Geodesic
Splitting of the lowest energy levels of the Schrödinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$
Teoretičeskaâ i matematičeskaâ fizika, Tome 87 (1991) no. 3, pp. 323-375 Cet article a éte moissonné depuis la source Math-Net.Ru

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For the equation $h\partial u/\partial t=h^2\Delta u/2-V(x)u$ with positive potential $V(x)$, global exponential asymptotic behavior of the fundamental solution is obtained by the method of the tunnel canonical operator. In the case of a potential with degenerate points of global minimum, the behavior of the solutions to the Cauchy problem is investigated at times of order $t=h^{-(1+\varkappa)}$, $\varkappa>0$. The developed theory is used to obtain exponential asymptotics of the lowest eigenfunctions of the Schrödinger operator $-h^2\Delta/2-V(x)$ and to estimate the tunnel effect. 
@article{TMF_1991_87_3_a0,
     author = {S. Yu. Dobrokhotov and V. N. Kolokoltsov and V. P. Maslov},
     title = {Splitting of the lowest energy levels of the {Schr\"odinger} equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {323--375},
     year = {1991},
     volume = {87},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1991_87_3_a0/}
}
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S. Yu. Dobrokhotov; V. N. Kolokoltsov; V. P. Maslov. Splitting of the lowest energy levels of the Schrödinger equation and asymptotic behavior of the fundamental solution of the equation $hu_t=h^2\Delta u/2-V(x)u$. Teoretičeskaâ i matematičeskaâ fizika, Tome 87 (1991) no. 3, pp. 323-375. http://geodesic.mathdoc.fr/item/TMF_1991_87_3_a0/

[1] Maslov V. P., DAN SSSR, 258:5 (1981), 1112–1120

[2] Maslov V. P., Tr. MIAN SSSR, 163, 1984, 150–180 | MR | Zbl

[3] Maslov V. P., Teoriya vozmuschenii i asimptoticheskie metody, MGU, M., 1965 | MR

[4] Maslov V. P., Asimptoticheskie metody i teoriya vozmuschenii, Nauka, M., 1988 | MR

[5] Maslov V. P., Kompleksnyi metod VKB v nelineinykh uravneniyakh, Nauka, M., 1977 | MR

[6] Maslov V. P., ch. 1, ZhVMiMF, 1961, no. 1, 114–128 ; ч. 2, No 4, 638–663 | MR | Zbl

[7] Varadhan S. R., Commun. Pure and Appl. Math., 19:3 (1966), 231–281 | DOI | MR

[8] Borovkov A. A., Teoriya veroyatn. i ee prilozh., 12:4 (1967), 635–654 | MR | Zbl

[9] Simon B., Ann. Inst. H. Poincare, 38 (1983), 295–307 | MR

[10] Polyakov A. M., Nucl. Phys., 120 (1977), 429–456 | DOI | MR

[11] Gildener E., Patrascigin A., Phys. Rev., D16 (1977), 425–443

[12] Harrel E. M., Commun. in Math. Phys., 75 (1980), 239–261 | DOI | MR | Zbl

[13] Harrel E. M., Ann. Phys. (N. Y.), 119 (1979), 351–369 | DOI | MR | Zbl

[14] Jona-Lasinio G., Martinelli L., Scoppola E., Commun. in Math. Phys., 80 (1981) | DOI | MR | Zbl

[15] Combes J. M., Duclos P., Seiler R., Commun. in Math. Phys., 92:2 (1983) | DOI | MR | Zbl

[16] Pankratova T. F., DAN SSSR, 276:4 (1984), 795–799 | MR | Zbl

[17] Radzharaman, Solitony i instantony v kvantovoi teorii polya, Mir, M., 1985

[18] Simon B., Ann. of Math., 120 (1984), 89–118 | DOI | MR | Zbl

[19] Heffler B., Sjöstrand J., I, Commun. in P.D.E., 9:4 (1984), 337–408 ; II, Ann. Inst. H. Poincare, 42:2 (1985), 127–212 ; IV, Commun. in P.D.E., 13:3 (1985), 245–340; VI, Ann. Inst. H. Poincare, 46:4 (1987), 353–372 | DOI | MR | MR | MR

[20] Witten E., J. Diff. Geom., 17 (1982), 661–692 | DOI | MR | Zbl

[21] Landau L. D., Lifshits E. M., Kvantovaya mekhanika (Nerelyativistskaya teoriya), Nauka, M., 1975 | MR

[22] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970 | MR

[23] Kifer Yu. I., Izv. AN SSSR. Ser. matem., 38:5 (1974), 1091–1115 | MR

[24] Babich V. M., DAN SSSR, 289:4 (1986), 836–839 | MR | Zbl

[25] Molchanov S. A., UMN, 30:1 (1975), 3–59 | MR | Zbl

[26] Maslov V. P., Kompleksnye markovskie tsepi i kontinualnyi integral Feinmana, Nauka, M., 1977 | MR

[27] Maslov V. P., Fedoryuk M. V., Kvaziklassicheskoe priblizhenie dlya uravnenii kvantovoi mekhaniki, Nauka, M., 1976 | MR

[28] Kolokoltsov V. N., Maslov V. P., ch. 1, Funkts. analiz i ego prilozh., 1989, no. 1, 1–14 ; ч. 2, No 4, 53–62 | MR | Zbl | MR | Zbl

[29] Maslov V. P., Asimptoticheskie metody resheniya psevdodifferentsialnykh uravnenii, Nauka, M., 1987 | MR

[30] Maslov V. P., UMN, 42:3 (1987), 39–48 | MR | Zbl

[31] Palis Zh., di Melu R., Geometricheskaya teoriya dinamicheskikh sistem, Mir, M., 1985 | MR

[32] Sternberg S., Amer. J. Math., 79:4 (1957), 809–824 | DOI | MR | Zbl

[33] Berezin F. A., Shubin M. A., Uravnenie Shredingera, MGU, M., 1983 | MR

[34] Maslov V. P., Fedoryuk M. V., Mat. zametki, 30:5 (1981), 763–768 | MR | Zbl

[35] Titchmarsh E., Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, T. 2, Mir, M., 1961 | MR

[36] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki. T. 4. Analiz operatorov, Mir, M., 1981 | MR

[37] Coleman S., “The uses of instantons”, The Whys of Subnuclear Physics, ed. A. Zichichi, New York, 1979, 805–916 | DOI | MR

[38] Madelung O., Teoriya tverdogo tela, Nauka, M., 1980

[39] Gershtein S. S., Ponomarev L. I,, Puzynina T. P., ZhETF, 48:2 (1965)

[40] Fedoryuk M. V., Asimptotika, integraly i ryady, Nauka, M., 1987 | MR