Representation of the Green's function of Schrödinger's

A. A. Arsen'ev

Sbornik. Mathematics, Tome 197 (2006) no. 11, pp. 1559-1568

Résumé

The Cauchy problem is considered for the many-dimensional Schrödinger equation describing a particle in constant electric and magnetic fields and the field of a potential equal to the sum of a decreasing and an almost periodic function. An approximation of the Green's function of the Cauchy problem for such an equation by a path integral over Gaussian wave packets is put forward. Bibliography: 21 titles.

Bibliographie

[1] D. Fujiwara, “A construction of the fundamental solution for the Schrödinger equations”, J. Anal. Math., 35 (1979), 41–96 | DOI | MR | Zbl

[2] D. Fujiwara, “Remarks on convergence of the Feynman path integrals”, Duke Math. J., 47:3 (1980), 559–600 | DOI | MR | Zbl

[3] G. B. Folland, Harmonic analysis in phase space, Princeton Univ. Press, Princeton, NJ, 1989 | MR | Zbl

[4] S. H. Fricke, A. B. Balantekin, T. Uzer, “Uniform approximation and coherent state path integrals”, J. Math. Phys., 32:11 (1991), 3125–3129 | DOI | MR

[5] Wei-Min Zhang, Da Hsuan Feng, R. Gilmore, “Coherent states: Theory and some applications”, Rev. Modern Phys., 62:4 (1990), 867–927 | DOI | MR

[6] M. A. de Gosson, Extended Weyl calculus and application to the phase-space Schrödinger equation, arXiv: math.SG/0503709

[7] M. A. de Gosson, Symplectically covariant Schrödinger equation in phase space, arXiv: math-ph/0505073

[8] G. A. Hagedorn, “Semiclassical quantum mechanics: The $\hbar\to0$ limit for coherent states”, Comm. Math. Phys., 71:1 (1980), 77–93 | DOI | MR

[9] G. A. Hagedorn, “Raising and lowering operators for semiclassical wave packets”, Ann. Phys., 269:1 (1998), 77–104 | DOI | MR | Zbl

[10] G. A. Hagedorn, A. Joye, “Exponentially accurate semiclassical dynamics: Propagation, localization, Ehrenfest times, scattering, and more general states”, Ann. Henri Poincaré, 1:5 (2000), 837–883 | DOI | MR | Zbl

[11] M. F. Herman, “Dynamics by semiclassical methods”, Annu. Rev. Phys. Chem., 45 (1994), 83–111 | DOI

[12] S. L. Robinson, “Semiclassical mechanics for time-dependent Wigner functions”, J. Math. Phys., 34:6 (1993), 2185–2205 | DOI | MR | Zbl

[13] V. V. Dodonov, V. I. Manko, “Invarianty i evolyutsiya nestatsionarnykh kvantovykh sistem”, Tr. In-ta fiziki AN SSSR, 183 (1987), 182–286 | MR

[14] J. Bolte, R. Glaser, “A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators”, Comm. Math. Phys., 247:2 (2004), 391–419 ; arXiv: math-ph/0204018 | DOI | MR | Zbl

[15] A. Martinez, V. Sordoni, “Microlocal WKB expansions”, J. Funct. Anal., 168:2 (1999), 380–402 | DOI | MR | Zbl

[16] D. Chruściński, K. Młodawski, “Wigner function and Schrödinger equation in phase-space representation”, Phys. Rev. A, 71 (2005), 052104 ; arXiv: quant-ph/0501163 | DOI | MR

[17] T. Ichinose, S. Takanobu, “Estimate of the difference between the Kac operator and the Schrödinger semigroup”, Comm. Math. Phys., 186:1 (1997), 167–197 | DOI | MR | Zbl

[18] G. R. Aidagulov, “Primenenie preobrazovaniya Fure–Gaussa k resheniyu zadachi Koshi dlya uravneniya Shrëdingera”, ZhVM i MF, 42:12 (2002), 1810–1815 | MR | Zbl

[19] G. R. Aidagulov, “Primenenie preobrazovaniya Fure–Gaussa k resheniyu zadachi Koshi dlya uravneniya Shrëdingera: teoreticheskii analiz chislennogo algoritma”, Vychislit. metody i program., 4:2 (2003), 16–20; http://num-meth.srcc.msu.su/zhurnal/tom_2003/v4r119.html

[20] G. R. Aidagulov, “Metod podvizhnoi setki dlya resheniya nestatsionarnogo uravneniya Shrëdingera”, Vychislit. metody i program., 5 (2004), 18–30 ; http://num-meth.srcc.msu.su/zhurnal/tom_2004/v5r102.html

[21] F. Treves, “Parametrices for a class of Schrödinger equations”, Comm. Pure Appl. Math., 48:1 (1995), 13–78 | MR | Zbl