Solvable models of quantum beating
https://doi.org/10.17586/2220-8054-2018-9-2-162-170
Abstract
We review some results about the suppression of quantum beating in a one dimensional nonlinear double well potential. We implement a single particle double well potential model, making use of nonlinear point interactions. We show that there is complete suppression of the typical beating phenomenon characterizing the linear quantum case.
Keywords
Supporting agencies: R.C. and L.T. acknowledge the support of the FIR 2013 project “Condensed Matter in Mathematical Physics”, Ministry of University and Research of Italian Republic (code RBFR13WAET). C.N. would like to acknowledge support from the CNRS-PICS project “MANUS” (Modelling and Numerics of Spintronics and Graphenes, 2016 – 2018).
About the Authors
R. Carlone
Universita “Federico II” di Napoli, Dipartimento di Matematica e Applicazioni “R. Caccioppoli”
Italy
Italy
MSA, via Cinthia, I-80126, Napoli
R. Figari
Universita “Federico II” di Napoli, Dipartimento di Fisica e INFN Sezione di Napoli
Italy
Italy
MSA, via Cinthia, I-80126, Napoli
C. Negulescu
Universite de Toulouse & CNRS, UPS, Institut de Math´ematiques de Toulouse
France
France
UMR 5219´ F-31062 Toulouse
L. Tentarelli
Sapienza Universita di Roma, Dipartimento di Matematica
Italy
Italy
Piazzale Aldo Moro, 5, 00185, Roma
References
1. Davies E.B. Symmetry breaking for a nonlinear Schrodinger equation.¨ Comm. Math. Phys., 1979, 64(3), P. 191–210.
2. Davies E.B. Nonlinear Schrodinger operators and molecular structure.¨ J. Phys. A, 1995, 28(14), P. 4025–4041.
3. Herbauts I.M., Dunstan D.J. Quantum molecular dynamics study of the pressure dependence of the ammonia inversion transition. Physical Review A, 2007, 76(6), article number 062506.
4. Jona-Lasinio G., Presilla C. and Toninelli C. Environment induced localization and superselection rules in a gas of pyramidal molecules. Mathematical physics in mathematics and physics (Siena, 2000), P. 207–217, Fields Inst. Commun. 30, Amer. Math. Soc., Providence, RI, 2001.
5. Sacchetti A. Nonlinear time-dependent one-dimensional Schrodinger equation with double-well potential.¨ SIAM J. Math. Anal., 2004, 35(5), P. 1160–1176.
6. Grecchi V., Martinez A. and Sacchetti A. Splitting instability: the unstable double wells. J. Phys. A, 1996, 29(15), P. 4561–4587.
7. Grecchi V., Martinez A. and Sacchetti A. Destruction of the beating effect for a nonlinear Schrdinger equation.¨ Comm. Math. Phys., 2002, 227(1), P. 191–209.
8. Sacchetti A. Nonlinear time-dependent Schrodinger equations: the Gross-Pitaevskii equation with double-well potential.¨ J. Evol. Equ., 2004, 4(3), P. 345–369.
9. Sacchetti A. Nonlinear time-dependent one-dimensional Schrodinger equation with double-well potential.¨ SIAM J. Math. Anal., 2004, 35(5), P. 1160–1176.
10. Carlone R., Figari R. and Negulescu C. The quantum beating and its numerical simulation. J. Math. Anal. Appl., 2017, 450(2), P. 1294– 1316.
11. Albeverio S., Gesztesy F., Høegh-Krohn R. and Holden H. Solvable models in quantum mechanics. AMS Chelsea Publishing, Providence, RI, 2005.
12. Adami R. and Teta A. A class of nonlinear Schrodinger equations with concentrated nonlinearity.¨ J. Funct. Anal., 2001, 180(1), P. 148–175.
13. Carlone R.,Correggi M. and Figari R. Two-dimensional time-dependent point interactions. Functional analysis and operator theory for quantum physics. P. 189–211, EMS Ser. Congr. Rep., Eur. Math. Soc., Zu¨rich, 2017.
14. Carlone R., Correggi M. and Tentarelli L. Well-posedness of the two-dimensional nonlinear Schrodinger equation with concentrated¨ nonlinearity. preprint arXiv:1702.03651 [math-ph], 2017.
15. Carlone R., Fiorenza A. and 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.
16. Adami R., Dell’Antonio G., Figari R. and Teta A. The Cauchy problem for the Schrodinger equation in dimension three with concentrated¨ nonlinearity. Ann. Inst. H. Poincar Anal. Non Lineaire´ , 2003, 20(3), P. 477–500.
17. Adami R., Dell’Antonio G., Figari R. and Teta A. Blow-up solutions for the Schrodinger equation in dimension three with a concentrated¨ nonlinearity. Ann. Inst. H. Poincar Anal. Non Lineaire´ , 2004, 21(1), P. 121–137.
18. Adami R. and Teta A. A simple model of concentrated nonlinearity. Mathematical results in quantum mechanics (Prague, 1998), P. 183– 189, Oper. Theory Adv. Appl. 108, Birkhuser, Basel, 1999.
Review
For citations:
Carlone R., Figari R., Negulescu C., Tentarelli L. Solvable models of quantum beating. Nanosystems: Physics, Chemistry, Mathematics. 2018;9(2):162–170. https://doi.org/10.17586/2220-8054-2018-9-2-162-170
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