메뉴 건너뛰기




Volumn , Issue , 2010, Pages 169-197

On the mathematical description of the effective behaviour of one-dimensional Bose-Einstein condensates with defects

Author keywords

[No Author keywords available]

Indexed keywords


EID: 84876915226     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter
Times cited : (3)

References (60)
  • 1
    • 70350671744 scopus 로고    scopus 로고
    • Colliding solitons for the nonlinear Schrödinger equation
    • Abu Salem W., Fröhlich J., Sigal I.M.: Colliding solitons for the nonlinear Schrödinger equation, Commun. Math. Phys. 291, 151-176 (2009)
    • (2009) Commun. Math. Phys. , vol.291 , pp. 151-176
    • Abu Salem, W.1    Fröhlich, J.2    Sigal, I.M.3
  • 2
    • 23744501331 scopus 로고    scopus 로고
    • Towards a rigorous derivation of the cubic NLSE in dimension one
    • Adami R., Bardos C., Golse, F., Teta A.: Towards a rigorous derivation of the cubic NLSE in dimension one, Asymp. An. 40, 2, 93-108 (2004).
    • (2004) Asymp. An. , vol.40 , pp. 93-108
    • Adami, R.1    Bardos, C.2    Golse, F.3    Teta, A.4
  • 3
    • 84892077972 scopus 로고    scopus 로고
    • Fast solitons on star graphs
    • electronic preprint arXiv:1004.2455
    • Adami R., Cacciapuoti C., Finco D., Noja D.: Fast solitons on star graphs, electronic preprint arXiv:1004.2455 (2010).
    • (2010)
    • Adami, R.1    Cacciapuoti, C.2    Finco, D.3    Noja, D.4
  • 4
    • 34249868461 scopus 로고    scopus 로고
    • Rigorous derivation of the cubic NLS in dimension one
    • Adami R., Golse, F, Teta A.: Rigorous derivation of the cubic NLS in dimension one. J. Stat. Phys, 127, 1193-1220 (2007).
    • (2007) J. Stat. Phys , vol.127 , pp. 1193-1220
    • Adami, R.1    Golse, F.2    Teta, A.3
  • 5
    • 25144486116 scopus 로고    scopus 로고
    • The transition from diffusion to blow-up for a NLS equation in dimension one
    • Adami R., Sacchetti A.: The transition from diffusion to blow-up for a NLS equation in dimension one. J.Phys. A: Math. Theor. 38, 8379-8392 (2005).
    • (2005) J.Phys. A: Math. Theor. , vol.38 , pp. 8379-8392
    • Adami, R.1    Sacchetti, A.2
  • 6
    • 79957945199 scopus 로고    scopus 로고
    • Existence of dynamics for a 1-d NLS equation perturbed with a generalized point defect
    • Adami R., Noja D.: Existence of dynamics for a 1-d NLS equation perturbed with a generalized point defect, J. Phys. A: Math. Theor. 42, 495302-19pp (2009).
    • (2009) J. Phys. A: Math. Theor. , vol.42 , pp. 495302-495319
    • Adami, R.1    Noja, D.2
  • 7
    • 84892099582 scopus 로고    scopus 로고
    • Stability, instability and symmetry breaking of the ground states for a nonlinear Schrödinger equation with a δ′ interaction
    • in preparation
    • Adami R., Noja D.: Stability, instability and symmetry breaking of the ground states for a nonlinear Schrödinger equation with a δ′ interaction, in preparation (2010).
    • (2010)
    • Adami, R.1    Noja, D.2
  • 8
    • 0003487664 scopus 로고
    • Theory of linear operators in Hilbert spaces
    • New York, Ungar
    • Akhiezer N., Glazman I.: Theory of linear operators in Hilbert spaces, New York, Ungar, (1963).
    • (1963)
    • Akhiezer, N.1    Glazman, I.2
  • 9
    • 0000128397 scopus 로고
    • Fundamental solutions of the Heat and Schrödinger Equations with point interaction
    • Albeverio S., Brzézniak Z., Dabrowski L.: Fundamental solutions of the Heat and Schrödinger Equations with point interaction. J. Funct. Anal. 130, 220-254 (1995).
    • (1995) J. Funct. Anal. , vol.130 , pp. 220-254
    • Albeverio, S.1    Brzézniak, Z.2    Dabrowski, L.3
  • 10
    • 0003871449 scopus 로고
    • Solvable Models in Quantum Mechanics
    • New York: Springer-Verlag
    • Albeverio S., Gesztesy F., Høgh-Krohn R., Holden H.: Solvable Models in Quantum Mechanics. New York: Springer-Verlag 1988.
    • (1988)
    • Albeverio, S.1    Gesztesy, F.2    Høgh-Krohn, R.3    Holden, H.4
  • 11
    • 0003741854 scopus 로고    scopus 로고
    • Perturbations of Differential Operators
    • Cambridge University Press
    • Albeverio S., Kurasov P.: Singular Perturbations of Differential Operators, Cambridge University Press 2000.
    • (2000)
    • Albeverio, S.1    Kurasov, P.2
  • 12
    • 84892069182 scopus 로고    scopus 로고
    • Propagation of chaos for many-boson systems in one dimension with a point pair-interaction
    • electronic preprint arXiv:0906.3047
    • Ammari Z., Breteaux S.: Propagation of chaos for many-boson systems in one dimension with a point pair-interaction, electronic preprint arXiv:0906.3047 (2009).
    • (2009)
    • Ammari, Z.1    Breteaux, S.2
  • 13
    • 0020591567 scopus 로고
    • Nonlinear scalar field equations. I. Existence of a ground state
    • Berestycki H., Lions P-L.: Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal 82, 313-346 (1983).
    • (1983) Arch. Ration. Mech. Anal , vol.82 , pp. 313-346
    • Berestycki, H.1    Lions, P.-L.2
  • 14
    • 0003426666 scopus 로고    scopus 로고
    • Hilbert spaces operators in Quantum Physics
    • Springer, New York
    • Blank J., Exner P., Havlicek M.: Hilbert spaces operators in Quantum Physics, Springer, New York (2008).
    • (2008)
    • Blank, J.1    Exner, P.2    Havlicek, M.3
  • 15
    • 0002811905 scopus 로고
    • On the theory of superfluidity
    • Eng. Trans. J. Phys. (USSR) 11, 23 (1947)
    • Bogoliubov N.N.: On the theory of superfluidity, Izv. Akad. Nauk USSR 11, 77 (1947). Eng. Trans. J. Phys. (USSR) 11, 23 (1947).
    • (1947) Izv. Akad. Nauk USSR , vol.11 , pp. 77
    • Bogoliubov, N.N.1
  • 16
    • 0042985041 scopus 로고
    • Planck Gesetz und Lichtquantenhypothese
    • Bose S.: Planck Gesetz und Lichtquantenhypothese. Zeitschrift für Physik 26, 178-181 (1924).
    • (1924) Zeitschrift für Physik , vol.26 , pp. 178-181
    • Bose, S.1
  • 17
    • 0007603725 scopus 로고
    • Soliton defect collisions in the nonlinear Schrödinger equation
    • Cao Xiang D., Malomed A. B.: Soliton defect collisions in the nonlinear Schrödinger equation. Phys. Lett. A 206, 177-182 (1995).
    • (1995) Phys. Lett. A , vol.206 , pp. 177-182
    • Cao Xiang, D.1    Malomed, A.B.2
  • 18
    • 10844227674 scopus 로고    scopus 로고
    • Semilinear Schrödinger Equations
    • Courant Lecture Notes in Mathematics AMS, Providence
    • Cazenave, T.: Semilinear Schrödinger Equations, vol. 10 Courant Lecture Notes in Mathematics AMS, Providence (2003).
    • (2003) , vol.10
    • Cazenave, T.1
  • 19
    • 0000090159 scopus 로고
    • Orbital stability of standing waves for some nonlinear Schrödinger equations
    • Cazenave, T. Lions, P.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys. 85, 549-561 (1982).
    • (1982) Commun. Math. Phys. , vol.85 , pp. 549-561
    • Cazenave, T.1    Lions, P.2
  • 20
    • 70449509363 scopus 로고    scopus 로고
    • Fast soliton scattering by attracting delta impurities
    • Datchev K., Holmer J.: Fast soliton scattering by attracting delta impurities. Commun. Part. Diff. Eq. 34, 1074-1113 (2009).
    • (2009) Commun. Part. Diff. Eq. , vol.34 , pp. 1074-1113
    • Datchev, K.1    Holmer, J.2
  • 22
    • 33745763559 scopus 로고    scopus 로고
    • Stationary states for nonlinear one dimensional Schrödinger equation with singular potential
    • Della Casa F., Sacchetti A.: Stationary states for nonlinear one dimensional Schrödinger equation with singular potential. Physica D 209, 60-68 (2006).
    • (2006) Physica D , vol.209 , pp. 60-68
    • Della Casa, F.1    Sacchetti, A.2
  • 23
  • 24
    • 0002146671 scopus 로고
    • Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung
    • Einstein A.: Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung. Sitzungber. Preuss. Akad. Wiss. Berlin, 1, 3-14 (1925).
    • (1925) Sitzungber. Preuss. Akad. Wiss. Berlin , vol.1 , pp. 3-14
    • Einstein, A.1
  • 25
    • 35349015162 scopus 로고    scopus 로고
    • Stationary and nonstationary fluid flow of a Bose-Einstein condensate through a penetrable barrier
    • Engels, P, Atherton C.: Stationary and nonstationary fluid flow of a Bose-Einstein condensate through a penetrable barrier, Phys. Rev. Lett. 99, 160405 (2007).
    • (2007) Phys. Rev. Lett. , vol.99 , pp. 160405
    • Engels, P.1    Atherton, C.2
  • 26
    • 33846807694 scopus 로고    scopus 로고
    • Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many body systems
    • Erdos L., Schlein B., Yau H-T.: Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many body systems. Invent. Math. 167, 515-614 (2007).
    • (2007) Invent. Math. , vol.167 , pp. 515-614
    • Erdos, L.1    Schlein, B.2    Yau, H.-T.3
  • 27
    • 33750551506 scopus 로고    scopus 로고
    • Derivation of the Gross-Pitaevskii equation for the dynamics of the Bose Einstein condensate
    • to appear
    • Erdos L., Schlein B., Yau H-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of the Bose Einstein condensate., Ann. Math., to appear (2006).
    • (2006) Ann. Math.
    • Erdos, L.1    Schlein, B.2    Yau, H.-T.3
  • 28
    • 77950570065 scopus 로고    scopus 로고
    • Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential
    • Erdos L., Schlein B., Yau H.-T.: Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential. J. Amer. Math. Soc. 22, 1099-1156 (2009).
    • (2009) J. Amer. Math. Soc. , vol.22 , pp. 1099-1156
    • Erdos, L.1    Schlein, B.2    Yau, H.-T.3
  • 29
    • 0035541766 scopus 로고    scopus 로고
    • Potential approximations to a δ′: an inverse Klauder phenomenon with norm-resolvent convergence
    • Exner P. Neidhardt H., Zagrebnov V.A.: Potential approximations to a δ′: an inverse Klauder phenomenon with norm-resolvent convergence, Commun. Math. Phys. 224, 593612 (2001).
    • (2001) Commun. Math. Phys. , vol.224 , pp. 593612
    • Exner, P.1    Neidhardt, H.2    Zagrebnov, V.A.3
  • 30
    • 0001890328 scopus 로고
    • Sul moto dei neutroni nelle sostanze idrogenate
    • Fermi, E.: Sul moto dei neutroni nelle sostanze idrogenate, Ric. Scient. 7, 2, 13-52 (1936).
    • (1936) Ric. Scient. , vol.7 , Issue.2 , pp. 13-52
    • Fermi, E.1
  • 31
    • 0037437726 scopus 로고    scopus 로고
    • Stability of solitary waves for nonlinear Schrodinger equations with inhomogeneous nonlinearities
    • Fibich G., Wang X.P.: Stability of solitary waves for nonlinear Schrodinger equations with inhomogeneous nonlinearities. Physica D 175, 96-108 (2003).
    • (2003) Physica D , vol.175 , pp. 96-108
    • Fibich, G.1    Wang, X.P.2
  • 32
    • 44449145089 scopus 로고    scopus 로고
    • Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential
    • Fukuizumi R., Jeanjean L.: Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential. DCDS 21, 121-136 (2008).
    • (2008) DCDS , vol.21 , pp. 121-136
    • Fukuizumi, R.1    Jeanjean, L.2
  • 33
    • 52149096486 scopus 로고    scopus 로고
    • Nonlinear Schrödinger equation with a point defect
    • Fukuizumi R., Ohta M., Ozawa T.: Nonlinear Schrödinger equation with a point defect. Ann. I.H.Poincaré, AN 25, 837-845 (2008).
    • (2008) Ann. I.H.Poincaré, AN , vol.25 , pp. 837-845
    • Fukuizumi, R.1    Ohta, M.2    Ozawa, T.3
  • 34
    • 2342576894 scopus 로고    scopus 로고
    • Strong NLS soliton-defect interactions
    • Goodman R. H., Holmes P. J.,Weinstein M. I.: Strong NLS soliton-defect interactions. Physica D 192, 215-248 (2004).
    • (2004) Physica D , vol.192 , pp. 215-248
    • Goodman, R.H.1    Holmes, P.J.2    Weinstein, M.I.3
  • 35
    • 0000468151 scopus 로고
    • Stability theory of solitary waves in the presence of symmetry
    • Grillakis M. Shatah J. Strauss W.: Stability theory of solitary waves in the presence of symmetry. I, J.Funct.Anal. 74, 160-197 (1987).
    • (1987) I, J.Funct.Anal. , vol.74 , pp. 160-197
    • Grillakis, M.1    Shatah, J.2    Strauss, W.3
  • 36
    • 29444432472 scopus 로고
    • Stability theory of solitary waves in the presence of symmetry
    • Grillakis M. Shatah J. Strauss W.: Stability theory of solitary waves in the presence of symmetry. II, J.Funct.Anal. 94, 308-348 (1990).
    • (1990) II, J.Funct.Anal. , vol.94 , pp. 308-348
    • Grillakis, M.1    Shatah, J.2    Strauss, W.3
  • 37
    • 34547337563 scopus 로고    scopus 로고
    • Fast soliton scattering by delta impurities
    • Holmer J., Marzuola J., Zworski M.: Fast soliton scattering by delta impurities. Commun. Math. Phys 274, 187-216 (2007).
    • (2007) Commun. Math. Phys , vol.274 , pp. 187-216
    • Holmer, J.1    Marzuola, J.2    Zworski, M.3
  • 39
    • 4344685967 scopus 로고    scopus 로고
    • Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation
    • Jackson R.K, Weinstein M.I.: Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation. J. Stat. Phys. 116, 881-905 (2004).
    • (2004) J. Stat. Phys. , vol.116 , pp. 881-905
    • Jackson, R.K.1    Weinstein, M.I.2
  • 40
    • 84881470722 scopus 로고    scopus 로고
    • Stability of standing waves for nonlinear Schrödinger equation with attractive delta potential and repulsive nonlinearity
    • Kaminaga M., Ohta M.: Stability of standing waves for nonlinear Schrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J. 26, 3948 (2009).
    • (2009) Saitama Math. J. , vol.26 , pp. 3948
    • Kaminaga, M.1    Ohta, M.2
  • 41
    • 0039844705 scopus 로고    scopus 로고
    • Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions
    • Ketterle W., van Druten N. J.: Bose-Einstein condensation of a finite number of particles trapped in one or three dimensions. Phys. Rev. A. 54, 656-660 (1996).
    • (1996) Phys. Rev. A. , vol.54 , pp. 656-660
    • Ketterle, W.1    van Druten, N.J.2
  • 42
    • 43049140512 scopus 로고    scopus 로고
    • Instability of bound staters of a nonlinear Schrödinger equation with a Dirac potential
    • Le Coz S., Fukuizumi R., Fibich G., Ksherim B, Sivan Y.: Instability of bound staters of a nonlinear Schrödinger equation with a Dirac potential. Physica D 237, 1103-1128 (2007).
    • (2007) Physica D , vol.237 , pp. 1103-1128
    • Le Coz, S.1    Fukuizumi, R.2    Fibich, G.3    Ksherim, B.4    Sivan, Y.5
  • 43
    • 26344439240 scopus 로고    scopus 로고
    • Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State
    • See also: Lieb E.H.:Exact Analysis of an Interacting Bose Gas. II, Phys. Rev. 130, 1616-1624 (1963).
    • Lieb E.H., Liniger W.: Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev. 130, 1605-1616 (1963). See also: Lieb E.H.:Exact Analysis of an Interacting Bose Gas. II, Phys. Rev. 130, 1616-1624 (1963).
    • Phys. Rev. , vol.130 , pp. 1605-1616
    • Lieb, E.H.1    Liniger, W.2
  • 44
    • 0037193018 scopus 로고    scopus 로고
    • Proof of Bose-Einstein Condensation for Dilute Trapped Gases
    • Lieb E.H., Seiringer R.: Proof of Bose-Einstein Condensation for Dilute Trapped Gases, Phys. Rev. Lett. 88, 170409 (2002).
    • (2002) Phys. Rev. Lett. , vol.88 , pp. 170409
    • Lieb, E.H.1    Seiringer, R.2
  • 45
    • 33745832074 scopus 로고    scopus 로고
    • The mathematics of the Bose gas and its condensation
    • Oberwolfach Seminar
    • Lieb E.H., Seiringer R., Solovej J.-P., Yngvason J.: The mathematics of the Bose gas and its condensation, Oberwolfach Seminar, 2005.
    • (2005)
    • Lieb, E.H.1    Seiringer, R.2    Solovej, J.-P.3    Yngvason, J.4
  • 47
    • 84891969240 scopus 로고    scopus 로고
    • Strenghtened convergence of marginals to the cubic NLS
    • SISSA preprint 35/2007/MP
    • Michelangeli A.: Strenghtened convergence of marginals to the cubic NLS, SISSA preprint 35/2007/MP (2007).
    • (2007)
    • Michelangeli, A.1
  • 48
    • 4244011551 scopus 로고
    • Bose-Einstein condensation and liquid helium
    • Penrose O., Onsager L.: Bose-Einstein condensation and liquid helium. Phys. Rev. 104, 576-584 (1956).
    • (1956) Phys. Rev. , vol.104 , pp. 576-584
    • Penrose, O.1    Onsager, L.2
  • 49
    • 0003687441 scopus 로고
    • Methods of modern Mathematical Physics
    • Academic Press
    • Reed R., Simon B.: Methods of modern Mathematical Physics, Vol I, Academic Press, (1980)
    • (1980) , vol.I
    • Reed, R.1    Simon, B.2
  • 50
    • 36148967767 scopus 로고    scopus 로고
    • Spectral splitting method for nonlinear Schrödinger equations with singular potential
    • Sacchetti A.: Spectral splitting method for nonlinear Schrödinger equations with singular potential. J. Comp. Phys. 227, 1483-1499 (2007).
    • (2007) J. Comp. Phys. , vol.227 , pp. 1483-1499
    • Sacchetti, A.1
  • 51
    • 70450060025 scopus 로고    scopus 로고
    • Universal critical power for nonlinear Schrödinger equations with a symmetric double well potential
    • Sacchetti A.: Universal critical power for nonlinear Schrödinger equations with a symmetric double well potential, Phys. Rev. Lett. 103 194101 (2009).
    • (2009) Phys. Rev. Lett. , vol.103 , pp. 194101
    • Sacchetti, A.1
  • 52
    • 54049087752 scopus 로고    scopus 로고
    • The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions
    • Seiringer R., Yin J.: The Lieb-Liniger Model as a Limit of Dilute Bosons in Three Dimensions, Commun. Math. Phys. 284, 459 (2008).
    • (2008) Commun. Math. Phys. , vol.284 , pp. 459
    • Seiringer, R.1    Yin, J.2
  • 53
    • 0001420204 scopus 로고
    • Instability of nonlinear ground states
    • Shatah J., Strauss W.: Instability of nonlinear ground states. Commun. Math. Phys. 100, 173-190 (1985).
    • (1985) Commun. Math. Phys. , vol.100 , pp. 173-190
    • Shatah, J.1    Strauss, W.2
  • 54
    • 0003405464 scopus 로고    scopus 로고
    • The nonlinear Schrödinger equation
    • Springer, New York
    • Sulem C., Sulem P.: The nonlinear Schrödinger equation, Springer, New York (1999).
    • (1999)
    • Sulem, C.1    Sulem, P.2
  • 55
    • 58649105782 scopus 로고    scopus 로고
    • Why are solitons stable?
    • Tao T.: Why are solitons stable?. Bull.Am.Math.Soc. 46, 1-33 (2009).
    • (2009) Bull.Am.Math.Soc. , vol.46 , pp. 1-33
    • Tao, T.1
  • 57
    • 0000686130 scopus 로고
    • Modulational stability of ground states of nonlinear Schrödinger equations
    • Weinstein M.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J.Math.Anal. 16, 472-491 (1985).
    • (1985) SIAM J.Math.Anal. , vol.16 , pp. 472-491
    • Weinstein, M.1
  • 58
    • 84990553584 scopus 로고
    • Lyapunov stability of ground states of nonlinear dispersive evolution equations
    • Weinstein M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl.Math. 39, 51-68 (1986).
    • (1986) Commun. Pure Appl.Math. , vol.39 , pp. 51-68
    • Weinstein, M.1
  • 59
    • 14544268141 scopus 로고    scopus 로고
    • Bound and resonance states of the nonlinear Schrödinger equation in simple model systems
    • Witthaut D., Mossmann S., Korsch H. J.: Bound and resonance states of the nonlinear Schrödinger equation in simple model systems. J.Phys.A 38, 1777-1792 (2005).
    • (2005) J.Phys.A , vol.38 , pp. 1777-1792
    • Witthaut, D.1    Mossmann, S.2    Korsch, H.J.3
  • 60
    • 0141860984 scopus 로고    scopus 로고
    • Korteweg-de Vries and Nonlinear Schrödinger equations: Qualitative theory
    • LNP, Springer Verlag, Berlin-Heidelberg
    • Zhidkov P.E.: Korteweg-de Vries and Nonlinear Schrödinger equations: Qualitative theory, LNP, Springer Verlag, Berlin-Heidelberg, (2001)
    • (2001)
    • Zhidkov, P.E.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.