×

zbMATH — the first resource for mathematics

Résonances en limite semi-classique. (Resonances in semi-classical limit). (French) Zbl 0631.35075
This paper (in fact it is a book of 228 pages!) gives a general theory of resonances for the Schrödinger operator. Very deep results are obtained with applications to Bender-Wu formulas and Zeeman effect.
Reviewer: D.Robert

MSC:
35P25 Scattering theory for PDEs
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J10 Schrödinger operator, Schrödinger equation
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] J.E. AVRON , I. HERBST , B. SIMON : Schrödinger operators with magnetic fields III , Atoms in homogeneous magnetic field, Comm. Math. Phys. 79, p. 529-572, ( 1981 ). Article | MR 85h:81084 | Zbl 0464.35086 · Zbl 0464.35086
[2] J.E. AVRON : Bender-Wu formulas and classical trajectories : higher dimensions and degeneracies , Int. J. of Quantum Chemistry 21 ( 1982 ), 119-124.
[3] J.E. AVRON : Bender-Wu formulas for the Zeeman effect in hydrogen , Ann. of Physics 131, ( 1981 ), 73-94. MR 82c:81030
[4] J. AGUILAR , J.M. COMBES : A class of analytic perturbations for one-body Schrödinger Hamiltonians , Comm. Math. Phys. 22 ( 1971 ), 269-279. Article | MR 49 #10287 | Zbl 0219.47011 · Zbl 0219.47011
[5] E. BALSLEV , J.M. COMBES : Spectral properties of many-body Schrödinger operators with dilation analytic interactions , Comm. Math. Phys. 22 ( 1971 ), 280-294. Article | MR 49 #10288 | Zbl 0219.47005 · Zbl 0219.47005
[6] R. BEALS , C. FEFFERMAN : Spatially inhomogeneous pseudo-differential operators , Comm. P.A. Math., 27 ( 1974 ), 1-24. MR 50 #5234 | Zbl 0279.35071 · Zbl 0279.35071
[7] R. BEALS : A general calculus of pseudo-differential operators , Duke Math. J., 42, n^\circ 1 ( 1975 ), 1-42. Article | MR 51 #3972 | Zbl 0343.35078 · Zbl 0343.35078
[8] C. BENDER , T. WU : Anharmonic oscillator , Phys. Rev., 184 ( 1969 ), 1231-1260. MR 41 #4951
[9] C. BENDER , T. WU : Phys. Rev. D7 , ( 1973 ), 1620.
[10] T. BANKS , C. BENDER , T. WU : Coupled anharmonic oscillators I . Equal mass case, Phys. Rev. D., Vol.8, n^\circ 10, (Nov. 1973 ), 3346-3366. MR 50 #15673
[11] T. BANKS , C. BENDER : Coupled anharmonic oscillators II: Unequal mass case , Phys. Rev. D., Vol. 8, n^\circ 10 (Nov. 1973 ), 3366-3378. MR 50 #15674
[12] L. BOUTET DE MONVEL , J. SJÖSTRAND : Sur la singularité des noyaux de Bergman et de Szegö , Astérisque 34-35, ( 1976 ), 123-164. Numdam | Zbl 0344.32010 · Zbl 0344.32010
[13] L. BOUTET DE MONVEL , V. GUILLEMIN : The spectral theory of Toeplitz operators , Ann. of Math. Studies, n^\circ 99, ( 1981 ), Princeton University Press. MR 85j:58141 | Zbl 0469.47021 · Zbl 0469.47021
[14] E. COLEMAN : The use of instantons , Conference d’Erice.
[15] J.M. COMBES , J. DUCLOS , R. SEILER : On the shape resonance , à paraître. · Zbl 0629.47044
[16] H. CYCON : Resonances defined by modified dilations , à paraître.
[17] N. DENCKER : The Weyl calculus with locally temperate metrics and weights , Préprint. · Zbl 0621.47045
[18] S. GRAFFI , V. GRECCHI , E.M. HARRELL , SILVERSTONE : The 1/R expansion for H+2 : Analyticity, Summability and Asymptotics , Préprint. · Zbl 0614.46068
[19] S. GRAFFI : Séminaire d’Analyse de Bologne ( 1982 - 1983 ).
[20] E. HARRELL , B. SIMON : The mathematical theory of resonances whose widths are exponentially small , Duke Math. J., Vol. 47, n^\circ 4, Déc. 1980 . Article | MR 82e:81007 | Zbl 0455.35091 · Zbl 0455.35091
[21] B. HELFFER , D. ROBERT : Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles , J. Funct. Anal., 53, n^\circ 3, ( 1983 ), 246-268. MR 85i:47052 | Zbl 0524.35103 · Zbl 0524.35103
[22] B. HELFFER , J. SJÖSTRAND : Multiple wells in the semi-classical limit I , Comm. in PDE, 9(4) ( 1984 ), 337-408. Zbl 0546.35053 · Zbl 0546.35053
[23] B. HELFFER , J. SJÖSTRAND : Puits multiples en limite semi-classique II, Interaction moléculaire, symétries, perturbation , Ann. de l’I.H.P., Vol. 42, n^\circ 2, 1985 , p. 127-212. Numdam | MR 87a:35142 | Zbl 0595.35031 · Zbl 0595.35031
[24] B. HELFFER , J. SJÖSTRAND : Multiple wells in the semi-classical limit III , non resonant wells, Math. Nachr., ( 1985 ). Zbl 0597.35023 · Zbl 0597.35023
[25] B. HELFFER , J. SJÖSTRAND : Puits multiples en mécanique semi-classique IV , étude du complexe de Witten, Comm. P.D.E., 10(3), ( 1985 ), 245-340. Zbl 0597.35024 · Zbl 0597.35024
[26] B. HELFFER , J. SJÖSTRAND : Puits multiples en mécanique semi-classique V , étude des mini puits, A paraître. Zbl 0628.35024 · Zbl 0628.35024
[27] I. HERBST : Dilation analyticity in a constant electric field , Comm. Math. Phys., 64 ( 1979 ), 279-298. Article | MR 81a:81020 | Zbl 0447.47028 · Zbl 0447.47028
[28] L. HÖRMANDER : Fourier integral operators I. , Acta Math. 127 ( 1971 ), 79-183. MR 52 #9299 | Zbl 0212.46601 · Zbl 0212.46601
[29] L. HÖRMANDER : The Weyl calculus of pseudo-differential operators , Comm. P.A.M., 32 ( 1979 ), 359-443. Zbl 0388.47032 · Zbl 0388.47032
[30] A. MARTINEZ : Estimations de l’effet tunnel pour le double puits , Publications Mathématiques d’Orsay, 85 T 17. · Zbl 0613.35022
[31] A. MELIN , J. SJÖSTRAND : Fourier integral operators with complex valued phase functions , Springer L.N. in Math., n^\circ 459, 120-223. MR 55 #4290 | Zbl 0306.42007 · Zbl 0306.42007
[32] M. REED , B. SIMON : Methods of modern mathematical physics , Vol. IV, Academic Press, New-York, 1978 . Zbl 0401.47001 · Zbl 0401.47001
[33] B. SIMON : Resonances and complex scaling , a rigorous overview, Int. J. Quantum Chemistry, 14 ( 1978 ), 529-542.
[34] B. SIMON : Semi-classical analysis of low lying eigenvalues I , Ann. I.H.P., 38, n^\circ 3, ( 1983 ), 295-307. Numdam | MR 85m:81040a | Zbl 0526.35027 · Zbl 0526.35027
[35] B. SIMON : Large orders of summability of eigenvalue perturbation theory: a mathematical overview , Int. J. Quantum Chemistry, 21 ( 1982 ), 3-25.
[36] B. SIMON : Coupling constant analyticity for the anharmonic oscillator , Ann. of Physics, 58 ( 1970 ), 76-136. MR 54 #4397
[37] J. SJÖSTRAND : Singularités analytiques microlocales , Astérisque n^\circ 95 ( 1982 ). MR 84m:58151 | Zbl 0524.35007 · Zbl 0524.35007
[38] J. SJÖSTRAND : Tunnel effects for semi-classical Schrödinger operators , Contribution to the workshop and symposium on hyperbolic equations and related topics, Katuda and Kyoto, August 27 - Sept. 5, 1984 , à paraître. Zbl 0669.35083 · Zbl 0669.35083
[39] TITSCHMARCH : Eigenfunction expansions II , Oxford at the Clarendon Press, 1958 . Zbl 0097.27601 · Zbl 0097.27601
[40] B.R. VAINBERG : On the analytic properties of the resolvent for a certain class of operator-pencils , Mat. Sb., 77(119), ( 1968 ), n^\circ 2, Math. USSR, Sb., 6 ( 1968 ), N^\circ 2.
[41] B.R. VAINBERG : On exterior elliptic problems ... , Mat. Sb., 92(134), ( 1973 ), n^\circ 2, Math. USSR. Sb., 21 ( 1973 ), n^\circ 2. Zbl 0294.35031 · Zbl 0294.35031
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.