Riv. Mat. Univ. Parma, Vol. 10, No. 2, 2019

Claudia Negulescu [a]

Analytical and numerical aspects of the linear and non-linear Schrödinger equation

Pages: 351-446
Accepted in revised form: 3 April 2019
Mathematics Subject Classification (2010): 34L40, 34E05, 34E20, 35Q41, 42B20, 65L20, 65M06, 81S22.
Keywords: Schrödinger equation, Multi-scale problems, Asymptotic analysis, Asymptotic-preserving schemes.
[a]: Université Paul Sabatier, Institut de Mathématique de Toulouse, 118 route de Narbonne, 31062 Toulouse, France

Abstract: The subject-matter of these lecture notes, presented in 2016 at the Ravello Summer School on ''Mathematical Physics'' as well as in 2018 at the Università degli Studi di Napoli Federico II, as a PhD Course at the Doctoral School ''Mathematics and Applications'', is a presentation of several numerical techniques for the simulation of the linear respectively non-linear Schrödinger equation, arising in a variety of physical and biological contexts. In particular, the author is interested in the design and ensuing mathematical and numerical study of multi-scale schemes, as for ex. Asymptotic-Preserving schemes, for the resolution of the Schrödinger equation in several regimes, as for example the semi-classical regime. The lectures are based on articles, which were chosen to illustrate different techniques in the construction of such efficient numerical schemes for singular perturbation problems. However, the here developped techniques can also be applied for various other singular perturbation problems and these notes can serve as an introduction to more elaborate works on the subject

References
[1]
M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications, New-York, 2013.
[2]
R. Adami, G. Dell'Antonio, R. Figari and A. Teta, The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 477-500. MR1972871
[3]
R. Adami and L. Erdős, Rate of decoherence for an electron weakly coupled to a phonon gas, J. Stat. Phys. 132 (2008), 301-328. MR2415105
[4]
R. Adami, R. Figari, D. Finco and A. Teta, On the asymptotic behaviour of a quantum two-body system in the small mass ratio limit, J. Phys. A 37 (2004), 7567-7580. MR2089400
[5]
R. Adami, R. Figari, D. Finco and A. Teta, On the asymptotic dynamics of a quantum system composed by heavy and light particles, Comm. Math. Phys. 268 (2006), 819-852. MR2259215
[6]
R. Adami, M. Hauray and C. Negulescu, Decoherence for a heavy particle interacting with a light particle: new analysis and numerics, Commun. Math. Sci. 14 (2016), 1373-1415. MR3506806
[7]
R. Adami and C. Negulescu, A numerical study of quantum decoherence, Commun. Comput. Phys. 12 (2012), 85-108. DOI
[8]
R. Adami and A. Teta, A class of nonlinear Schrödinger equations with concentrated nonlinearity, J. Funct. Ana. 180 (2001), 148-175. MR1814425
[9]
R. Adami and A. Teta, A simple model of concentrated nonlinearity, Oper. Theory Adv. Appl., 108, Birkhäuser, Basel, 1999, 183-189. MR1708796
[10]
G. P. Agrawal, Nonlinear fiber optics, Elsevire, New York, 2006. DOI
[11]
S. Albeverio, Z. Brzeźniak and L. Dabrowski, Fundamental solutions of the heat and Schrödinger equations with point interactions, J. Funct. Anal. 130 (1995), 220-254. MR1331982
[12]
S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable models in quantum mechanics, 2nd ed., AMS Chelsea Publishing, Providence R.I, 2005. MR2105735
[13]
A. Arnold, Mathematical properties of quantum evolution equations, in ''Quantum Transport - Modelling, Analysis and Asymptotics'', Lecture Notes in Math., 1946, Springer-Verlag, Berlin, 2008, 45-109. MR2497875
[14]
A. Arnold, N. Ben Abdallah and C. Negulescu, WKB-based schemes for the oscillatory 1D Schrödinger equation in the semiclassical limit, SIAM J. Numer. Anal. 49 (2011), 1436-1460. MR2831055
[15]
A. Arnold and C. Negulescu, Stationary Schrödinger equation in the semi-classical limit: numerical coupling of oscillatory and evanescent regions, Numer. Math. 138 (2018), 501-536. MR3748311
[16]
W. Bao, D. Jaksch and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys. 187 (2003), 318-342. MR1977789
[17]
W. Bao, S. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput. 25 (2003), 27-64. MR2047194
[18]
L. Barletti, L. Brugnano, G. Frasca Caccia and F. Iavernaro, Energy-conserving methods for the nonlinear Schrödinger equation, Appl. Math. Comput. 318 (2018), 3-18. MR3713844
[19]
A. Bátkai, P. Csomós and G. Nickel, Operator splittings and spatial approximations for evolution equations, J. Evol. Equ. 9 (2009), 613-636. MR2529739
[20]
N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys. 37 (1996), 3306-3333. MR1401227
[21]
N. Ben Abdallah, P. Degond and P. A. Markowich, On a one-dimensional Schrödinger-Poisson scattering model, Z. Angew. Math. Phys. 48 (1997), 135-155. MR1439739
[22]
N. Ben Abdallah and O. Pinaud, Multiscale simulation of transport in an open quantum system: resonances and WKB interpolation, J. Comput. Phys. 213 (2006), 288-310. MR2203442
[23]
A. Bertoni, Simulation of electron decoherence induced by carrier-carrier scattering, J. Comput Electron. 2 (2003), 291-295. DOI
[24]
Ph. Blanchard, D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu and H. D. Zeh, Decoherence and the appearance of a classical world in quantum theory, Springer-Verlag, Berlin Heidelberg, 1996. Zbl 0855.00003
[25]
I. Bloch, Quantum coherence and entanglement with ultracold atoms in optical lattices, Nature 453 (2008), 1016-1022. DOI
[26]
H.-P. Breuer and F. Petruccione, The theory of open quantum systems, reprint of the 2002 ed., Oxford University Press, Oxford, 2007. Zbl 1223.81001
[27]
C. Cacciapuoti, R. Carlone and R. Figari, Decoherence induced by scattering: a three-dimensional model, J. Phys. A 38 (2005), 4933-4946. MR2148634
[28]
A. O. Caldeira and A. J. Leggett, Influence of damping on quantum interference: an exactly soluble model, Phys. Rev. A 31 (1985), 1059. DOI
[29]
E. Cancès, C. Le Bris and Y. Maday, Méthodes mathématiques en chimie quantique. Une introduction, Math. Appl. (Berlin), 53, Springer, Berlin, 2006. MR2426947
[30]
H. Cartarius, D. Haag, D. Dast and G. Wunner, Nonlinear Schrödinger equation for a $${\mathcal{PT}}$$-symmetric delta-function double well, J. Phys. A 45 (2012), 444008. MR2991875
[31]
R. Carlone, R. Figari and C. Negulescu, The quantum beating and its numerical simulation, J. Math. Anal. Appl. 450 (2017), 1294-1316. MR3639102
[32]
T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de métodos matemáticos, 26, Instituto de Matemática, UFRJ, 1993. BooksGoogle
[33]
T. Cazenave, Semilinear Schrödinger equations, Courant Lect. Notes Math., 10, AMS, Providence, RI, 2003. MR2002047
[34]
F. F. Chen, Introduction to plasma physics and controlled fusion: Plasma physics, Vol. 1, Springer Verlag, New York, 2006. BooksGoogle
[35]
J. Clark, The reduced effect of a single scattering with a low-mass particle via a point interaction, J. Funct. Anal. 256 (2009), 2894-2916. MR2502427
[36]
C. Cohen-Tannoudji, B. Diu and F. Laloë, Mécanique quantique I, Hermann éditeurs des sciences et des arts, Paris, 1997.
[37]
S. Descombes and M. Thalhammer, An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime, BIT 50 (2010), 729-749. MR2739463
[38]
G. Dujardin and E. Faou, Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential, Numer. Math. 108 (2007), 223-262. MR2358004
[39]
G. Dujardin and E. Faou, Long time behavior of splitting methods applied to the linear Schrödinger equation, C. R. Math. Acad. Sci. Paris 344 (2007), 89-92. MR2288596
[40]
D. Dürr, R. Figari and A. Teta, Decoherence in a two-particle model, J. Math. Phys. 45 (2004), 1291-1309. MR2043829
[41]
D. Dürr and H. Spohn, Decoherence Through Coupling to the Radiation Field, in ''Decoherence: Theoretical, Experimental and Conceptual Problems'', Lect. Notes in Phys., 538, Springer-Verlag, Berlin Heidelberg, 2000, 77-86. Article
[42]
W. E, Principles of multiscale modeling, Cambridge University Press, Cambridge, 2011. MR2830582
[43]
R. Fukuizumi and A. Sacchetti, Bifurcation and stability for nonlinear Schrödinger equations with double well potential in the semiclassical limit, J. Stat. Phys. 145 (2011), 1546-1594. MR2863720
[44]
L. Gauckler and C. Lubich, Splitting integrators for nonlinear Schrödinger equations over long times, Found. Comput. Math. 10 (2010), 275-302. MR2628827
[45]
B. Gaveau and L. S. Schulman, Explicit time-dependent Schrödinger propagators, J. Phys. A 19 (1986), 1833-1846. MR0851483
[46]
A. K. Ghatak, R. L. Gallawa and I. C. Goyal, Modified airy function and WKB solutions to the wave equation, NIST Monograph 176, 1991. Article
[47]
A. K. Ghatak, R. L. Gallawa and I. C. Goyal, Accurate solutions to Schrödinger's equation using modified Airy functions, IEEE J. Quantum Electron. 28 (1992), 400-403. DOI
[48]
V. Grecchi, A. Martinez and A. Sacchetti, Destruction of the beating effect for a non-linear Schrödinger equation, Comm. Math. Phys. 227 (2002), 191-209. MR1903844
[49]
A. Griffin, D. W. Snoke and S. Stringari, eds., Bose-Einstein condensation, Cambridge University Press, New York, 1995. DOI
[50]
E. Hansen and A. Ostermann, Dimension splitting for evolution equations, Numer. Math. 108 (2008), 557-570. MR2369204
[51]
H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, Englewood Cliffs NJ, 1984.
[52]
M. H. Holmes, Introduction to perturbation methods, Springer-Verlag, New York, 1995. MR1351250
[53]
K. Hornberger and J. E. Sipe, Collisional decoherence reexamined, Phys. Rev. A 68 (2003), 012105, 16 pp. DOI
[54]
K. Hornberger, S. Uttenthaler, B. Brezger, L. Hackermüller, M. Arndt and A. Zeilinger, Collisional decoherence observed in matter wave interferometry, Phys. Rev. Lett. 90 (2003), 160401, 4 pp. DOI
[55]
K. Hornberger and B. Vacchini, Monitoring derivation of the quantum linear Boltzmann equation, Phys. Rev. A 77 (2008), 022112, 18 pp. DOI
[56]
F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. I. The $$h$$-version of the FEM, Comput. Math. Appl. 30 (1995), 9-37. MR1353516
[57]
F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. II. The $$h$$-$$p$$ version of the FEM, SIAM J. Numer. Anal. 34 (1997), 315-358. MR1445739
[58]
T. Jahnke and C. Lubich, Numerical integrators for quantum dynamics close to the adiabatic limit, Numer. Math. 94 (2003), 289-314. MR1974557
[59]
S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput. 21 (1999), 441-454. MR1718639
[60]
E. Joos and H. D. Zeh, The emergence of classical properties through interaction with the environment, Z. Phys. B 59 (1985), 223-243. DOI
[61]
C. Kharif, E. Pelinovsky and A. Slunyaev, Rogue waves in oceans, Springer-Verlag, Berlin, 2009. MR2841079
[62]
L. D. Landau and E. M. Lifschitz, Quantum Mechanics, Butterworth-Heinemann, 1981. Article
[63]
R. E. Langer, On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order, Trans. Amer. Math. Soc. 33 (1931), 23-64. MR1501574
[64]
C. Le Bris, Systèmes multi-échelles. Modélisation et simulation, Springer-Verlag, Berlin, 2005. MR2227763
[65]
C. S. Lent and D. J. Kirkner, The quantum transmitting boundary method, J. Appl. Phys. 67 (1990), 6353-6359. DOI
[66]
K. Lorenz, T. Jahnke and C. Lubich, Adiabatic integrators for highly oscillatory second-order linear differential equations with time-varying eigendecomposition, BIT 45 (2005), 91-115. MR2164227
[67]
C. Lubich, From quantum to classical molecular dynamics: reduced models and numerical analysis, Zur. Lect. Adv. Math., European Mathematical Society (EMS), Zürich, 2008. MR2474331
[68]
M. Lundstrom, Fundamentals of carrier transport, Cambridge University Press, Cambridge, 2000. DOI
[69]
M. Lundstrom and J. Guo, Nanoscale transistors, Springer Verlag, Boston, MA, 2006. DOI
[70]
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor equations, Springer-Verlag, Wien, 1990. MR1063852
[71]
E. Middlemas and J. Knisley Soliton solutions of a variation of the nonlinear Schrödinger equation, in ''Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference'', J. Rychtár, S. Gupta, R. Shivaji, M. Chhetri, eds., Springer Proceedings in Mathematics & Statistics series, 64, Springer, New York, NY, 2013, 39-53. DOI
[72]
I. Mitra and S. Roy, Relevance of quantum mechanics in circuit implementation of ion channels in brain dynamics, arXiv:q-bio/0606008, preprint, 2006.
[73]
J. D. Murray, Mathematical biology, Springer-Verlag, Berlin Heidelberg, 2002. MR1908418
[74]
A. H. Nayfeh, Perturbation methods, John Wiley & Sons, New York-London-Sydney, 1973. MR0404788
[75]
C. Negulescu, Numerical analysis of a multiscale finite element scheme for the resolution of the stationary Schrödinger equation, Numer. Math. 108 (2008), 625-652. MR2369207
[76]
C. Negulescu, N. Ben Abdallah and M. Mouis, An accelerated algorithm for 2D simulations of the quantum ballistic transport in nanoscale MOSFETs, J. Comput. Phys. 225 (2007), 74-99. MR2346672
[77]
R. Omnès, The interpretation of quantum mechanics, Princeton Ser. Phys., Princeton University Press, Princeton, 1994. MR1291602
[78]
M. Onorato, S. Residori, U. Bortolozzo, A. Montina and F. T. Arecchi, Rogue waves and their generating mechanisms in different physical contexts, Phys. Rep. 528 (2013), 47-89. MR3070399
[79]
A. R. Osborne, Nonlinear ocean waves and the inverse scattering transform, International Geophysics Series, 97, Elsevier/Academic Press, Boston, MA, 2010. MR2654218
[80]
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci., 44, Springer-Verlag, New-York, 1983. MR0710486
[81]
H. L. Pécseli, Waves and oscillations in plasmas, Series in Plasma Physics, CRC Press, Boca Raton, FL, 2013. DOI
[82]
C. J. Pethick and H. Smith, Bose-Einstein condensation in dilute gases, Cambridge University Press, Cambridge, 2008. DOI
[83]
V. Petviashvili and O. Pokhotelov, Solitary waves in plasmas and in the atmosphere, Gordon and Breach Science Publishers, Philadelphia, PA, 1992. MR1261190
[84]
M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York-London, 1972. MR0493419
[85]
M. Reed and B. Simon, Methods of modern mathematical physics. III. Scattering theory, Academic Press, New York-London, 1979. MR0529429
[86]
I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys. 291 (2009), 31-61. MR2530155
[87]
P. H. Rutherford, Nonlinear growth of the tearing mode, Phys. Fluids 16 (1973), 1903-1908, DOI
[88]
A. Sacchetti, Universal critical power for nonlinear Schrödinger equations with a symmetric double well potential, Phys. Rev. Lett. 103 (2009), 194101, 4 pp. DOI
[89]
M. Schechter, Operator methods in quantum mechanics, Dover Publications, Inc., Mineola, NY, 2002. MR1969612
[90]
B. Schlein, Derivation of effective evolution equations from many-body quantum mechanics, Riv. Math. Univ. Parma 8 (2017), 83-108. MR3706142
[91]
M. Schlosshauer, Decoherence and the quantum-to-classical transition, Springer-Verlag, Berlin Heidelberg, 2007. DOI
[92]
P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52 (1995), 2493-2496. DOI
[93]
A. J. Smith and A. R. Baghai-Wadji, A numerical technique for solving Schrödingers equation in molecular electronic applications, in ''Smart Structures, Devices, and Systems IV'', Proc. SPIE vol. 7268, 2008. DOI
[94]
B. Sportisse, An analysis of operator splitting techniques in the stiff case, J. Comput. Phys. 161 (2000), 140-168. MR1762076
[95]
C. A. Stafford, D. M. Cardamone and S. Mazumadar, The quantum interference effect transistor, Nanotechnology 18 (2007), 424014. DOI
[96]
C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation, Appl. Math. Sci., 139, Springer-Verlag, New-York, 1999. MR1696311
[97]
J. L. Vázquez, The porous medium equation. Mathematical theory, Oxford Math. Monogr., Oxford University Press, Oxford, 2007. MR2286292
[98]
W. Zhang, N. Konstantinidis, K. A. Al-Hassanieh and V. V. Dobrovitski, Modelling decoherence in quantum spin systems, J. Phys. Condens. Matter 19 (2007), 083202. DOI

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