016, 145, P. 68–82.
26. Tentarelli L. NLS ground states on metric graphs with localized nonlinearities. J. Math. Anal. Appl., 2016, 433(1), P. 291–304.
27. Carlone R., Exner P. Dynamics of an electron confined to a “hybrid plane” and interacting with a magnetic field. Rep. Math. Phys., 2011, 67(2), P. 211–227.
28. Carlone R., Posilicano A. A quantum hybrid with a thin antenna at the vertex of a wedge. Phys. Lett. A, 2017, 381(12), P. 1076–1080.
29. Exner P., Seba P. Quantum motion on a half–line connected to a plane.ˇ J. Math. Phys., 1987, 28(2), P. 386–391.
30. Exner P., Seba P. Resonance statistics in a microwave cavity with a thin antenna.ˇ Phys. Lett. A, 1997, 228(3), P. 146–150.
31. Exner P., Seba P. A “hybrid plane” with spin-orbit interaction.ˇ Russ. J. Math. Phys., 2007, 14(4), P. 430–434.
32. Adami R., Teta A. A Class of Nonlinear Schrodinger Equations with Concentrated Nonlinearity.¨ J. Funct. Anal., 2001, 180(1), P. 148–175.
33. Adami R., Dell’Antonio G., Figari R., Teta A. The Cauchy problem for the Schrodinger equation in dimension three with concentrated¨ nonlinearity. Ann. Inst. H. Poincare Anal. Non Lin´ eaire´ , 2003, 20(3), P. 477–500.
34. Adami R., Dell’Antonio G., Figari R., Teta A. Blow-up solutions for the Schrodinger equation in dimension three with a concentrated¨ nonlinearity. Ann. Inst. H. Poincare Anal. Non Lin´ eaire´ , 2004, 21(1), P. 121–137.
35. Cacciapuoti C., Finco D., Noja D., Teta A. The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit. Lett. Math. Phys., 2014, 104(12), P. 1557–1570.
36. Cacciapuoti C., Finco D., Noja D., Teta A. The point-like limit for a NLS equation with concentrated nonlinearity in dimension three. J. Funct. Anal., 2017, 273(5), P. 1762–1809.
37. M. Correggi, G. Dell’Antonio, Decay of a Bound State under a Time-Periodic Perturbation: A Toy Case. J. Phys. A: Math. Gen., 2005, 38, P. 4769–4781.
38. Correggi M., Dell’Antonio G., Figari R., Mantile A. Ionization for Three Dimensional Time-Dependent Point Interactions. Comm. Math. Phys., 2005, 257(1), P. 169–192.
39. Costin O., Costin R.D., Lebowitz J.L., Rokhlenko A. Evolution of a model quantum system under time periodic forcing: conditions for complete ionization. Comm. Math. Phys., 2001, 221(1), P. 1–26.
40. Sayapova M.R., Yafaev D.R. The evolution operator for time-dependent potentials of zero radius. Trudy Mat. Inst. Steklov, 1983, 159, P. 167–174.
41. Carlone R., Correggi M., Figari R. Two-dimensional time-dependent point interactions. Functional Analysis and Operator Theory for Quantum Physics, 189–211, Eur. Math. Soc., Zu¨rich, 2017.
42. Carlone R., Correggi M., Tentarelli L. Well-posedness of the Two-dimensional Nonlinear Schrodinger Equation with Concentrated Non-¨ linearity, preprint arXiv:1702.03651 [math-ph], (2017).
43. Carlone R., Fiorenza A., Tentarelli L. The action of Volterra integral operators with highly singular kernels on Holder continuous, Lebesgue¨ and Sobolev functions. J. Funct. Anal., 2017, 273(3), P. 1258-1294.
44. Albeverio S., Brzezniak Z., Dabrowski L. Time-dependent propagator with point interaction. J. Phys. A, 1994, 27(14), P. 4933–4943.
45. Samko S.G., Kilbas A.A., Marichev O.I. Fractional integrals and derivatives. Theory and applications. Gordon and Breach Science Publishers, Yverdon, 1993.
46. Erdelyi A., Magnus W., Oberhettinger F., Tricomi F.G.´ Higher transcendental functions. Vol. III, Robert E. Krieger Publishing Co., Melbourne, Fla., 1981.
47. Adami R. A Class of Schrodinger Equations with Concentrated Nonlinearity¨ . Ph.D. Thesis, Universita degli Studi di Roma “La Sapienza”,` 2000.
48. Gradshteyn I.S., Ryzhik I.M. Tables of Integrals, Series and Products. Elsevier/Academic Press, Amsterdam, 2007.
49. Abramovitz M., Stegun I.A. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55, Washington D.C., 1964.
50. Gorenflo R., Vessella S. Abel integral equations, Lecture Notes in Mathematics, vol. 1461, Springer-Verlag, Berlin, 1991, Analysis and applications.
51. Miller R.K. Nonlinear Volterra Integral Equations. Mathematics Lecture Note Series, W.A. Benjamin Inc., Menlo Park, Calif., 1971.
52. Miller R.K., Sell G.R. Existence, uniqueness and continuity of solutions of integral equations. Ann. Mat. Pura Appl. (4), 1968, 80, P. 135–152.
53. Kufner A., Persson L.E. Weighted inequalities of Hardy type. World Scientific Publishing Co., River Edge, NJ, 2003.
Review
For citations:
Carlone R., Correggi M., Tentarelli L. An introduction to the two-dimensional Schrodinger equation with nonlinear¨ point interactions. Nanosystems: Physics, Chemistry, Mathematics. 2018;9(2):187-195. https://doi.org/10.17586/2220-8054-2018-9-2-187-195
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