Takeuchi: Symétrisations indépendantes du temps pour certains opérateurs du type de Schrödinger. I

Takeuchi, Jiro:
Symétrisations indépendantes du temps pour certains opérateurs du type de Schrödinger. I
Bollettino dell'Unione Matematica Italiana Serie 8 5-B (2002), fasc. n.1, p. 1-53, Unione Matematica Italiana (French)
pdf (436 Kb), djvu (599 Kb). | MR1881443 | Zbl 1121.35119

Sunto

Si danno condizioni sufficienti e condizioni necessarie affinché il problema di Cauchy per alcuni operatori di tipo Schrödinger sia ben posto in spazi di Sobolev. Gli operatori qui considerati sono operatori di Schrödinger con potenziali vettoriali complessi, una generalizzazione degli operatori di 2-evoluzione nel senso di Petrowsky, e alcuni sistemi tipo Leray-Volevich di operatori lineari a derivate parziali. Il metodo che usiamo in questo articolo è la simmetrizazione $L^{2}$ degli operatori non dipendenti dal tempo, che abbiamo già usato nelle Note [52]-[54].

Referenze Bibliografiche

[1] A. BABA, The $L^2$-wellposed Cauchy problem for Schrödinger type equations, Tsukuba J. Math., 16 (1992), 235-256. | MR 1178678 | Zbl 0784.35018

[2] A. BABA, The $H^\infty$-wellposed Cauchy problem for Schrödinger type equations, Tsukuba J. Math., 18 (1994), 101-117. | MR 1287832 | Zbl 0815.35006

[3] G. D. BIRKHOFF, Quantum mechanics and asymptotic series, Bull. Amer. Math. Soc., 39 (1933), 681-700. | fulltext mini-dml | MR 1562719 | Zbl 0008.08902

[4] A. P. CALDERÓN-R. VAILLANCOURT, A class of bounded pseudo-differential operators, Proc. Nat. Acad. Sci. U.S.A., 69 (1972), 1185-1187. | MR 298480 | Zbl 0244.35074

[5] H. CARTAN, Formes Différentielles, Cours de Mathématiques II, Collection Méthodes, Hermann, Paris, 1967. | MR 231303 | Zbl 0184.12701

[6] P. CONSTANTIN-J. C. SAUT, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. | MR 928265 | Zbl 0667.35061

[7] R. COURANT-D. HILBERT, Methods of Mathematical Physics, vol. I, Interscience Publishers, New York, 1953. | MR 65391 | Zbl 0051.28802

[8] S. DOI, On the Cauchy problem for Schrödinger type equations and the regularity of solutions, J. Math. Kyoto Univ., 34 (1994), 319-328. | fulltext mini-dml | MR 1284428 | Zbl 0807.35026

[9] S. DOI, Remarks on the Cauchy problem for Schrödinger type equations, Comm. Partial Differential Equations, 21 (1996), 163-178. | MR 1373768 | Zbl 0853.35025

[10] L. GARDING-T. KOTAKE-J. LERAY, Uniformisation et développement asymptotique de la solution du problème de Cauchy linéaire à données holomorphes, Bull. Soc. Math. France, 92 (1964), 263-361. | fulltext mini-dml | MR 196280 | Zbl 0147.08101

[11] S. HARA, A necessary condition for $H^\infty$-wellposed Cauchy problem of Schrödinger type equations with variable coefficients, J. Math. Kyoto Univ., 32 (1992), 287-305. | fulltext mini-dml | MR 1173967 | Zbl 0794.35113

[12] N. HAYSHI-K. NAKAMITSU-M. TSUTSUMI, On solutions of the initial value problem for the nonlinear Schrödinger equations, J. Funct. Anal., 71 (1987), 218-245. | MR 880978 | Zbl 0657.35033

[13] E. HILLE-R. S. PHILLIPS, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, 1957. | MR 89373 | Zbl 0078.10004

[14] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators, vol. III, Springer-Verlag, Berlin, 1985. | Zbl 0601.35001

[15] G. HUFFORD, On the characteristic matrix of a matrix of differential operators, J. Diff. Equations, 1 (1965), 27-38. | MR 176228 | Zbl 0143.13302

[16] W. ICHINOSE, Some remarks on the Cauchy problem for Schrödinger type equations, Osaka J. Math., 21 (1984), 565-581. | fulltext mini-dml | MR 759481 | Zbl 0572.35021

[17] W. ICHINOSE, Sufficient condition on $H^\infty$ wellposedness for Schrödinger type equations, Comm. Partial Differential Equations, 9 (1984), 33-48. | MR 735147 | Zbl 0563.35066

[18] W. ICHINOSE, The Cauchy problem for Schrödinger type equations with variables coefficients, Osaka J. Math., 24 (1987), 853-886. | fulltext mini-dml | MR 927063 | Zbl 0654.35042

[19] W. ICHINOSE, On $L^2$ well-posedness of the Cauchy problem for Schrödinger type equations on the Riemannian manifold and Maslov theory, Duke Math. J., 56 (1988), 549-588. | fulltext mini-dml | MR 948533 | Zbl 0713.58055

[20] W. ICHINOSE, A note on the Cauchy problem for Schrödinger type equations on the Riemannian manifold, Math. Japonica, 35 (1990), 205-213. | MR 1049082 | Zbl 0707.35126

[21] W. ICHINOSE, On the Cauchy problem for Schrödinger type equations and Fourier integral operators, J. Math. Kyoto Univ., 33 (1993), 583-620. | fulltext mini-dml | MR 1239080 | Zbl 0802.35125

[22] W. ICHINOSE, On a necessary condition for $L^2$ well-posedness of the Cauchy problem for some Schrödinger type equations with a potential term, J. Math. Kyoto Univ., 33 (1993), 647-663. | fulltext mini-dml | MR 1239083 | Zbl 0802.35126

[23] K. KAJITANI, The Cauchy problem for Schrödinger type equations with variable coefficients, J. Math. Soc. Japan, 50 (1998), 179-202. | fulltext mini-dml | MR 1484618 | Zbl 0917.35129

[24] K. KAJITANI, The Cauchy problem for Schrödinger type equations, Bull. Sc. Math., 2e série, 119 (1995), 459-473. | MR 1354247 | Zbl 0856.35024

[25] T. KATO, Nonlinear Schrödinger equations, Schrödinger Operators, Springer Lecture Notes in Physics, 345 (1989), 218-263. | Zbl 0698.35131

[26] T. KATO, On the Cauchy problemfor the (generalized) Korteweg-de Vries equation, Studies in Appl. Math. Adv. in Math., Suppl. Studies, 8 (1983), 93-128. | MR 759907 | Zbl 0549.34001

[27] H. KUMANO-GO, Pseudo-Differential Operators, MIT Press, 1981. | Zbl 0489.35003

[28] J. LERAY, Hyperbolic Differential Equations, Inst. Adv. Study, Princeton, (1953). | MR 63548 | Zbl 0588.35002

[29] J. LERAY, Lagrangian Analysis and Quantum Mechanics. A mathematical structure related to asymptotic expansions and the Maslov index, MIT Press, 1981. | MR 644633 | Zbl 0483.35002

[30] V. P. MASLOV, Theory of Perturbations and Asymptotic Methods, Moskow, 1965 (en russe); French translation from Russian, Dunod, Paris, 1970.

[31] M. MIYAKE, On Cauchy-Kowalewski's theorem for general systems, Publ. Res. Inst. Math. Sci. Kyoto Univ., 15 (1979), 315-337. | fulltext mini-dml | MR 555658 | Zbl 0426.35007

[32] S. MIZOHATA, Some remarks on the Cauchy problem, J. Math. Kyoto Univ., 1 (1961), 109-127. | fulltext mini-dml | MR 170112 | Zbl 0104.31903

[33] S. MIZOHATA, On Kowalewskian systems, Russian Math. Surveys, 29-7 (1974), 223-235. | MR 402242 | Zbl 0305.35003

[34] S. MIZOHATA, On some Schrödinger type equations, Proc. Japan Acad., 57 (1981), 81-84. | fulltext mini-dml | MR 605288 | Zbl 0469.35039

[35] S. MIZOHATA, Sur quelques équations du type Schödinger, Séminaire J. Vaillant 1980-1981, Univ. Paris-VI. | Zbl 0469.35040

[36] S. MIZOHATA, Sur quelques équations du type Schödinger, Journées «Équations aux Dérivées Partielles», Saint-Jean-de-Monts, Soc. Math. France, 1981. | fulltext mini-dml | Zbl 0469.35040

[37] S. MIZOHATA, On the Cauchy Problem, Notes and Reports in Math., 3, Academic Press, 1985. | MR 860041 | Zbl 0616.35002

[38] I. G. PETROWSKY, Über das Cauchysche Problem für ein System linearer partieller Differentialgleichungen im Gebiete der nicht-analytischen Funktionen, Bull. Univ. État Moscou, 1 (1938), 1-74. | Jbk 64.1156.02

[39] E. SCHRÖDINGER, Quantisierung als Eigenwertproblem (Vierte Mitteilung), Ann. der Physik, 81 (1926), 109-139. | Jbk 52.0966.03

[40] E. SCHRÖDINGER, Gesammelte Abhandlungen, 3: Beiträge zur Quantentheorie Herausgegeben von der Österreichischen Akademie der Wissenschaften, Verlag der Österreichischen Akademie der Wissenschaften, Wien, 1984. | MR 889674 | Jbk 48.0009.04

[41] J. TAKEUCHI, A necessary condition for the well-posedness of the Cauchy problem for certain class of evolution equations, Proc. Japan Acad., 50 (1974), 133-137. | fulltext mini-dml | MR 367491 | Zbl 0308.35061

[42] J. TAKEUCHI, Some remarks on my paper «On the Cauchy problem for some nonkowalewskian equations with distinct characteristic roots», J. Math. Kyoto Univ., 24 (1984), 741-754. | fulltext mini-dml | MR 775984 | Zbl 0572.35020

[43] J. TAKEUCHI, On the Cauchy problem for systems of linear partial differential equations of Schrödinger type, Bull. Iron and Steel Technical College, 18 (1984), 25-34. | MR 872970

[44] J. TAKEUCHI, A necessary condition for $H^\infty$-wellposedness of the Cauchy problem for linear partial differential operators of Schrödinger type, J. Math. Kyoto Univ., 25 (1985), 459-472. | fulltext mini-dml | MR 807492 | Zbl 0587.35014

[45] J. TAKEUCHI, Le problème de Cauchy pour quelques équations aux dérivées partielles du type de Schrödinger, C. R. Acad. Sci. Paris, Série I, 310 (1990), 823-826. | MR 1058504 | Zbl 0721.35024

[46] J. TAKEUCHI, Le problème de Cauchy pour quelques équations aux dérivées partielles du type de Schrödinger, II, C. R. Acad. Sci. Paris, Série I, 310 (1990), 855-858. | MR 1060600 | Zbl 0803.35118

[47] J. TAKEUCHI, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, III, C. R. Acad. Sci. Paris, Serie I, 312 (1991), 341-344. | MR 1094197 | Zbl 0803.35119

[48] J. TAKEUCHI, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, IV, C. R. Acad. Sci. Paris, Serie I, 312 (1991), 587-590. | MR 1101038 | Zbl 0803.35120

[49] J. TAKEUCHI, Le probleme de Cauchy pour certains systemes de Leray-Volevi du type de Schrodinger, V, C. R. Acad. Sci. Paris, Serie I, 312 (1991), 799-802. | MR 1108494 | Zbl 0803.35121

[50] J. TAKEUCHI, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, VI, C. R. Acad. Sci. Paris, Serie I, 313 (1991), 761-764. | MR 1139834 | Zbl 0803.35122

[51] J. TAKEUCHI, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, VII, C. R. Acad. Sci. Paris, Serie I, 314 (1992), 527-530. | MR 1159395 | Zbl 0834.35109

[52] J. TAKEUCHI, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, VIII; symetrisations independantes du temps, C. R. Acad. Sci. Paris, Serie I, 315 (1992), 1055-1058. | Zbl 0768.35066

[53] J. TAKEUCHI, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, IX; symetrisations independantes du temps, C. R. Acad. Sci. Paris, Serie I, 316 (1993), 1025-1028. | Zbl 0776.35062

[54] J. TAKEUCHI, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, X; symetrisations independantes du temps, Proc. Japan Acad., Ser. A, 69 (1993), 189-192. | fulltext mini-dml | Zbl 0828.35115

[55] J. TAKEUCHI, Le probleme de Cauchy pour certaines equations aux derivees partielles du type de Schrodinger, Thèse de Doctorat de l'Université Paris-VI, Octobre 1995, pp. 115. | Zbl 0834.35109

[56] S. TARAMA, On the $H^\infty$-wellposed Cauchy problem for some Schrodinger type equations, Mem. Fac. Eng. Kyoto Univ., 55 (1993), 143-153. | MR 1251240

[57] L. R. VOLEVICH, On general systems of differential equations, Soviet Math. Dokl., 1 (1960), 458-461. | MR 131045 | Zbl 0107.30603

[58] L. R. VOLEVICH, A problem of linear programming arising in differential equations, Uspehi Mat. Nauk., 18-3 (1963), 155-162 (en russe). | MR 162046 | Zbl 0178.11001

[59] C. WAGSCHAL, Diverses formulations du problème de Cauchy pour un système d'équations aux dérivées partielles, J. Math. Pures et Appl., 53 (1974), 51-69. | MR 361415 | Zbl 0265.35017

[60] K. YAJIMA, On smoothing property of Schrodinger propagators, Functional Analytic Methods for partial differential equations, Proc. Intern. Conf. on Functional Analysis and its Applications, Springer Lecture Notes in Math., 1450 (1990), 20-35. | MR 1084599 | Zbl 0725.35084

[61] K. YOSIDA, Functional Analysis, Springer-Verlag, 1965.

bdim: Biblioteca Digitale Italiana di Matematica

Un progetto SIMAI e UMI

Referenza completa

Sunto

Referenze Bibliografiche