×

zbMATH — the first resource for mathematics

Weakly nonlinear time-adiabatic theory. (English) Zbl 1337.81056
Author’s abstract: We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schrödinger equations in which either the nonlinear coupling constant or, equivalently, the solution is asymptotically small. To this end, a notion of criticality is introduced at which the linear bound states stay adiabatically stable, but nonlinear effects start to show up at leading order in the form of a slowly varying nonlinear phase modulation. In addition, we prove that in the same regime a class of nonlinear bound states also stays adiabatically stable, at least in terms of spectral projections.
Reviewer: Ma Wen-Xiu (Tampa)

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q55 NLS equations (nonlinear Schrödinger equations)
70H11 Adiabatic invariants for problems in Hamiltonian and Lagrangian mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Avron, J.E.; Elgart, A., Adiabatic theorem without a gap condition, Commun. Math. Phys., 203, 445-463, (1999) · Zbl 0936.47047
[2] Berry, M.V., Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A, 392, 45-57, (1984) · Zbl 1113.81306
[3] Berry, M.V., Histories of adiabatic quantum transitions, Proc. R. Soc. Lond. Ser. A, 429, 61-72, (1990)
[4] Bohm A., Mostafazadeh A., Koizumi H., Niu Q., Zwanziger J.: The Geometric phase in quantum systems: foundations, mathematical concepts, and applications in molecular and condensed matter physics. Springer, Berlin (2003) · Zbl 1039.81003
[5] Born, M.; Fock, V., Beweis des adiabatensatzes, Z. Phys. A Hadrons Nuclei, 51, 165-180, (1928) · JFM 54.0994.03
[6] Carles, R., Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9, 937-964, (2011) · Zbl 1285.35105
[7] Carles, R.; Fermanian Kammerer, C., A nonlinear adiabatic theorem for coherent states, Nonlinearity, 24, 2143-2164, (2011) · Zbl 1231.81040
[8] Carles, R.; Markowich, P.A.; Sparber, C., Semiclassical asymptotics for weakly nonlinear Bloch waves, J. Stat. Phys., 117, 343-375, (2004) · Zbl 1104.81049
[9] Erdös, L.; Schlein, B.; Yau, H.-T., Rigorous derivation of the Gross-Pitaevskii equation, Phys. Rev. Lett., 98, 040404, (2007)
[10] Gang, Z., Grech, P.D.: An adiabatic theorem for the Gross-Pitaevskii equation. Preprint (2015) · Zbl 1373.81434
[11] Grech, P.D.: Adiabatic dynamics in closed and open quantum systems. PhD thesis (ETH Zürich) (2011). doi:10.3929/ethz-a-6665029
[12] Gustafson, S.; Phan, T.V., Stable directions for degenerate excited states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 43, 1716-1758, (2011) · Zbl 1230.35129
[13] Hislop, P.D., Exponential decay of two-body eigenfunctions: A review, Electron. J. Differ. Eqs. Conf., 04, 265-288, (2000) · Zbl 0999.81017
[14] Joye, A., General adiabatic evolution with a gap condition, Commun. Math. Phys., 275, 139-162, (2007) · Zbl 1176.47032
[15] Kato, T., On the adiabatic theorem of quantum mechanics, J. Phys. Soc. Jpn., 5, 435-439, (1950)
[16] Nenciu, G., On the adiabatic theorem of quantum mechanics, J. Phys. A Math. Gen., 13, l15-l18, (1980) · Zbl 0435.47025
[17] Nenciu, G., Linear adiabatic theory. exponential estimates, Commun. Math. Phys., 152, 479-496, (1993) · Zbl 0768.34038
[18] Nirenberg L.: Topics in Nonlinear Functional Analysis. Courant Institute, New York (1974) · Zbl 0286.47037
[19] Pickl, P., A simple derivation of Mean field limits for quantum systems, Lett. Math. Phys., 97, 151-164, (2011) · Zbl 1242.81150
[20] Pillet, C.-A.; Wayne, C.E., Invariant manifolds for a class of dispersive, Hamiltonian equations, J. Diff. Equ., 141, 310-326, (1997) · Zbl 0890.35016
[21] Pitaevskii, L., Stringari, S.: Bose-Einstein Condensation. Internat. Series of Monographs on Physics vol 116, Clarendon Press, Oxford (2003) · Zbl 1110.82002
[22] Rauch, J.: Hyperbolic Partial Differential Equations and Geometric Optics. Graduate Studies in Mathematics vol. 133, American Math. Soc. (2012) · Zbl 1252.35004
[23] Rose, H.A.; Weinstein, M.I., On the bound states of the nonlinear Schrödinger equation with a linear potential, Physica D, 30, 207-218, (1998) · Zbl 0694.35202
[24] Salem, W.K.A., Solitary wave dynamics in time-dependent potentials, J. Math. Phys., 49, 032101, (2008) · Zbl 1153.81428
[25] Schmid, J.: Adiabatic theorems with and without spectral gap condition for non-semisimple spectral values. In: Mathematical Results in Quantum Mechanics (Proceedings of the QMath12 Conference), pp. 355-362, World Scientific Berlin (2013) · Zbl 1339.81044
[26] Simon, B., Holonomy, the quantum adiabatic theorem, and berry’s phase, Phys. Rev. Lett., 51, 2167-2170, (1983)
[27] Sulem C., Sulem P.-L.: The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse. Springer, New York (1999) · Zbl 0928.35157
[28] Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering for nonintegrable equations, Commun. Math. Phys., 133, 119-146, (1990) · Zbl 0721.35082
[29] Soffer, A.; Weinstein, M.I., Selection of ground states for nonlinear Schrödinger equations, Rev. Math. Phys., 16, 977-995, (2004) · Zbl 1111.81313
[30] Tao, T.: Nonlinear Dispersive Equations. CBMS Regional Conference Series in Mathematics 106, Amer. Math. Soc., Providence (2006) · Zbl 1106.35001
[31] Teschl, G.: Mathematical Methods in Quantum Mechanics, With Applications to Schrödinger Operators. Graduate Studies in Mathematics 99, Amer. Math. Soc., Providence (2009) · Zbl 1231.81040
[32] Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Lecture Notes in Mathematics vol. 1821, Springer, Berlin, Heidelberg (2003) · Zbl 1053.81003
[33] Tsai, T.-P., Asymptotic dynamics of nonlinear Schrödinger equations with many bound states, J. Diff. Equ., 192, 225-282, (2003) · Zbl 1038.35128
[34] Tsai, T.-P.; Yau, H.-T., Asymptotic dynamics of nonlinear Schrödinger equations: resonance dominated and dispersion dominated solutions, Commun. Pure Appl. Math., 55, 0153-0216, (2002) · Zbl 1031.35137
[35] Weinstein, M.I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, 472-491, (1985) · Zbl 0583.35028
[36] Yukalov, V.I., Adiabatic theorems for linear and nonlinear Hamiltonians, Phys. Rev. A, 79, 052117, (2009)
[37] Xia, D.; Chang, M.-C.; Niu, Q., Berry phase effects on electronic properties, Rev. Mod. Phys., 82, 1959-2007, (2010) · Zbl 1243.82059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.