zbMATH — the first resource for mathematics

Wigner measures and codimension two crossings. (English) Zbl 1061.81078
Summary: This article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels. Codimension two crossings are the simplest energy level crossings, which affect the Born-Oppenheimer approximation in the zeroth order term. The model we study is a two-level Schrödinger equation with a Laplacian as kinetic operator and a matrix-valued linear potential, whose eigenvalues cross, if the two nucleonic coordinates equal zero. We discuss the case of well-localized initial data and obtain a description of the wavefunction’s two-scaled Wigner measure and of the weak limit of its position density, which is valid globally in time.

81V55 Molecular physics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text: DOI
[1] Colin de Verdière, Ann. I.H.P. Phys. Theor. 71 pp 95– (1999)
[2] Dimassi, M. and Sjöstrand, J.,Spectral Asymptotics in the Semi-Classical Limit, (Cambridge University Press, Cambridge, 1999). · Zbl 0926.35002
[3] Exner, J. Math. Phys. 42 pp 4707– (2001)
[4] Fermanian Kammerer, C. R. Acad. Sci. Paris 331 pp Ser: 1, 515– (2000) · Zbl 0964.35009
[5] Fermanian Kammerer, C., ”A non-commutative Landau-Zener formula, prépublication de l’Université de Cergy-Pontoise,” (2002).
[6] Fermanian Kammerer, Bull. S. M. F. 130 pp 123– (2002)
[7] Gérard, Commun. Partial Differ. Equ. 16 pp 1761– (1991)
[8] Gérard, P., ”Mesures semi-classiques et ondes de Bloch,” Exposé de l’Ecole Polytechnique, E.D.P., Exposé NoXVI (1991). · Zbl 0739.35096
[9] Gérard, Duke Math. J. 71 pp 559– (1993)
[10] Gérard, Commun. Pure Appl. Math. 50 pp 323– (1997)
[11] Hagedorn, Commun. Math. Phys. 136 pp 433– (1991)
[12] Hagedorn, Mem. Am. Math. Soc. 111 pp 536– (1994)
[13] Hagedorn, Ann. I.H.P. Phys. Theor. 68 pp 85– (1998)
[14] Hagedorn, Rev. Math. Phys. 11 pp 41– (1999)
[15] Joye, Asymptotic Anal. 9 pp 209– (1994)
[16] Landau, L.,Collected Papers of L. Landau(Pergamon, New York, 1965).
[17] Lions, Rev. Mat. Iberoam. 9 pp 553– (1993) · Zbl 0801.35117
[18] Martin, Rev. Math. Phys. 7 pp 193– (1995)
[19] Miller, L., ”Propagation d’ondes semi-classiques à travers une interface et mesures deux-microlocales,” Thèse de l’Ecole Polytechnique, 1995.
[20] Spohn, Commun. Math. Phys. 224 pp 113– (2001)
[21] Zener, Proc. R. Soc. London 137 pp 696– (1932)
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.