no. 4
Semi-classical eigenstates at the bottom of a multidimensional well
Annales de l'I.H.P. Physique théorique, Tome 62 (1995) no. 4, pp. 361-382 
@article{AIHPA_1995__62_4_361_0,
     author = {Pankratova, T. F.},
     title = {Semi-classical eigenstates at the bottom of a multidimensional well},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     pages = {361--382},
     year = {1995},
     publisher = {Gauthier-Villars},
     volume = {62},
     number = {4},
     mrnumber = {1343782},
     zbl = {0835.35120},
     language = {en},
     url = {https://www.numdam.org/item/AIHPA_1995__62_4_361_0/}
}
TY  - JOUR
AU  - Pankratova, T. F.
TI  - Semi-classical eigenstates at the bottom of a multidimensional well
JO  - Annales de l'I.H.P. Physique théorique
PY  - 1995
SP  - 361
EP  - 382
VL  - 62
IS  - 4
PB  - Gauthier-Villars
UR  - https://www.numdam.org/item/AIHPA_1995__62_4_361_0/
LA  - en
ID  - AIHPA_1995__62_4_361_0
ER  - 
%0 Journal Article
%A Pankratova, T. F.
%T Semi-classical eigenstates at the bottom of a multidimensional well
%J Annales de l'I.H.P. Physique théorique
%D 1995
%P 361-382
%V 62
%N 4
%I Gauthier-Villars
%U https://www.numdam.org/item/AIHPA_1995__62_4_361_0/
%G en
%F AIHPA_1995__62_4_361_0
Pankratova, T. F. Semi-classical eigenstates at the bottom of a multidimensional well. Annales de l'I.H.P. Physique théorique, Tome 62 (1995) no. 4, pp. 361-382. https://www.numdam.org/item/AIHPA_1995__62_4_361_0/

[1] M.V. Fedoryuk, Asymptotics of the Point Spectrum of the Operator W'' + λ2p(x) W, Mat. Sb., Vol. 68, (110), 1965, pp. 81-110 (Russian).

[2] S.Yu. Slavyanov, Asymptotics of Singular Sturm-Liuville Problems with Respect to a Large Parameter in the Case of Close Transition Points, Differentsial'nye Uravneniya, Vol. 5, 1969, pp. 313-325, English transl. in Differential Equations, Vol. 5, 1969. | MR | Zbl

[3] A.G. Alenitsyn, The Spectrum Splitting Generated by a Potential Barrier in Problems with a Symmetrical Potential, Differentsial'nye Uravneniya, Vol. 18, 1982, pp. 1971-1975, English transl. in Differential Equations, Vol. 18, 1982. | Zbl | MR

[4] B. Simon, Semiclassical Analysis of Low Lying Eigenvalues, I. Non-Degenerate Minima: Asymptotic Expansions, Ann. Inst. Henry Poincaré, Vol. 38, 1983, pp. 295-307. | Numdam | Zbl | MR | EuDML

[5] B. Simon, Semiclassical Analysis of Low Lying Eigenvalues, II. Tunneling, Ann. Math., Vol. 120, 1984, pp. 89-118. | Zbl | MR

[6] V.P. Maslov, The Complex WKB-Method in Nonlinear Equations, Nauka, Moscow, 1981 (Russian).

[7] G. Jona-Lasinio, F. Martinelli and E. Scoppola, New Approach to the Semiclassical Limit of Quantum Mechanics, Comm. Math. Phys., Vol. 80, 1981, pp. 223-254. | Zbl | MR

[8] E.M. Harrel, Double Wells, Comm. Math. Phys., Vol. 75, 1980, pp. 239-261. | Zbl | MR

[9] T.F. Pankratova, Quasimodes and Splitting of Eigenvalues, Soviet Math. Dokl., Vol. 29, 1984, pp. 597-601. | Zbl | MR

[10] E. Delabaere and H. Dillinger, Contribution à la resurgence quantique, Thèse de doctorat de Math., Université de Nice-Sophia-Antipolis.

[11] B. Helffer and J. Sjöstrand, Multiple Wells in the Semi-Classical Limit I, Comm. in Partial Differential Equations, Vol. 9, 1984, pp. 337-408. | Zbl | MR

[12] B. Helffer and J. Sjöstrand, Puits Multiples en Limite Semi-Classical II. Interaction moléculaire. Symétries. Perturbation, Ann. Inst. Henry Poincaré, Vol. 42, 1985, pp. 127- 212. | Zbl | MR | Numdam

[13] B. Helffer and J. Sjöstrand, Multiple Wells in the Semi-Classical Limit III. Interaction Through Non-Resonant Wells, Math. Nachr., Vol. 124, 1985, pp. 263-313. | Zbl

[14] G. Jona-Lasinio, F. Martinelli and E. Scoppola, Multiple Tunneling in d-Dimensions: a Quantum Particle in a Hierarchical Potential, Ann. Inst. Henry Poincaré, Vol. 42, 1985, pp. 73-108. | Zbl | MR | Numdam

[15] S. Yu Slavyanov and N.A. Veshev, The Quantization of Lower States of an Unharmonic Oscillator with Methods of the Classical Dynamics., Problemy Matfiziki, vyp 13, 1991 (Russian).

[16] S. Graffi and T. Paul, The Schrödinger Equation and Canonical Perturbation Theory, Comm. Math. Phys., Vol. 108, 1987, pp. 25-40. | Zbl | MR

[17] J. Bellissard and M. Vittot, Heisenberg's Picture and Noncommutative Geometry of the Semiclassical Limit in Quantum Mechanics, Ann. Inst. Henry Poincaré, Vol. 52, 1990, pp. 175-235. | Zbl | MR | Numdam

[18] S.Yu. Dobrokhotov, V.N. Kolokoltsov and V.P. Maslov, The Splitting of the Low Lying Energy Levels of the Schrödinger Operator and the Asymptotics of the Fundamental Solution of the Equation hut = (h2Δ/2 - V (x))u with a Symmetric Potential, Teoret. Mat. Fiz., Vol. 87, 1991, pp. 323-375, English transl. in Theoret. and Math. Phys., Vol. 87, 1991. | Zbl | MR

[19] S.Yu. Dobrokhotov, V.N. Kolokoltsov and V.P. Maslov, Quantization of the Bellman Equation. Exponential Asymptotics and Tunneling, Advances in Soviet Mathematics, Vol. 13, 1992, pp. 1-46. | Zbl | MR

[20] S.Yu. Dobrokhotov and V.N. Kolokoltsov, Splitting Amplitudes of the Lowest Energy Levels of the Schrödinger Operator with Double well Potential, Teoret. Mat. Fiz., Vol. 94, 1993, pp. 346-434, English transl. in Theoret. and Math. Phys., Vol. 94, 1993. | Zbl | MR

[21] V.I. Arnol'D, Modes and Quasi-Modes, Functional. Anal. i. Prilozhen, Vol. 6, 1972, pp. 12-20, English transl. in Functional Anal. Appl., Vol. 6, 1972 (Russian). | Zbl

[22] V.I. Arnol'D, Mathematical Methods in Classical Mechanics, Nauka, Moscow, 1989, (Russian). | Zbl | MR

[23] V.I. Arnol'D, V.V. Kozlov and A.I. Neishtadt, Dynamical Systems-3. Mathematical Aspects in Classical and Celestial Mechanics, Itogi Nauki i Tekhnik. Modern Problems in Mathematics. Fundamental Orientations, Viniti ANSSSR Eds., Vol. 3, Moscow, 1985 (Russian). | Zbl

[24] J. Williamson, On the Algebraic Problem Concerning the Normal Forms of Linear Dynamical Systems, Amer. J. Math., Vol. 58, 1936, pp. 141-163. | Zbl | MR | JFM

[25] J. Williamson, The Exponential Representation of Canonical Matrices, Amer. J. Math., Vol. 61, 1939, pp. 897-911. | Zbl | MR

[26] J. Sjöstrand, Analytic Wavefront Sets and Operators with Multiple Characteristics, Hokkaido Math. J., Vol. 12, 1983, pp. 392-433. | Zbl | MR

[27] Nguyên Hu'U Du'C and F. Pham, Germes de configurations legendriennes stables et functions d'Airy-Weber generalisees, Annales de l'Institut Fourier, Grenoble, Vol. 41, 1991, pp. 905-936. | Zbl | MR | Numdam

[28] L.V. Yantorovich and G.P. Akilov, Functional Analysis, Nauka, 2nd rev., Moscow, 1977, English transl., Pergamon Press, 1982. | Zbl | MR